The Hopf algebra of finite topologies and T-partitions

The Hopf algebra of finite topologies and T-partitions Loïc Foissy, Claudia Malvenuto To cite this version: Loïc Foissy, Claudia Malvenuto. The Hopf algebra of finite topologies and T-partitions. 2014. ฀hal01017229v2฀ HAL Id: hal-01017229 https://hal.archives-ouvertes.fr/hal-01017229v2 Preprint submitted on 3 Oct 2014 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS LOÏC FOISSY AND CLAUDIA MALVENUTO ABSTRACT. A noncommutative and noncocommutative Hopf algebra on finite topologies HT is introduced and studied (freeness, cofreeness, self-duality. . .). Generalizing Stanley’s definition of P -partitions associated to a special poset, we define the notion of T -partitions associated to a finite topology, and deduce a Hopf algebra morphism from HT to the Hopf algebra of packed words WQSym. Generalizing Stanley’s decomposition by linear extensions, we deduce a factorization of this morphism, which induces a combinatorial isomorphism from the shuffle product to the quasi-shuffle product of WQSym. It is strongly related to a partial order on packed words, here described and studied. KEYWORDS. Finite topologies, combinatorial Hopf algebras, packed words, partitions. AMS CLASSIFICATION. 16T05, 06A11, 54A10. Contents Introduction 1. Reminders 1.1. WQSym and FQSym 1.2. Special posets 1.3. Infinitesimal bialgebras 2. Topologies on a finite set 2.1. Notations and definitions 2.2. Two products on finite topologies 2.3. A coproduct on finite topologies 2.4. Link with special posets 2.5. Pictures and duality 3. Ribbon basis 3.1. Definition 3.2. The products and the coproduct in the basis of ribbons 4. Generalized T-partitions 4.1. Definition 4.2. Linear extensions 4.3. From linear extensions to T-partitions 4.4. Links with special posets 4.5. The order on packed words References 1 2 4 4 5 7 8 8 9 11 13 15 16 16 17 19 19 22 25 29 29 32 2 LOÏC FOISSY AND CLAUDIA MALVENUTO Introduction In his thesis [16], Stanley introduced the notion of (P, ω, m)-partition associated to a (P, ω) poset. More precisely, a (P, ω) poset, or equivalently a special poset, is a finite set (P, ≤P , ≤) with two orders, the second being total, see section 1.2 for examples. A (P, ω, m)-partition, or, briefly, a P -partition, associated to a special poset P is a map f : P −→ N, such that: (1) If i ≤P j in P, then f (i) ≤ f (j). (2) If i ≤P j and i > j in P, then f (i) < f (j). Stanley proved [16, 8] that the set of P -partitions of P can be decomposed into a disjoint family of subsets indexed by the set of linear extensions of the partial order ≤P . Special posets are organized as a Hopf algebra HSP , described in [11] as a subobject of the Hopf algebra of double posets, that is to say finite sets with two partial orders. Linear extensions are used to define a Hopf algebra morphism L from HSP to the Malvenuto-Retenauer Hopf algebra of permutations FQSym [9, 10, 1]. Considering P -partitions which are packed words (which allows to find all P -partitions), it is possible to define a Hopf algebra morphism Γ from HSP to WQSym, the Hopf algebra of packed words. Then Stanley’s decomposition allows to define an injective Hopf algebra morphism ϕ : FQSym −→ WQSym, such that the following diagram commutes: L // FQSym HSP ❑ ❑❑❑ ❑❑❑ ϕ Γ ❑❑❑ %%  WQSym Our aim in this the present text is a generalization of Stanley’s theorem on P partitions and its applications to combinatorial Hopf algebras. We here replace special posets by special preposets (P, ≤P , ≤), where ≤P is a preorder, that is to say a reflexive and transitive relation, and ≤ is a total order. By Alexandroff’s correspondence, these correspond to topologies on finite sets [n] = {1, . . . , n}. A construction of a Hopf algebra on finite topologies (up to homeomorphism) is done in [4], where one also can find a brief historic of the subject. We apply the same construction here and obtain a Hopf algebra HT on finite topologies, which is noncommutative and noncocommutative. It is algebraically studied in section 2; we prove its freeness and cofreeness (proposition 5 and theorem 7), show that the Hopf algebra of special posets is both a subalgebra and a quotient of HT via the construction of a family of Hopf algebra morphisms θq (proposition 8). A (degenerate) Hopf pairing is also defined on HT , with the help of Zelevinsky’s pictures, extending the pairing on special posets of [11]. The set of topologies on a given set is totally ordered by the refinement; using this ordering and a Möbius inversion, we define another basis of HT , called the ribbon basis. The product and the coproducts are described in this new basis (theorem 12). The notions of T -partitions and linear extensions of a topology are defined in section 4. A T -partition of a topology T on the set [n] is introduced in definition THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 3 13. Namely, if ≤T is the preorder associated to the topology T , a generalized T -partition of T is a surjective map f : [n] −→ [p] such that: • if i ≤T j, then f (i) ≤ f (j). The T -partition f is strict if: • If i ≤T j, i > j and not j ≤T i, then f (i) < f (j). • If i < j < k, i ≤T k, k ≤T i and f (i) = f (j) = f (k), then i ≤T j, j ≤T i, j ≤T k and k ≤T i. The last condition, which is empty for special posets, is necessary to obtain an equivalent of Stanley’s decomposition, as it will be explained later. We now identify any T -partition f associated to the topology T on [n] with the word f (1) . . . f (n). A family of Hopf algebra morphisms Γ(q1 ,q2 ,q3 ) from HT to WQSym, parametrized by triples of scalars, is defined in proposition 14. In particular, for any finite topology T: X X Γ(1,1,1) (T ) = f, Γ(1,0,0) (T ) = f. f generalized T -partition of T f strict T -partition of T Linear extensions are introduced in definition 15. They are used to defined a Hopf algebra morphism L : HT −→ WQSym, up to a change of the product of WQSym: one has to replace its usual product by the shifted shuffling product , used in [5]. We then look for an equivalent of Stanley’s decomposition theorem of P -partitions, reformulated in terms of Hopf algebras, that is to say we look for a Hopf algebra morphism ϕ(q1 ,q2 ,q3 ) making the following diagram commute: L // (WQSym, , ∆) HT ◆ ◆◆◆ ◆◆◆ ϕ(q1 ,q2 ,q3 ) ◆◆ Γ(q1 ,q2 ,q3 ) ◆◆◆ ''  (WQSym, ., ∆) We prove in proposition 21 that such a ϕ(q1 ,q2 ,q3 ) exists if, and only if, (q1 , q2 , q3 ) = (1, 0, 0) or (0, 1, 0), which justifies the introduction of strict T -partitions. The morphism ϕ(1,0,0) is defined in proposition 19, with the help of a partial order on packed words introduced in definition 17; the set decomposition of T -partitions is stated in corollary 20. Finally, the partial order on packed words is studied in section 4.5, with a combinatorial application in corollary 26. The text is organized as follows. The first section recalls the construction of the Hopf algebras WQSym, FQSym and HSP . The second section deals with the Hopf algebra of topologies and its algebraic study; the ribbon basis is the object of the third section. The equivalent of Stanley’s decomposition, from a combinatorial and a Hopf algebraic point of view, is the object of the last section, together with the study of the partial order on packed words. Aknowledgment. The research leading these results was partially supported by the French National Research Agency under the reference ANR-12-BS01-0017. Notations. • We work on a commutative base field K, of any characteristic. Any vector space, coalgebra, algebra. . . of this text is taken over K. 4 LOÏC FOISSY AND CLAUDIA MALVENUTO • For all n ≥ 0, we put [n] = {1, . . . , n}. In particular, [0] = ∅. We denote by N>0 the set of strictly positive integers. 1. Reminders 1.1. WQSym and FQSym. Let us first recall the construction of WQSym [13]. • A packed word is a word f whose letters are in N>0 , such that for all 1 ≤ i ≤ j, j appears in f =⇒ i appears in f . Here are the packed words of length ≤ 3: 1 = ∅; (1); (12), (21), (11); (123), (132), (213), (231), (312), (321), (122), (212), (221), (112), (121), (211), (111). • Let f = f (1) . . . f (n) be a word whose letters are in N>0 . There exists a unique increasing bijection φ from {f (1), . . . , f (n)} into a set [m]. The packed word P ack(f ) is φ(f (1)) . . . φ(f (n)). • If f is a word whose letters are in N>0 , and I is a subset of N>0 , then f|I is the subword of f obtained by keeping only the letters of f which are in I. As a vector space, a basis of WQSym is given by the set of packed words. Its product is defined as follows: if f and f ′ are packed words of respective lengths n and n′ : X f ′′ . f.f ′ = f ′′ packed word of length n + n′ , P ack(f ′′ (1)...f ′′ (n))=f, P ack(f ′′ (n+1)...f ′′ (n+n′ ))=f ′ For example: (112).(12) = (11212) + (11213) + (11214) + (11223) + (11224) + (11234) + (11312) + (11323) + (11324) + (11423) + (22312) + (22313) + (22314) + (22413) + (33412). The unit is the empty packed word 1 = ∅. If f is a packed word, its coproduct in WQSym is defined by: max(f ) ∆(f ) = X f|[k] ⊗ P ack(f|N>0 \[k] ). k=0 For example: ∆((511423)) = 1 ⊗ (511423) + (1) ⊗ (4312) + (112) ⊗ (321) + (1123) ⊗ (21) + (11423) ⊗ (1) + (511423) ⊗ 1. Then (WQSym, ., ∆) is a graded, connected Hopf algebra. We denote by j the involution on packed words defined in the following way: if f = f (1) . . . f (n) is a packed word of length n, there exists a unique decreasing bijection ϕ from {f (1), . . . , f (n)} into a set [l]. We put j(f ) = ϕ(f (1)) . . . ϕ(f (n)). For example, j((65133421)) = (12644356). The extension of j to WQSym is a Hopf algebra isomorphism from (WQSym, ., ∆) to (WQSym, ., ∆op ). THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 5 In particular, permutations are packed words. Note that the subspace of WQSym generated by all the permutations is a coalgebra, but not a subalgebra: for example, (1).(1) = (12) + (21) + (11). On the other side, the subspace of WQSym generated by packed words which are not permutations is a biideal, and the quotient of WQSym by this biideal is the Hopf algebra of permutations FQSym [9, 1]. As a vector space, a basis of FQSym is given by the set of all permutations; if σ and σ ′ are two permutations of respective lengths n and n′ , X X ǫ ◦ (σ ⊗ τ ), σ.σ ′ = σ ′′ = σ′′ ∈Sn+n′ , P ack(σ′′ (1)...σ′′ (n))=σ, P ack(σ′′ (n+1)...σ′′ (n+n′ ))=σ′ ǫ∈Sh(n,n′ ) where Sh(n, n′ ) is the set of (n, n′ )-shuffles, that is to say permutations ǫ ∈ Sn+n′ such that ǫ(1) < . . . < ǫ(n) and ǫ(n + 1) < . . . < ǫ(n + n′ ). For example: (132).(21) = (13254) + (14253) + (15243) + (14352) + (15342) + (15432) + (24351) + (25341) + (25431) + (35421). If σ ∈ Sn , its coproduct is given by: ∆(σ) = n X σ|[k] ⊗ P ack(σ|N>0 \[k] ). k=0 For example: ∆((51423)) = 1 ⊗ (51423) + (1) ⊗ (4312) + (12) ⊗ (321) + (123) ⊗ (21) + (1423) ⊗ (1) + (51423) ⊗ 1. The canonical epimorphism from WQSym to FQSym is denoted by ̟. We shall need the standardisation map, which associates a permutation to any packed word. If f = f (1) . . . f (n) is a packed word, Std(f ) is the unique permutation σ ∈ Sn such that for all 1 ≤ i, j ≤ n: f (i) < f (j) =⇒ σ(i) < σ(j), (f (i) = f (j) and i < j) =⇒ σ(i) < σ(j). In particular, if f is a permutation, Std(f ) = f . Here are examples of standardization of packed words which are not permutations: Std(11) = (12), Std(122) = (123), Std(212) = (213), Std(221) = (231), Std(112) = (123), Std(121) = (132), Std(211) = (312), Std(111) = (123). 1.2. Special posets. Let us briefly recall the construction of the Hopf algebra on special posets [11, 3]. A special (double) poset is a family (P, ≤, ≤tot ), where P is a finite set, ≤ is a partial order on P and ≤tot is a total order on P . For example, here are the special posets of cardinality ≤ 3: they are represented by the Hasse graph of ≤, the total order ≤tot is given by the indices of the vertices. q q 1 = ∅ ; q 1 ; q 1 q 2 , q 21 , q 12 ; q 1 q 2 q 3 , qq 21 q 3 , qq 31 q 2 , qq 12 q 3 , qq 32 q 1 , qq 13 q 2 , q1 q2 q3 qq 3 qq 2 qq 3 qq 1 qq 2 qq 1 2 q q3 1 q q3 1 q q2 q2 q 3 , 1 ∧q q 3 , 1 ∧q q 2 , q 21 , q 31 , q 12 , q 32 , q 13 , q 23 . q 3 q 1 , ∨q1 , ∨q2 , ∨q3 , 2 ∧q 6 LOÏC FOISSY AND CLAUDIA MALVENUTO If P = (P, ≤, ≤tot ) and Q = (Q, ≤, ≤tot ) are two special posets, we define a special posets P.Q in the following way: • As a set, P.Q = P ⊔ Q. • If i, j ∈ P , then i ≤ j in P.Q if, and only if, i ≤ j in P , and i ≤tot j in P.Q if, and only if, i ≤tot j in P . • If i, j ∈ Q, then i ≤ j in P.Q if, and only if, i ≤ j in Q, and i ≤tot j in P.Q if, and only if, i ≤tot j in Q. • If i ∈ P and j ∈ Q, then i and j are not comparable for ≤, and i ≤tot j. 2q q3 5 q q6 q q For example, q 31 q 2 . ∨q1 = q 31 q 2 ∨q4 . The vector space generated by the set of (isoclasses) of special posets is denoted by HSP . This product is bilinearly extended to HSP , making it an associative algebra. The unit is the empty special poset 1 = ∅. If P is a special poset and I ⊆ P , then by restriction I is a special poset. We shall say that I is an ideal of P if for all i, j ∈ P : (i ∈ I and i ≤ j) =⇒ j ∈ I. We give HSP the coproduct defined by: X ∆(P ) = (P \ I) ⊗ I. I ideal of P For example: 4 3 q 4q 4q qq 3 q q1 3 q q1 3 q q1 3q q1 q q q q ∨ q ∨ q ∆( 2 ) = 2 ⊗1+1⊗ ∨q2 + ∨q2 ⊗ q 1 + q 21 ⊗ q 1 + q 21 ⊗ q 1 q 2 + q 12 ⊗ q 21 + q 1 ⊗ q 32 q 1 . Let P = (P, ≤, ≤tot ) be a special poset. A linear extension of P is a total order ≤′ extending the partial order ≤. Let ≤′ be a linear extension of P . Up to a unique isomorphism, we can assume that P = [n] as a totally ordered set. For any i ∈ [n], we denote by σ(i) the index of i in the total order ≤′ . Then σ ∈ Sn , and we now identify ≤′ and σ. The following map is a surjective Hopf algebra morphism:   HSP −→ FQSymX L: P −→ σ.  σ linear extension of P For example: q1 q2 q3 q 3 ) = (312) + (321), L(1 ∧q q 3 ) = (132) + (231), L(1 ∧q q 2 ) = (123) + (213), L(2 ∧q 1 q q2 1q q3 2q q3 L( ∨q1 ) = (123) + (132), L( ∨q2 ) = (213) + (312), L( ∨q3 ) = (231) + (321), q3 q3 q2 q q q L( q 21 ) = (123), L( q 12 ) = (213), L( q 13 ) = (231), q2 q1 q1 q q q L( q 31 ) = (132), L( q 32 ) = (312), L( q 23 ) = (321). Let P be a special poset. With the help of the total order of P, we identify P with the set [n], where n is the cardinality of P. A P -partition of P is a map f : P −→ [n] such that: (1) If i ≤P j in P, then f (i) ≤ f (j). (2) If i ≤P j and i > j in P, then f (i) < f (j). THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 7 We represent a P -partition of P by the word f (1) . . . f (n). Obviously, if w = w1 . . . wn is a word, it is a P -partition of the special poset P if, and only if, P ack(w) is a P -partition of P. We define:   HSP −→ WQSym X Γ: P −→ w.  w packed word, P -partition of P For example: Γ( q 1 ) = (1), Γ( q 1 q 2 ) = (12) + (21) + (11), q Γ( q 21 ) = (12) + (11), q Γ( q 12 ) = (21), 2 q q3 Γ( ∨q1 ) = (123) + (132) + (122) + (112) + (121) + (111), 1 q q3 Γ( ∨q2 ) = (213) + (312) + (212) + (211), 1 q q2 Γ( ∨q3 ) = (231) + (321) + (221). We shall prove in section 4 that Γ is a Hopf algebra morphism. Remark. There is a natural surjective Hopf algebra morphism ̺ from WQSym to the Hopf algebra of quasisymmetric functions QSym [14]. For any special poset P: X ̺ ◦ Γ(P) = xf (1) . . . xf (n) ∈ QSym ⊆ Q[[x1 , x2 , . . .]]. f P -partition of w So ̺ ◦ Γ(P) is the generating function of P in the sense of [16]. We shall also prove that Stanley’s decomposition theorem can be reformulated in the following way: let us consider the map   FQSym −→ WQSymX ϕ: w. σ −→  w packed word, Std(w) = σ Then ϕ is an injective Hopf algebra morphism, such that ϕ◦L = Γ. Combinatorially speaking, for any special poset P: G {P -partition of P} = {w | P ack(w) = σ}. σ linear extension of P 1.3. Infinitesimal bialgebras. An infinitesimal bialgebra [7] is a triple (A, m, ∆) such that: • (A, m) is a unitary, associative algebra. • (A, ∆) is a counitary, coassociative algebra. • For all x, y ∈ A, ∆(xy) = (x ⊗ 1)∆(y) + ∆(x)(1 ⊗ y) − x ⊗ y. The standard examples are the tensor algebras T (V ), with the concatenation product and the deconcatenation coproduct. By the rigidity theorem of [7], these are essentially the unique examples: Theorem 1. Let A be a graded, connected, infinitesimal bialgebra. Then A is isomorphic to T (P rim(A)) as an infinitesimal bialgebra. 8 LOÏC FOISSY AND CLAUDIA MALVENUTO 2. Topologies on a finite set 2.1. Notations and definitions. Let X be a set. Recall that a topology on X is a family T of subsets of X, called the open sets of T , such that: (1) ∅, X ∈ T . (2) The union of an arbitrary number of elements of T is in T . (3) The intersection of a finite number of elements of T is in T . Let us recall from [2] the bijective correspondence between topologies on a finite set X and preorders on X: (1) Let T be a topology on the finite set X. The relation ≤T on X is defined by i ≤T j if any open set of T which contains i also contains j. Then ≤T is a preorder, that is to say a reflexive, transitive relation. Moreover, the open sets of T are the ideals of ≤T , that is to say the sets I ⊆ X such that, for all i, j ∈ X: (i ∈ I and i ≤T j) =⇒ j ∈ I. (2) Conversely, if ≤ is a preorder on X, the ideals of ≤ form a topology on X denoted by T≤ . Moreover, ≤T≤ =≤, and T≤T = T . Hence, there is a bijection between the set of topologies on X and the set of preorders on X. (3) Let T be a topology on X. The relation ∼T on X, defined by i ∼T j if i ≤T j and j ≤T i, is an equivalence on X. Moreover, the set X/ ∼T is partially ordered by the relation defined by i ≤T j if i ≤ j. Consequently, we shall represent preorders on X (hence, topologies on X) by the Hasse diagram of X/ ∼T , the vertices being the equivalence classes of ∼T . For example, here are the topologies on [n] for n ≤ 3: q q 1 = ∅ ; q 1 ; q 1 q 2 , q 21 , q 12 , q 1, 2 ; q q q q q q 1 q 2 q 3 , q 21 q 3 , q 31 q 2 , q 12 q 3 , q 32 q 1 , q 13 q 2 , q1 q2 q3 qq 3 qq 2 qq 3 qq 1 qq 2 qq 1 2 q q3 1 q q3 1 q q2 q2 q 3 , 1 ∧q q 3 , 1 ∧q q 2 , q 21 , q 31 , q 12 , q 32 , q 13 , q 23 , q 3 q 1 , ∨q1 , ∨q2 , ∨q3 , 2 ∧q q q q q 2 q 1, 3 q 2, 3 q 1, 2 q 3 , q 1, 3 q 2 , q 2, 3 q 1 , q 31, 2 , q 21, 3 , q 12, 3 , q 1, , q 2 , q 1 , q 1, 2, 3 . 3 The number tn of topologies on [n] is given by the sequence A000798 in [15]: n tn 1 1 2 3 4 5 6 7 8 9 4 29 355 6 942 209527 9 535 241 642 779 354 63 260 289 423 n tn 10 11 12 8 977 053 873 043 1 816 846 038 736 192 519 355 571 065 774 021 G The set of topologies on [n] will be denoted by Tn , and we put T = Tn . n≥0 If T is a finite topology on a set X, then ι(T ) = {X \ O | O ∈ T } is also a finite topology, on the same set X. Consequently, ι defines a involution of the set T. The preorder associated to ι(T ) is ≤ι(T ) =≥T . Notations. Let f be a packed word of length n. We define a preorder ≤f on [n] by: i ≤f j if f (i) ≤ f (j). THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 9 The associated topology is denoted by Tf . The open sets of this topology are the subsets f −1 ({i, . . . , max(f )}), 1 ≤ i ≤ max(f ), and ∅. For example: qq 1, 2, 5 T(331231) = q 43, 6 . 2.2. Two products on finite topologies. Notations. Let O ⊆ N and let n ∈ N. The set O(+n) is the set {k + n | k ∈ O}. Definition 2. Let T ∈ Tn and T ′ ∈ Tn′ . (1) The topology T .T ′ is the topology on [n + n′ ] which open sets are the sets O ⊔ O′ (+n), with O ∈ T and O′ ∈ T ′ . (2) The topology T ↓ T ′ is the topology on [n + n′ ] which open sets are the sets O ⊔ [n′ ](+n), with O ∈ T , and O′ (+n), with O′ ∈ T ′ . q4 q q 5q 3 2∧ 2q q3 q1 2 q q3 q4 2 q q3 q1 Example. ∨q1 . q 2 = ∨q1 q 5 and ∨q1 ↓ q 2 = ∨q1 . Proposition 3. These two products are associative, with ∅ = 1 as a common unit. Proof. Obviously, for any T ∈ T, 1.T = T .1 = 1 ↓ T = T ↓ 1 = T . Let T ∈ Tn , and T ′ ∈ Tn′ . The preorder associated to T .T ′ is: {(i, j) | i ≤T j} ⊔ {(i + n, j + n) | i ≤T ′ j}. The preorder associated to T ↓ T ′ is: {(i, j) | i ≤T j} ⊔ {(i + n, j + n) | i ≤T ′ j} ⊔ {(i, j) | 1 ≤ i ≤ n < j ≤ n + n′ }. Let T ∈ Tn , T ′ ∈ Tn′ and T ′′ ∈ Tn′′ . The preorders associated to (T .T ′ ).T ′′ and to T .(T ′ .T ′′ ) are both equal to: {(i, j) | i ≤T j} ⊔ {(i + n, j + n) | i ≤T ′ j} ⊔ {(i + n + n′ , j + n + n′ ) | i ≤T ′′ j}. So (T .T ′ ).T ′′ = T .(T ′ .T ′′ ). The preorders associated to (T ↓ T ′ ) ↓ T ′′ and to T ↓ (T ′ ↓ T ′′ ) are both equal to: {(i, j) | i ≤T j} ⊔ {(i + n, j + n) | i ≤T ′ j} ⊔ {(i + n + n′ , j + n + n′ ) | i ≤T ′′ j} ⊔{(i, j) | 1 ≤ i ≤ n < j ≤ n + n′ + n′′ } ⊔ {(i, j) | n < i ≤ n + n′ < j ≤ n + n′ + n′′ }. So (T ↓ T ′ ) ↓ T ′′ = T ↓ (T ′ ↓ T ′′ ).  Definition 4. (1) We denote by HT the vector space generated by T. It is graded, the elements of Tn being homogeneous of degree n. We extend the two products defined earlier on HT . (2) Let T ∈ T, different from 1. (a) We shall say that T is indecomposable if it cannot be written as T = T ′ .T ′′ , with T ′ , T ′′ 6= 1. (b) We shall say that T is ↓-indecomposable if it cannot be written as T = T ′ ↓ T ′′ , with T ′ , T ′′ 6= 1. (c) We shall say that T is bi-indecomposable if it is both indecomposable and ↓-indecomposable. Note that (HT , ., ↓) is a 2-associative algebra [7], that is to say an algebra with two associative products sharing the same unit. Proposition 5. (1) The associative algebra (HT , .) is freely generated by the set of indecomposable topologies. 10 LOÏC FOISSY AND CLAUDIA MALVENUTO (2) The associative algebra (HT , ↓) is freely generated by the set of ↓-indecomposable topologies. (3) The 2-associative algebra (HT , ., ↓) is freely generated by the set of biindecomposable topologies. Proof. 1. An easy induction on the degree proves that any T ∈ T can be written as T = T1 . . . . .Tk , with T1 , . . . , Tk indecomposable. Let us assume that T = T1 . · · · .Tk = T1′ . · · · .Tl′ , with T1 , . . . , Tk , T1′ , . . . , Tl′ indecomposable topologies. Let m be the smallest integer ≥ 1 such that for all 1 ≤ i ≤ m < j ≤ n, i and j are not comparable for ≤T . By definition of the product ., for all i ≤ deg(T1 ), for all j > deg(T1 ), i and j are not comparable for ≤T , so m ≤ deg(T1 ). Let T ′ be the restriction of the topology T1 to {1, . . . , m} and T ′′ be the restriction of the topology T1 to {m + 1, . . . , deg(T1 )}, reindexed to {1, . . . , deg(T1 ) − m}. By definition of m, T1 = T ′ .T ′′ . As T1 is indecomposable, T ′ = 1 or T ′′ = 1; as m ≥ 1, T ′′ = 1, so T ′ = T1 . Similarly, T ′ = T1′ = T1 . The restriction of T to {m + 1, . . . , deg(T )}, after a reindexation, gives T2 . · · · .Tk = T2′ . · · · .Tl′ . We conclude by an induction on the degree of T . 2. Similar proof. For the unicity of the decomposition, use the smallest integer m ≥ 1 such that for all i ≤ m < j ≤ n, i ≤T j. 3. First step. Let T ∈ Tn , n ≥ 1. Let us assume that T is not ↓-indecomposable. Then T = T ′ ↓ T ′′ , with T ′ , T ′′ 6= 1, so 1 ≤T n: this implies that T is indecomposable. Hence, one, and only one, of the following assertions holds: • T is indecomposable and not ↓-indecomposable. • T is not indecomposable and ↓-indecomposable. • T is bi-indecomposable. Second step. Let (A, . ↓) be a 2-associative algebra, and let aT ∈ A for any bi-indecomposable T ∈ T. Let us prove that there exists a unique morphism of 2associative algebras φ : HT −→ A, such that φ(T ) = aT for all bi-indecomposable T ∈ T. The proof will follow, since HT satisfies the universal property of the free 2-associative algebra generated by the bi-indecomposable elements. We define φ(T ) for T ∈ T by induction on deg(T ) in the following way: (1) φ(1) = 1A . (2) If T is bi-indecomposable, then φ(T ) = aT . (3) If T is indecomposable and not ↓-indecomposable, we write uniquely T = T1 ↓ . . . ↓ Tk , with k ≥ 2, T1 , . . . , Tk ∈ T, ↓-indecomposable. Then φ(T ) = φ(T1 ) ↓ . . . ↓ φ(Tk ). (4) If T is not indecomposable and ↓-indecomposable, we write uniquely T = T1 . · · · .Tk , with k ≥ 2, T1 , . . . , Tk ∈ T, indecomposable. Then φ(T ) = φ(T1 ). · · · .φ(Tk ). By the first step, φ is well-defined. By the unicity of the decomposition into decomposables or ↓-indecomposables, φ is a morphism of 2-associative algebras.  We denote by F (X) the generating formal series of all topologies on [n], by FI (X) the formal series of indecomposable topologies on [n], by F↓I (X) the formal series of ↓-indecomposable topologies on [n], and by FBI (X) the formal series on THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 11 bi-indecomposable topologies on [n]. Then: FI (X) = F↓I (X) = F (X) − 1 −2 + 3F (X) − F (X)2 , FBI (X) = . F (X) F (X) This gives: n I, ↓ I BI 1 2 1 3 1 2 n I, ↓ I BI 3 4 5 6 7 8 9 22 292 6 120 193 594 9 070 536 622 336 756 61 915 861 962 15 229 5 298 177 661 8 605 831 601 894 158 60 571 434 501 10 11 12 8 846 814 822 932 1 798 543 906 246 948 515 674 104 905 890 202 8 716 575 772 821 1 780 241 773 757 704 511 992 638 746 006 383 2.3. A coproduct on finite topologies. Notations. (1) Let X be a finite, totally ordered set of cardinality n, and T a topology on X. There exists a unique increasing bijection φ from X to [n]. We denote by Std(T ) the topology on [n] defined by: Std(T ) = {φ(O) | O ∈ T }. It is an element of Tn . (2) Let X be a finite set, and T be a topology on X. For any Y ⊆ X, we denote by T|Y the topology induced by T on Y , that is to say: T|Y = {O ∩ Y | O ∈ T }. Note that if Y is an open set of T , T|Y = {O ∈ T | O ⊆ Y }. Proposition 6. Let T ∈ Tn , n ≥ 1. We put: X ∆(T ) = Std(T|[n]\O ) ⊗ Std(T|O ). O∈T Then: (1) (HT , ., ∆) is a graded Hopf algebra. (2) (HT , ↓, ∆) is a graded infinitesimal bialgebra. (3) The involution ι defines a Hopf algebra isomorphism from (HT , ., ∆) to (HT , ., ∆op ). Proof. Let T ∈ Tn , n ≥ 0. Then: (∆ ⊗ Id) ◦ ∆(T ) X = Std((T|[n]\O )|([n]\O)\O′ ) ⊗ Std((T|[n]\O )|O′ ) ⊗ Std(T|O ) O∈T , O′ ∈T|[n]\O = X Std(T|[n]\(O⊔O′ ) ) ⊗ Std(T|O′ ) ⊗ Std(T|O ). O∈T , O′ ∈T|[n]\O If O ∈ T and O′ ∈ T|[n]\O , then O ⊔ O′ is an open set of T . Conversely, if O1 ⊆ O2 are open sets of T , then O2 \ O1 ∈ T|[n]\O1 . Putting O1 = O and O2 = O ⊔ O′ : X (∆ ⊗ Id) ◦ ∆(T ) = Std(T|[n]\O2 ) ⊗ Std(T|O2 \O1 ) ⊗ Std(T|O1 ). O1 ⊆O2 ∈T 12 LOÏC FOISSY AND CLAUDIA MALVENUTO Moreover: X (Id ⊗ ∆) ◦ ∆(T ) = Std(T|[n]\O ) ⊗ Std(T|O\O′ ) ⊗ Std(T|O′ ) O∈T , O′ ∈T|O If O is an open set of T and O′ is an open set of T|O , then O′ is an open set of T . Hence, putting O1 = O′ and O2 = O: X (Id ⊗ ∆) ◦ ∆(T ) = Std(T|[n]\O2 ) ⊗ Std(T|O2 \O1 ) ⊗ Std(T|O1 ). O1 ⊆O2 ∈T This proves that ∆ is coassociative. It is obviously homogeneous of degree 0. Moreover, ∆(1) = 1 ⊗ 1 and for any T ∈ Tn , n ≥ 1: X Std(T|[n]\O ) ⊗ Std(T|O ). ∆(T ) = T ⊗ 1 + 1 ⊗ T + ∅(O([n] So ∆ has a counit. Let T ∈ Tn , T ′ ∈ Tn′ , n, n′ ≥ 0. By definition of T .T ′ : X ∆(T .T ′ ) = Std((T .T ′ )|[n+n′ ]\O.O′ ) ⊗ Std((T .T ′ )|O.O′ ) O∈T ,O′ ∈T ′ = X O∈T = Std(T|[n]\O ).Std(T[n′ ′ ]\O′ ) ⊗ Std(T|O ).Std(T|O′ ) ,O′ ∈T ′ X O∈T ,O′ ∈T ′ 
 ′ Std(T|[n]\O ) ⊗ Std(T|O ) . Std(T|[n ′ ]\O ′ ) ⊗ Std(T|O ′ ) = ∆(T ).∆(T ′ ). Hence, (HT , ., ∆) is a Hopf algebra. By definition of T ↓ T ′ : X

Std (T ↓ T ′ )|[n+n′ ]\(O↓[n′ ]) ⊗ Std (T ↓ T ′ )|O↓[n′ ] ∆(T ↓ T ′ ) = O∈T ,O6=∅ + X O′ ∈T ′ ,O′ 6=[n′ ]

Std (T ↓ T ′ )|[n+n′ ]\O′ (+n) ⊗ Std (T ↓ T ′ )|O′ (+n)

+ Std (T ↓ T ′ )|[n+n′ ]\[n′ ](+n) ⊗ Std (T ↓ T ′ )[n′ ](+n) X Std(T|[n]\O ) ⊗ Std(T|O ) ↓ T ′ = O∈T ,O6=∅ + X ′ ′ ′ T ↓ Std(T|[n ′ ]\O ′ ) ⊗ Std(T|O ′ ) + T ⊗ T O′ ∈T ′ ,O′ 6=[n′ ] = X O∈T ,O6=∅ + X
Std(T|[n]\O ) ⊗ Std(T|O ) ↓ (1 ⊗ T ′ ) O′ ∈T ′ ,O′ 6=[n′ ]   ′ ′ +T ⊗T′ (T ⊗ 1) ↓ Std(T|[n ′ ]\O ′ ) ⊗ Std(T|O ′ ) = (∆(T ) − T ⊗ 1) ↓ (1 ⊗ T ′ ) + (T ⊗ 1) ↓ (∆(T ) − 1 ⊗ T ′ ) + T ⊗ T ′ = ∆(T ) ↓ (1 ⊗ T ′ ) + (T ⊗ 1) ↓ ∆(T ) − T ⊗ T ′ . Hence, (HT , ↓, ∆) is an infinitesimal bialgebra. THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 13 For all T , T ′ ∈ T, ι(T .T ′ ) = ι(T ).ι(T ′ ). Moreover: X ∆(ι(T )) = Std(ι(T )|O ) ⊗ Std(ι(T )|[n]\O ) O∈T = X ι(Std(T|O )) ⊗ ι(Std(T|[n]\O )) O∈T = (ι ⊗ ι) ◦ ∆op (T ). So ι is a Hopf algebra morphism from HT to Hcop T .  As a consequence of theorem 1: Theorem 7. The graded, connected coalgebra (HT , ∆) is cofree, that is to say is isomorphic to the tensor algebra on the space of its primitive elements with the deconcatenation coproduct. Remark. Forgetting the total order on [n], that is to say considering isoclasses of finite topologies, we obtain the Hopf algebra of finite spaces of [4] as a quotient of HT ; the product ↓ induces the product ≻ on finite spaces. 2.4. Link with special posets. Let T ∈ T. We put: c(T ) = deg(T ) − ♯{equivalence classes of ∼T }. Note that c(T ) ≥ 0. Moreover, c(T ) = 0 if, and only if, the relation ∼T is the equality, or equivalently if the preorder ≤T is an order, that is to say if T is T0 [17]. If T ∈ Tn , n ≥ 0, is T0 , then for any open set O of T , T|O and T|[n]\O are also T0 . Moreover, if T and T ′ are T0 , then T .T ′ and T ′ ↓ T ′ also are. Hence, the subspace HT0 of HT generated by T0 topologies is a Hopf subalgebra. Considering T0 topologies as special posets, it is isomorphic to the Hopf algebra of special posets HSP : this defines an injective Hopf algebra morphism from HSP to HT . We now identify HSP with its image by this morphism, that is to say with HT0 . Note that HSP is stable under ↓, so is a Hopf 2-associative subalgebra of HT . Notation. Let T ∈ Tn , n ≥ 0. We denote by T the special poset Std([n]/ ∼T ), resulting on the set of equivalence classes of ∼T , where the elements of [n]/ ∼T , that is to say the equivalence classes of ∼T , are totally ordered by the smallest element of each class. In this way, T is a special poset. Examples. q 1, 2 q 3 = q 1 q 2 , q 1, 3 q 2 = q 1 q 2 , q 2, 3 q 1 = q 1 q 2 , q3 q q 1, 2 = q 21 , q2 q q 1, 3 = q 21 , q1 q q 2, 3 = q 12 , q 1, 2 q q 3 = q 12 , q 1, 3 q q 2 = q 12 , q 2, 3 q q 1 = q 21 ; q 1, 2 = q 1 , q 1, 2, 3 = q 1 . Proposition 8. Let q ∈ K. The following map is a surjective morphism of Hopf 2-associative algebras:  HT −→ HSP θq : T −→ q c(T ) T . 14 LOÏC FOISSY AND CLAUDIA MALVENUTO Proof. If T , T ′ ∈ T, then T .T ′ = T .T ′ and T ↓ T ′ = T ↓ T ′ . Moreover, deg(T .T ′ ) = deg(T ↓ T ′ ) = deg(T ) + deg(T ′ ), and the number of equivalence classes of ∼T .T ′ and ∼T ↓T ′ are both equal to the sum of the number of equivalence classes of ∼T and ∼′T . Hence, c(T .T ′ ) = c(T ↓ T ′ ) = c(T ) + c(T ′ ), and: ′ ′ θq (T .T ′ ) = q c(T .T ) T .T ′ = q c(T ) q c(T ) T .T ′ = θq (T ).θq (T ′ ), ′ ′ θq (T ↓ T ′ ) = q c(T ↓T ) T ↓ T ′ = q c(T ) q c(T ) T ↓ T ′ = θq (T ) ↓ θq (T ′ ), so θq is a 2-associative algebra morphism. If T ∈ Tn , n ≥ 1, then any open set of T is a union of equivalence classes of ∼T . So there is a bijection:  {open sets of T } −→ {ideals of T } O −→ Std(O/ ∼T ) Moreover, c(T ) = c(T|[n]\O ) + c(T|O ) = c(Std(T|[n]\O )) + c(Std(T|O )). If T has k equivalence classes, we obtain: ∆ ◦ θq (T ) = q c(T ) ∆(T ) X = q c(T ) (([n] \ O)/ ∼T ) ⊗ (O/ ∼T ) O∈T = X q c(Std(T|[n]\O )) q c(Std(T|O )) Std(T|[n]\O ) ⊗ Std(T|O ) O∈T = (θq ⊗ θq ) ◦ ∆(T ). If T ∈ T is T0 , then T = T and c(T ) = 0, so θq (T ) = T : θq is surjective.  We obtain a commutative diagram of Hopf 2-associative algebras: ;; HT ✇✇ ✇ ✇ θq ✇✇  - ✇✇ // HSP HSP ι Id Examples. θq ( q 1, 2 q 3 ) = q q 1 q 2 , θq ( q 1, 3 q 2 ) = q q 1 q 2 , θq ( q 2, 3 q 1 ) = q q 1 q 2 , q θq ( q 31, 2 ) = q q θq ( q 21, 3 ) = q q θq ( q 12, 3 ) = q q2 q1 , q2 q1 , q1 q2 , q 2 θq ( q 1, )=q 3 q 1, 3 θq ( q 2 ) = q q 3 θq ( q 2, )=q 1 q1 q2 , q1 q2 , q2 q1 . θq ( q 1, 2 ) = q q 1 , θq ( q 1, 2, 3 ) = q 2 q 1 . Remarks. (1) In particular, for any T ∈ T: θ0 (T ) =  T if T is T0 , 0 otherwise. (2) θq is homogeneous for the gradation of HT by the cardinality if, and only if, q = 0. It is always homogeneous for the gradation by the number of equivalence classes (note that this gradation is not finite-dimensional). THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 15 2.5. Pictures and duality. The concept of pictures between tableaux was introduced by Zelevinsky in [18], and generalized to pictures between double posets by Malvenuto and Reutenauer in [11]. We now generalize this for finite topologies, to obtain a Hopf pairing on HT . Notations. Let ≤T be a preorder on [n], and let i, j ∈ [n]. We shall write i <T j if (i ≤T j and not j ≤T i). Definition 9. Let T ∈ Tk , T ∈ Tl , and let f : [k] −→ [l]. We shall say that f is a picture from T to T ′ if: • • • • f is bijective; For all i, j ∈ [k], i <T j =⇒ f (i) < f (j); For all i, j ∈ [k], f (i) <T ′ f (j) =⇒ i < j; For all i, j ∈ [k], i ∼T j ⇐⇒ f (i) ∼T ′ f (j). The set of pictures between T and T ′ is denoted by P ic(T , T ′ ). Proposition 10. We define a pairing on HT by hT , T ′ i = ♯P ic(T , T ′ ) for all T , T ′ ∈ T. Then this pairing, when extended by linearity, is a symmetric Hopf pairing. Moreover, ι is an isometry for this pairing. Proof. First, if T , T ′ ∈ T, P ic(T ′ , T ) = {f −1 | f ∈ P ic(T , T ′ )}, so hT ′ , T i = hT , T ′ i: the pairing is symmetric. We fix T1 , T2 , T ∈ T, of respective degrees n1 , n2 and n. Let f ∈ P ic(T1 .T2 , T ). Let x ∈ [n1 + n2 ] \ [n1 ] and y ∈ [n1 + n2 ], such that f (x) ≤T f (y). Two cases can occur: • f (x) <T f (y). Then x < y, so y ∈ [n1 + n2 ] \ [n1 ]. • f (x) ∼T f (y). Then x ∼T1 .T2 y. By definition of T1 .T2 , [n1 + n2 ] \ [n1 ] is an open set of T1 .T2 , so y ∈ [n1 + n2 ] \ [n1 ]. In both cases, y ∈ [n1 + n2 ] \ [n1 ], so f ([n1 + n2 ] \ [n1 ]) is an open set of T , which we denote by Of . Moreover, by restriction, Std(f|[n1 ] ) is a picture between T1 and Std(T|[n]\Of ) and Std(f|[n1 +n2 ]\[n1 ] ) is a picture between T2 and Std(T|O ). We can define a map:  G  P ic(T1 .T2 , T ) −→ P ic(T1 , Std(T|[n]\O )) × P ic(T2 , Std(T|O )) φ: O∈T  f −→ (Std(f|[n1 ] ), Std(f|[n1 +n2 ]\[n1 ] )) This map is clearly injective. Let O ∈ T , (f1 , f2 ) ∈ P ic(T1 , Std(T|[n]\O )) × P ic(T2 , Std(T|O )). Let f be the unique bijection [n1 + n2 ] −→ [n] such that f ([n1 + n2 ] \ [n1 ]) = O, f1 = Std(f|[n1 ] ) and f2 = Std(f|[n1 +n2 ]\[n1 ] ). Let us prove that f is a picture between T1 .T2 and T . • If i <T1 .T2 j in [n1 + n2 ], then (i, j) ∈ [n1 ]2 or (i, j) ∈ ([n1 + n2 ] \ [n1 ])2 . In the first case, f1 (i) < f1 (j), so f (i) < f (j). In the second case, f2 (i − n1) < f2 (j − n1 ), so f (i) < f (j). • If f (i) <T f (j) in T , as f ([n1 + n2 ] \ [n1 ]) = O is an open set, (i, j) ∈ [n1 ]2 or (i, j) ∈ ([n1 +n2 ]\[n1 ])2 or (i, j) ∈ [n1 ]×([n1 +n2 ]\[n1 ]). In the first case, f1 (i) <Std(T|[n]\O ) f1 (j), so i < j. In the second case, f2 (i − n1 ) <Std(T|O ) f2 (j − n1 ), so i − n1 < j − n1 and i < j. In the last case, i < j. 16 LOÏC FOISSY AND CLAUDIA MALVENUTO • If i ∼T1 .T2 j, by definition of T1 .T2 , (i, j) ∈ [n1 ]2 or (i, j) ∈ ([n1 +n2 ]\[n1 ])2 . In the first case, f1 (i) ∼Std(T|[n]\O ) f1 (j), so f (i) ∼T f (j). In the second case, f2 (i − n1 ) ∼Std(T|O ) f2 (j − n1 ), so f (i) ∼T f (j). • If f (i) ∼T f (j), as O = f ([n1 + n2 ] \ [n1 ]) is an open set of T , both O and f ([n1 ]) = [n] \ O are stable under ∼T , so (i, j) ∈ [n1 ]2 or (i, j) ∈ ([n1 + n2 ] \ [n1 ])2 . In the first case, f1 (i) ∼Std(T|[n]\O ) f1 (j), so i ∼T1 j and i ∼T1 .T2 j. In the second case, f2 (i − n1 ) ∼Std(T|O ) f2 (j − n1 ), so i − n1 ∼T2 j − n1 and i ∼T1 .T2 j. Finally, φ is bijective. We obtain: hT1 .T2 , T i = ♯P ic(T1 .T2 , T ) X = ♯P ic(T1 , Std(T|[n]\O )) × ♯P ic(T2 , Std(T|O )) O∈T = X hT1 , Std(T|[n]\O )ihP ic(T2 , Std(T|O )i O∈T = hT1 ⊗ T2 , ∆(T )i. So h−, −i is a Hopf pairing. Let T , T ′ ∈ T. It is not difficult to show that P ic(ι(T ), ι(T ′ )) = j(P ic(T , T ′ )), so hι(T ), ι(T ′ )i = hT , T ′ i.  Remark. Here is the matrix of the pairing in degree 2: qq 1 q q 1 q 2 qq 21 2 1, 2 q1 q2 2 1 1 0 qq 2 1 1 0 0 1 q1 q2 1 0 1 0 q 1, 2 0 0 0 2 q q So this pairing is degenerated, as q 1 q 2 − q 21 − q 12 is in its kernel. The first values of the rank of the pairing in dimension n is given by the following array: n Rk(h−, −i|(HT )n ) dim(Ker(h−, −i|(HT )n )) 1 2 1 3 0 1 3 4 16 111 13 244 3. Ribbon basis 3.1. Definition. The set Tn of topologies on [n] is partially ordered by the refinement of topologies: if T , T ′ ∈ Tn , T ≤ T ′ if any open set of T is an open set of T ′ . For example, here is the Hasse graph of T2 : q1 q2 ② ❊❊❊ ② ❊❊ ②② ❊❊ ②② ❊ ②② qq 1 qq 2 1 ❊ 2 ❊❊ ② ② ❊❊ ② ❊❊ ②② ❊❊ ②② ②② q 1, 2 THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 17 Definition 11. We define a basis (RT )T ∈T of HT in the following way: for all T ∈ Tn , n ≥ 0, X T = RT ′ . T ′ ≤T Examples. If {i, j} = {1, 2}: q R q j = q ji − q i, j , R q i, j = q i, j , qi If {i, j, k} = {1, 2, 3}: q q R q i q j = q i q j − q ji − q ij + q i, j . R q i, j, k = q i, j, k , q R q k = q ki, j − q i, j, k , q i, j q k R q j, k = q j, − q i, j, k , i qi q k q R q i q j, k = q i q j, k − q j, − q ij, k + q i, j, k , i qk q q k q R q k = q ji − q j, − q ki, j + q i, j, k , i qq j i qj qk q q k j q qk q , Rj q q k = ∨qi − q ji − q ki + q j, i ∨qi qq i qq i qi q q k − q jk − q kj + q ij, k , R qi = j ∧q q k j ∧q qq j qj j q qk q q q k + q ki − q i, j q k + q ki, j = q ji q k − ∨qi − i ∧q R qj qi qk q q q q R q i q j q k = q i q j q k − q ji q k − q ki q j − q ij q k − q kj q i − q k q q q q q − q j, − q i,j k − q i,k j − q ij, k − q ji, k − q ki, j i qi qj qk i q qj i q qk j q qk q k + i ∧q q k + i ∧q q j + ∨qi + ∨qj + ∨qk + j ∧q q + q i,k j − q i, j, k, qi q q k q j − q jk q i + 2 q i, j, k . 3.2. The products and the coproduct in the basis of ribbons. Notations. (1) Let T ∈ Tn and let I, J ⊆ [n]. We shall write I ≤T J if for all i ∈ I, j ∈ J, i ≤T j. (2) Let T ∈ Tn and let I, J ⊆ [n]. We shall write I <T J if for all i ∈ I, j ∈ J, i <T j. Theorem 12. (1) Let T ∈ Tk , T ′ ∈ Tl , k, l ≥ 0. Then: X RT ′′ . RT .RT ′ = T ′′ ∈Tk+l , ′′ ′′ T|[k] =T , Std(T|[k+l]\[k] )=T ′ (2) Let T ∈ Tk , T ′ ∈ Tl , k, l ≥ 0. Then: X RT ↓ RT ′ = RT ′′ . T ′′ ∈Tk+l , ′′ ′′ T|[k] =T , Std(T|[k+l]\[k] )=T ′ , [k]≤T ′′ [k+l]\[k] (3) For all T ∈ Tn , n ≥ 0: ∆(RT ) = X O∈T , [n]\O<T O RStd(T|[n]\O ) ⊗ RStd(T|O ) . 18 LOÏC FOISSY AND CLAUDIA MALVENUTO Proof. 1. First step. We first prove that for any T ′′ ∈ Tk+l , T ′′ ≤ T .T ′ if, and ′′ ′′ only if, T|[k] ≤ T and Std(T|[k+l]\[k] ) ≤ T ′. ′′ ′′ =⇒. As T ′′ ≤ T .T ′ , we obtain T|[k] ≤ (T .T ′ )|[k] = T , and Std(T|[k+l]\[k] ) ≤ ′ ′ Std((T .T )|[k+l]\[k] ) = T . ′′ ⇐=. Let I be an open set of T ′′ . Then I1 = I ∩ [k] is an open set of T|[k] , so I1 is an open set of T . Moreover, I2 = I ∩ ([k + l] \ [k])(−k) is an open set of ′′ Std(T|[k+l]\[k] ), so I2 is an open set of T ′ . By definition of T .T ′ , I1 ⊔ I2 [k] = I is an open set of T .T ′ , so T ′′ ≤ T .T ′ . Second step. We define a product ⋆ on HT by the formula: X RT ′′ , RT ⋆ RT ′ = T ′′ ∈Tk+l , ′′ ′′ T|[k] =T ,Std(T|[k+l]\[k] )=T ′ for any T ∈ Tk , T ′ ∈ Tl , k, l ≥ 0. Then: X T ⋆T′ = RS ⋆ RS ′ S≤T ,S ′ ≤T ′ = X S≤T ,S ′ ≤T ′ RS ′′ S ′′ ∈Tk+l , ′′ ′′ S|[k] =S,Std(S|[k+l]\[k] )=S ′ X = X RS ′′ S ′′ ∈Tk+l , ′′ ′′ S|[k] ≤T ,Std(S|[k+l]\[k] )≤T ′ = X RS ′′ S ′′ ∈Tk+l , S ′′ ≤T .T ′ ′ = T .T . We used the first step for the fourth equality. So ⋆ = .. 2. First step. We first prove that for any T ′′ ∈ Tk+l , T ′′ ≤ T ↓ T ′ if, and only ′′ ′′ if, T|[k] ≤ T , Std(T|[k+l]\[k] ) ≤ T ′ and [k] ≤T ′′ [k + l] \ [k]. ′′ ′′ =⇒. By restriction, T|[k] ≤ T , Std(T|[k+l]\[k] ) ≤ T ′ . Let x ∈ [k] and y ∈ ′′ [k + l] \ [k], and let O be an open set of T which contains x. It is also an open set of T ↓ T ′ which contains x, so it contains [k + l] \ [k] by definition of T ↓ T ′ . So x ≤T ′′ y. ⇐=. As [k] ≤T ′′ [k + l] \ [k], any open sets of T ′′ which contains an element of [k] contains [k + l] \ [k]. Hence, there are two types of open sets in T ′′ : • Open sets O contained in [k + l] \ [k]. Then O(−k) is an open set of ′′ Std(T|[k+l]\[k] ), so it is an open set of T ′ , and finally O is an open set of ′ T ↓T . ′′ • Open sets O which contain [k + l] \ [k]. Then O ∩ [k] is an open set of T|[k] , so is an open set of T . Hence, O is an open set of T ↓ T ′ . We obtain in this way that T ′′ ≤ T ↓ T ′ . THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 19 Second step. Using the first step, we conclude as in the second step of the first point. 3. First step. Let O ∈ T , S ≤ T|O and S ′ ≤ T|[n]\O . It is not difficult to see ′ ′ that there exists a unique topology T ′ ∈ Tn such that S = T|O , S ′ = T|[n]\O and [n] \ O <T ′ O. It is: T ′ = {Ω ∪ O | Ω ∈ S ′ } ∪ S. Second step. We define a coproduct on HT by: X δ(RT ) = RStd(T|[n]\O ) ⊗ RStd(T|O ) , O∈T , [n]\O<T O for any T ∈ T. Then: δ(T ) = X δ(RT ′ ) T ′ ≤T = X T = ′ ≤T O∈T X ′, X ′ ′ ) RStd(T|[n]\O ) ⊗ RStd(T|O ([n]\O)<T ′ O X ′ ′ ) RStd(T|[n]\O ) ⊗ RStd(T|O O∈T T ′ ≤T , O∈T ′ , ([n]\O)<T ′ O = X X RS ⊗ RS ′ O∈T S≤Std(T|[n]\O ), S ′ ≤Std(T|O ) = X Std(T|[n]\O ) ⊗ Std(T|O ) O∈T = ∆(T ). We used the first step for the third equality. So δ = ∆.  4. Generalized T-partitions 4.1. Definition. Definition 13. Let T ∈ Tn . (1) A generalized T-partition of T is a surjective map f : [n] −→ [p] such that if i ≤T j in [n], then f (i) ≤ f (j) in [p]. If f is a generalized T-partition of T , we shall represent it by the packed word f (1) . . . f (n). (2) Let f a generalized T-partition of T . We shall say that f is a (strict) T-partition if for all i, j ∈ [n]: • i <T j and i > j implies that f (i) < f (j) in [p]. • If i < j < k, i ∼T k and f (i) = f (j) = f (k), then i ∼T j and j ∼T k. (3) The set of generalized T-partitions of T is denoted by P(T ); the set of (strict) T-partitions of T is denoted by Ps (T ). (4) If f ∈ P(T ), we put: ℓ1 (f ) = ♯{(i, j) ∈ [n]2 | i <T j, i < j, and f (i) = f (j)}, ℓ2 (f ) = ♯{(i, j) ∈ [n]2 | i <T j, i > j, and f (i) = f (j)}, ℓ3 (f ) = ♯{(i, j, k) ∈ [n]3 | i < j < k, i ∼T k, i ∼ | T j, j ∼ | T k and f (i) = f (j) = f (k)}. 20 LOÏC FOISSY AND CLAUDIA MALVENUTO Note that f is strict if, and only if, ℓ2 (f ) = ℓ3 (f ) = 0. 1q q5 Example. Let T = ∨q2, 4 q 3 . If f is a packed word of length 5, it is a generalized T -partition of T if, and only if, f (2) = f (4) ≤ f (1), f (5). Hence:   (11111), (11112), (11211), (11212), (11213), (11312), (21111),          (21112), (21113), (21211), (21212), (21213), (21311), (21312),  (21313), (21314), (21413), (22122), (22123), (31112), (31211), . P(T ) =        (31212), (31213), (31214), (31312), (31412), (32122), (32123),    (32124), (41213), (41312), (42123) Moreover, f is a strict T -partition of T if, and only if, f (2) = f (4) < f (1), f (2) = f (4) ≤ f (5), and f (3) 6= f (2), f (4). Hence:    (21211), (21212), (21213), (21311), (21312), (21313), (21314),  (21413), (31211), (31212), (31213), (31214), (31312), (31412), . Ps (T ) =   (32122), (32123), (32124), (41213), (41312), (42123) Remarks. (1) Let f ∈ P(T ). If i ∼T j in [n], then i ≤T j and j ≤T i, so f (i) ≤ f (j) and f (j) ≤ f (i), and finally f (i) = f (j). (2) If T ∈ T is T0 , then the set of strict T-partitions of T is the set of Ppartitions of the poset T , as defined in [6, 16]. We shall now omit the term "strict" and simply write "T-partitions". Proposition 14. Let q = (q1 , q2 , q3 ) ∈ K3 . We define a linear map Γq : HT −→ WQSym in the following way: for all T ∈ Tn , n ≥ 0, X ℓ (f ) ℓ (f ) ℓ (f ) q11 q22 q33 f (1) . . . f (n). Γq (T ) = f ∈P(T ) Then Γq is a homogeneous surjective Hopf algebra morphism. Moreover, j◦Γ(q2 ,q1 ,q3 ) ) = Γ(q1 ,q2 ,q3 ) ◦ ι. Proof. We shall use the following notations: if T ∈ T and f ∈ P(T ), we put ℓ (f ) ℓ (f ) ℓ (f ) ℓ(f ) = (ℓ1 (f ), ℓ2 (f ), ℓ3 (f )) and q ℓ(f ) = q11 q22 q33 . Let us first prove that Γq is an algebra morphism. Let T , T ′ ∈ T, of respective degree n and n′ . Let f = f (1) . . . f (n + n′ ) be a packed word. If 1 ≤ i ≤ n < j ≤ n + n′ , then i and j are not comparable for ≤T .T ′ . Hence, f ∈ P(T .T ′ ) if, and only if, f ′ = P ack(f (1) . . . f (n)) ∈ P(T ) and f ′′ = P ack(f (n + 1) . . . f (n + n′ )) ∈ P(T ′ ). Moreover, ℓ(f ) = ℓ(f ′ ) + ℓ(f ′′ ), so: X X q ℓ(f ) f (1) . . . f (n + n′ ) Γq (T .T ′ ) = f ′ ∈P(T ),f ′′ ∈P(T ′ ) = f ′ ∈P(T = P ack(f (1)...f (n))=f ′ P ack(f (n+1)...f (n+n′ ))=f ′′ ′ ′′ X q ℓ(f ) q ℓ(f X q ℓ(f ) q ℓ(f ) f ′ .f ′′ ),f ′′ ∈P(T ′ ) f ′ ∈P(T ),f ′′ ∈P(T ′ ) = Γq (T ).Γq (T ′ ). ) X f (1) . . . f (n + n′ ) ′ P ack(f (1)...f (n))=f P ack(f (n+1)...f (n+n′ ))=f ′′ ′ ′′ THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 21 Let us now prove that Γq is a coalgebra morphism. Let T ∈ Tn . We consider the two following sets: G A= P(T|[n]\O ) × P(T|O ), B = {(f, i) | f ∈ P(T ), 0 ≤ i ≤ max(f )}. O∈T Let (f, g) ∈ P(T|[n]\O ) × P(T|O ) ⊆ A. We define h : [n] −→ N by h(i) = f (i) if i∈ / O and h(i) = g(i) + max(f ) if i ∈ O. As f and g are packed words, h(1) . . . h(n) also is. Let us assume that i ≤T j. Three cases are possible: • i, j ∈ / O. As f (i) ≤ f (j), h(i) ≤ h(j). • i, j ∈ O. As g(i) ≤ g(j), h(i) ≤ h(j). • i∈ / O and j ∈ O. Then h(i) = f (i) ≤ max(f ) < max(f ) + g(j) = h(j). Consequently, h ∈ P(T ). Hence, we define a map θ : A −→ B, sending (f, g) to (h, max(f )). θ is injective: if θ(f, g) = θ(f ′ , g ′ ) = (h, k), then max(f ) = max(f ′ ) = k. Moreover: O = h−1 ({max(f ) + 1, . . . , max(h)}) = h−1 ({k + 1, . . . , max(h)}) = O′ . Then f = h|O = h|O′ = f ′ and g = P ack(h|[n]\O ) = P ack(h|[n]\O ) = g ′ . Finally, (f, g) = (f ′ , g ′ ). θ is surjective: let (h, k) ∈ B. We put O = h−1 ({k + 1, . . . max(h)}). Let i ∈ O and j ∈ [n], such that i ≤T j. As h ∈ P(T ), h(j) ≥ h(i) > k, so j ∈ O and O is an open set of T . Let f = h|[n]\O and g = P ack(h|O ). By restriction, f ∈ P(T|[n]\O ) and g ∈ P(T|O ). Moreover, as h is a packed word, max(f ) = k, and for all i ∈ O, g(i) = h(i) − k. This implies that θ(f, g) = (h, k). Let (f, g) ∈ P(T|[n]\O ) × P(T|O ) ⊆ A, and let θ(f, g) = (h, k). If i ∈ / O and j ∈ O, h(i) < h(j). Hence: ℓ1 (h) = ♯{(i, j) ∈ ([n] \ O)2 | i <T j, i < j and h(i) = h(j)} + ♯{(i, j) ∈ O2 | i <T j, i < j and h(i) = h(j)} = ♯{(i, j) ∈ ([n] \ O)2 | i <T j, i < j and f (i) = f (j)} + ♯{(i, j) ∈ O2 | i <T j, i < j and g(i) + max(f ) = g(j) + max(f )} = ℓ1 (f ) + ℓ1 (g). Similarly, ℓ2 (h) = ℓ2 (f ) + ℓ2 (g) and ℓ3 (h) = ℓ3 (f ) + ℓ3 (g). We obtain: X X
q ℓ(h) h|h−1 ({1,...,k}) ⊗ P ack h|h−1 ({k+1,...,max(h)}) ∆ ◦ Γq (T ) = h∈P(T ) 0≤k≤max(h) = X q ℓ(h) h|h−1 ({1,...,k}) ⊗ P ack h|h−1 ({k+1,...,max(h)}) X q ℓ(f ) q ℓ(g) f ⊗ g (h,k)∈B = (f,g)∈A = X O∈T   X f ∈P(T|[n]\O ) = (Γq ⊗ Γq ) ◦ ∆(T ).   q ℓ(f ) f  ⊗  X g∈P(T|O )  q ℓ(g) g 
22 LOÏC FOISSY AND CLAUDIA MALVENUTO Consequently, Γq is a Hopf algebra morphism. Let f be a packed word of length n, and Tf be the associated topology, as defined in section 2. Note that f ∈ P(Tf ); moreover, if g ∈ P(Tf ) is different from f , then max(g) < max(f ). Hence: Γq (Tf ) = f + (linear span of packed words g of maximum < max(f )) . In particular, if f = (1 . . . 1), Tw = q 1, . . . , n, and Γq (Tf ) = (1 . . . 1). By a triangularity argument, Γq is surjective. Let T ∈ T. It is not difficult to prove that P(T ) = j (P(ι(T ))). Moreover, if f ∈ P(ι(T )) and g = j(f ), then ℓ1 (f ) = ℓ2 (g), ℓ2 (f ) = ℓ1 (g) and ℓ3 (f ) = ℓ3 (g). So: X ℓ (g) ℓ (g) ℓ (g) q22 q11 q22 g = Γ(q1 ,q2 ,q3 ) (T ), j ◦ Γ(q2 ,q2 ,q3 ) ◦ ι(T ) = g∈P(T ) so j ◦ Γ(q2 ,q2 ,q3 ) ◦ ι = Γ(q1 ,q2 ,q3 ) .  Examples. Γq ( q 1 ) = (1), Γq ( q 1 q 2 ) = (12) + (21) + (11), q Γq ( q 21 ) = (12) + q1 (11), q Γq ( q 12 ) = (21) + q2 (11), Γq ( q 1, 2 ) = (11), 2 q q3 Γq ( ∨q1 ) = (123) + (132) + (122) + q1 (112) + q1 (121) + q12 (111), 1 q q3 Γq ( ∨q2 ) = (213) + (312) + (212) + q2 (112) + q1 (211) + q1 q2 (111), 1 q q2 Γq ( ∨q3 ) = (231) + (321) + (221) + q2 (121) + q2 (211) + q22 (111), q Γq ( q 31, 2 ) = (112) + q12 (111), q Γq ( q 21, 3 ) = (121) + q1 q2 q3 (111), q Γq ( q 12, 3 ) = (211) + q22 (111), q 3 Γq ( q 2, ) = (122) + q12 (111), 1 q 3 Γq ( q 1, ) = (212) + q1 q2 q3 (111), 2 q 1, 2 Γq ( q 3 ) = (221) + q22 (111). Remarks. (1) In particular: X f (1) . . . f (n), Γ(1,1,1) (T ) = f ∈P(T ) Γ(1,0,0) (T ) = X f (1) . . . f (n). f ∈Ps (T ) (2) The restriction of Γ(1,0,0) to HSP is the map Γ defined in section 1.2. 4.2. Linear extensions. Definition 15. Let T ∈ Tn , n ≥ 0. A linear extension of T is an ordered partition A = (A1 , . . . , Ak ) of [n] such that: • the equivalence classes of ∼T are A1 , . . . , Ak ; THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 23 • if, in the poset T , Ai ≤T Aj , then i ≤ j. The set of linear extensions of T is denoted by L(T ). Notations. If A = (A1 , . . . , Ak ) is a linear extension of T ∈ Tn , we bijectively associate to A the packed word f = f (1) . . . f (n) of length n such that for all i ∈ [n], for all j ∈ [k], f (i) = j if, and only if i ∈ Aj . We shall now use the packed word f instead of A. 3 q q 1, 5, 6 Example. Let T = ∨q2, 4 . The linear extensions of T are ({2, 4}, {3}, {1, 5, 6}) and ({2, 4}, {1, 5, 6}, {3}), or, expressed in packed words, (312133) and (213122). Remark. Let T ∈ Tn , n ≥ 0, and f = f (1) . . . f (n) be a packed word of length n. Then f ∈ L(T ) if, and only if: (1) for all i, j ∈ [n], f (i) = f (j) if, and only if, i ∼T j; (2) for all i, j ∈ [n], i <T j implies that f (i) < f (j). Hence, L(T ) ⊆ P (T ) and more precisley: L(T ) = {f ∈ P(T ) | max(f ) = ♯T } = {f ∈ Ps (T ) | max(f ) = ♯T }. Proposition 16. (1) Let f ′ , f ′′ be two packed words, of respective length n ′ and n . We put: X f. f ′ f ′′ = P ack(f (1)...f (n))=f ′ P ack(f (n+1)...f (n+n′ ))=f ′′ {f (1),...f (n)}∩{f (n+1),...,f (n+n′ )}=∅ This defines a product on WQSym, such that (WQSym, algebra. (2) Let L be the following map:   HT −→ WQSym X L: f. T −→  , ∆) is a Hopf f ∈L(T ) Then L is a surjective Hopf morphism from HT to (WQSym, Moreover, j ◦ L = L ◦ ι. , ∆). Proof. Let T , T ′ ∈ T, of respective degree n and n′ . Let f = f (1) . . . f (n + n′ ) be a packed word. If 1 ≤ i ≤ n < j ≤ n + n′ , then i and j are not comparable for ≤T .T ′ . Hence, f ∈ L(T .T ′ ) if, and only if: • f ′ = P ack(f (1) . . . f (n)) ∈ L(T ) and f ′′ = P ack(f (n + 1) . . . f (n + n′ )) ∈ L(T ′ ). • {f (1), . . . f (n)} ∩ {f (n + 1), . . . , f (n + n′ )} = ∅. So: X X f (1) . . . f (n + n′ ) L(T .T ′ ) = f ′ ∈L(T ),f ′′ ∈L(T ′ ) = X f ′ ∈L(T ),f ′′ ∈L(T ′ ) = L(T ) L(T ′ ). P ack(f (1)...f (n))=f ′ P ack(f (n+1)...f (n+n′ ))=f ′′ {f (1),...f (n)}∩{f (n+1),...,f (n+n′ )}=∅ f f′ 24 LOÏC FOISSY AND CLAUDIA MALVENUTO Let T ∈ Tn . We consider the two following sets: G A= L(T|[n]\O ) × L(T|O ), B = {(f, i) | f ∈ L(T ), 0 ≤ i ≤ max(f )}. O∈T Let (f, g) ∈ L(T|[n]\O ) × L(T|O ) ⊆ A. We define h : [n] −→ N by h(i) = f (i) if i∈ / O and h(i) = g(i) + max(f ) if i ∈ O. As f and g are packed words, h(1) . . . h(n) also is. Moreover, as O and [n] \ O are union of equivalence classes of ∼T , for all i, j ∈ [n], h(i) = h(j) if, and only if, i ∼T j. Let us assume that i ≤T j. Three cases are possible: • i, j ∈ / O. As f (i) ≤ f (j), h(i) ≤ h(j). • i, j ∈ O. As g(i) ≤ g(j), h(i) ≤ h(j). • i∈ / O and j ∈ O. Then h(i) = f (i) ≤ max(f ) < max(f ) + g(j) = h(j). Consequently, h ∈ L(T ). Hence, we define a map θ : A −→ B, sending (f, g) to (h, max(f )). The proof of the bijectivity of θ is similar to the proof for the case of generalized T-partitions. We obtain: X X
h|h−1 ({1,...,k}) ⊗ P ack h|h−1 ({k+1,...,max(h)}) ∆ ◦ L(T ) = h∈L(T ) 0≤k≤max(h) = X h|h−1 ({1,...,k}) ⊗ P ack h|h−1 ({k+1,...,max(h)}) X f ⊗g (h,k)∈B =
(f,g)∈A = X O∈T    X f ∈L(T|[n]\O ) = (L ⊗ L) ◦ ∆(T ).  f ⊗  X g∈L(T|O )  g Let f be a packed word. Then L(Tf ) = {f }, so L(Tf ) = f , that is L is surjective. As L is compatible with the products and the coproducts of both HT and (WQSym, , ∆), its image, that is to say (WQSym, , ∆), is a Hopf algebra, and L is a Hopf algebra morphism. It is not difficult to prove that for all T ∈ T, j(L(ι(T ))) = L(T ), which implies that j ◦ L ◦ ι = L.  Remark. The product is defined and used in [5]. Examples. q L( q 21 ) = (12), L( q 1 q 2 ) = (12) + (21), L( q 1 ) = (1), q L( q 12 ) = (21), L( q 1, 2, 3 ) = (111), L( q 1, 2 ) = (11), q2 q q3 L( q 1, 3 ) = (121), L( q 12, 3 ) = (211), L( q 1, 2 ) = (112), q 3 q 3 q 2 ) = (221), L( q 1, ) = (212), L( q 2, ) = (122), L( q 1, 3 2 1 2q q3 1q q3 1 q q2 L( ∨q1 ) = (123) + (132), L( ∨q2 ) = (213) + (312), L( ∨q3 ) = (231) + (321), q3 q3 q2 q q q L( q 21 ) = (123), L( q 12 ) = (213), L( q 13 ) = (231), qq 1 qq 1 qq 2 L( q 32 ) = (312), L( q 23 ) = (321). L( q 31 ) = (132), THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS Here is an example of product (112) 25 : (12) = (11234) + (11324) + (11423) + (22314) + (22413) + (33412). 4.3. From linear extensions to T-partitions. Definition 17. Let f, g be two packed words of the same length n. We shall say that g ≤ f if: • For all i, j ∈ [n], f (i) ≤ f (j) implies that g(i) ≤ g(j). • For all i, j ∈ [n], f (i) > f (j) and i < j implies that g(i) > g(j). • For all i, j ∈ [n], f (i) = f (j) implies that g(i) = g(j). The relation ≤ is a partial order on the set of packed words of length n. Here are the Hasse graphs of these posets if n = 2 or n = 3: (12) (21) ; (11) (123) ●● ●● ✇✇ ✇ ●● ✇ ●● ✇✇ ✇ ✇ (122) (112) ●● ●● ✇✇ ✇ ●● ✇ ●● ✇✇ ✇✇ (111) (132) (213) (231) (312) (121) (212) (221) (211) (321) Remark. This order is also introduced and used in [12]. Lemma 18. Let f, g be packed words of length n. Then g ≤ f if, and only if, g is a T-partition of Tf . Proof. =⇒. Let us assume that g ≤ f . If i ≤Tf j, then f (i) ≤ f (j), so g(i) ≤ g(j). If i <Tf j and i > j, then f (i) < f (j), so g(i) < g(j). We assume that i < j < k, i ∼Tf k and g(i) = g(j) = g(k). As i ∼T k, f (i) = f (k). If f (j) < f (i), as g ≤ f , we obtain g(j) < g(i): contradiction. So f (i) ≤ f (j). If f (j) > f (k), as g ≤ f , we obtain g(j) > g(k): contradiction. So f (j) ≤ f (k), and finally, f (i) = f (j) = f (k), so i ∼Tf j and j ∼Tf k. Hence, g ∈ Ps (Tf ). ⇐=. Let us assume that g ∈ Ps (Tf ). If f (i) ≤ f (j), then i ≤Tf j, so g(i) ≤ g(j). If f (i) = f (j), then i ∼Tf j, so g(i) = g(j). If f (i) < f (j) and i > j, then i <Tf j and i > j, so g(i) < g(j). This implies g ≤ f .  Proposition 19. We define:   WQSym ϕ(1,0,0) : f  −→ WQSym X −→ g. g≤f 26 LOÏC FOISSY AND CLAUDIA MALVENUTO Then ϕ(1,0,0) is a Hopf algebra isomorphism from (WQSym, such that the following diagram commutes: , ∆) to (WQSym, ., ∆) L // (WQSym, , ∆) HT ◆ ◆◆◆ ◆◆◆ ϕ(1,0,0) ◆◆ Γ(1,0,0) ◆◆◆  '' (WQSym, ., ∆) Proof. Let T ∈ Tn , n ≥ 0. First step. Let f ∈ L(T ) and g ≤ f ; let us prove that g ∈ Ps (T ). This is the generalization of lemma 18 to any finite topology. (1) If i ≤T j, then f (i) ≤ f (j), so g(i) ≤ g(j). (2) If i <T j and j < i, then f (i) < f (j), so g(i) < g(j). (3) Let us assume that i < j < k, i ∼T k and g(i) = g(j) = g(k). If i ∼ | T j, then f (i) 6= f (j). If f (i) > f (j), as g ≤ f , we obtain g(i) > g(j): this is a contradiction. So f (i) < f (j). As ∼T is an equivalence, j ∼ | T k, and we obtain in the same way f (j) < f (k). So f (i) < f (k), and i ∼ | T k: this is a contradiction. As a consequence, i ∼T j and j ∼T k. This proves that g ∈ Ps (T ). Second step. Let us now consider an element g ∈ Ps (T ). We want to prove that there exists a unique f ∈ L(T ) such that g ≤ f . For all p ∈ [max(g)], g −1 ({p}) is the union of equivalence classes of ∼T , and we put: g −1 ({p}) = Cp,1 ⊔ . . . ⊔ Cp,kp . Moreover, as g ∈ Ps (T ), necessarily the Cp,r are intervals. Hence, we assume that for all p: Cp,1 < . . . < Cp,kp , which means for all r < s, all the elements of Cp,r are smaller than all the elements of Cp,s . Unicity. Let us assume there exists f ∈ L(T ), such that g ≤ f . The linear extension f is constant on Cp,r : we put f (Cp,r ) = {cp,r }. As f ∈ L(T ), the cp,r are all distincts. (1) If r < s and cp,r > cp,s , choosing i ∈ Cp,r and j ∈ Cp,s , then i < j and f (i) > f (j), so, as g ≤ f , p = g(i) > g(j) = p: contradiction. So cp,r < cp,s . (2) If p < q and cq,s < cp,r , choosing i ∈ Cp,r and j ∈ Cq,s , then f (i) ≥ f (j), which implies that p = g(i) ≥ g(j) = q: contradiction. So cp,r < cq,s . We finally obtain: c1,1 < . . . < c1,k1 < . . . < cmax(g),1 < . . . < cmax(g),kmax(g) , which entirely determines f : f is unique. Existence. Let us consider the packed word f determined by: f (i) = k1 + . . . + kp−1 + r if x ∈ Cp,r . The values of f on two different subsets Cp,r are different, so i ∼T j if, and only if, f (i) = f (j). If i <T j, assuming i ∈ Cp,r and j ∈ Cq,s , then p = g(x) ≤ g(y) = q. When p < q, then f (i) < f (j). When p = q, if j < i we should have, as g ∈ Ps (T ), g(i) < g(j): contradiction. So i < j, and r < s, so f (x) < f (y). Finally, f ∈ L(T ). THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 27 If f (i) = f (j), then i and j are in the same Cp,r , so g(i) = g(j). If f (i) ≤ f (j), assuming that i ∈ Cp,r and j ∈ Cq,s , then p ≤ q, so p = g(x) ≤ g(y) = q. If f (i) < f (j) and i > j then p ≤ q; if p = q, then r > s, so f (i) > f (j): contradiction. So p < q, and p = g(i) < g(j) = q. We obtain that g ≤ f . As a conclusion: ϕ(1,0,0) ◦ L(T ) = X X g(1) . . . g(n) = f ∈L(T ) g≤f X g(1) . . . g(n) = Γ(1,0,0) (T ). g∈Ps (T ) So ϕ(1,0,0) ◦ L = Γ(1,0,0) . Third step. Let x, y ∈ WQSym. As L is surjective, there exists x′ , y ′ ∈ HT such that L(x′ ) = x and L(y ′ ) = y. Then: y) = ϕ(1,0,0) (L(x′ ) ϕ(1,0,0) (x L(y ′ )) = ϕ(1,0,0) ◦ L(x′ .y ′ ) = Γ(1,0,0) (x′ .y ′ ) = Γ(1,0,0) (x′ ).Γ(1,0,0) (y ′ ) = ϕ(1,0,0) ◦ L(x′ ).ϕ(1,0,0) ◦ L(y ′ ) = ϕ(1,0,0) (x).ϕ(1,0,0) (y). Moreover: ∆ ◦ ϕ(1,0,0) (x) = ∆ ◦ ϕ(1,0,0) ◦ L(x′ ) = ∆ ◦ Γ(1,0,0) (x′ ) = (Γ(1,0,0) ⊗ Γ(1,0,0) ) ◦ ∆(x′ ) = (ϕ(1,0,0) ⊗ ϕ(1,0,0) ) ◦ (L ⊗ L) ◦ ∆(x′ ) = (ϕ(1,0,0) ⊗ ϕ(1,0,0) ) ◦ ∆(L(x′ )) = (ϕ(1,0,0) ⊗ ϕ(1,0,0) ) ◦ ∆(x). So ϕ(1,0,0) is a Hopf algebra morphism. Let x ∈ WQSym. As Γ(1,0,0) is surjective, there exists x′ ∈ HT , such that Γ(1,0,0) (x′ ) = x. Then ϕ(1,0,0) (L(x′ )) = x: ϕ(1,0,0) is surjective. Since it is homogeneous and the homogeneous components of WQSym are finite dimensional, it is an isomorphism.  Examples. ϕ(1,0,0) ((123)) = (123) + (122) + (112) + (111), ϕ(1,0,0) ((132)) = (132) + (121), ϕ(1,0,0) ((213)) = (213) + (212), ϕ(1,0,0) ((231)) = (231) + (221), ϕ(1,0,0) ((312)) = (312) + (211), ϕ(1,0,0) ((321)) = (321), ϕ(1,0,0) ((112)) = (112) + (111), ϕ(1,0,0) ((121)) = (121), ϕ(1,0,0) ((211)) = (211), ϕ(1,0,0) ((122)) = (122) + (111), ϕ(1,0,0) ((212)) = (212), ϕ(1,0,0) ((221)) = (221), ϕ(1,0,0) ((111)) = (111). 28 LOÏC FOISSY AND CLAUDIA MALVENUTO Corollary 20. For any finite topology T of degree n: G G Ps (Tf ). {g | g ≤ f } = Ps (T ) = f ∈L(T ) f ∈L(T ) Proof. This is the first step of the proof of proposition 19.  So T -partitions only depend on linear extensions. In the case of special posets, that is to say of T0 topologies, the previous corollary is proved in [16, 6]. Remark. Let ϕ(0,1,0) = j ◦ ϕ(1,0,0) ◦ j. Then: ϕ(0,1,0) ◦ L = j ◦ ϕ(1,0,0) ◦ j ◦ L = j ◦ ϕ(1,0,0) ◦ L ◦ ι = j ◦ Γ(1,0,0) ◦ ι = Γ(0,1,0) . Moreover, for all packed word f : ϕ(0,1,0) (f ) = X g, g≤′ f where the order relation • For all i, j ∈ [n], • For all i, j ∈ [n], • For all i, j ∈ [n], ≤′ on packed words is defined by g ≤′ f if: f (i) ≤ f (j) implies that g(i) ≤ g(j). f (i) < f (j) and i < j implies that g(i) < g(j). f (i) = f (j) implies that g(i) = g(j). Proposition 21. Let q ∈ K3 . There exists a linear endomorphism ϕq of WQSym such that ϕq ◦ L = Γq if, and only if, q = (1, 0, 0) or q = (0, 1, 0). Proof. =⇒. If ϕq exists, then: q3 q3 q q ϕq ((123)) = ϕq ◦ L( q 21 ) = Γq ( q 21 ) = (123) + q1 (112) + q1 (122) + q13 (111), q3 q3 q q ϕq ((213)) = ϕq ◦ L( q 12 ) = Γq ( q 12 ) = (213) + q2 (112) + q1 (212) + q12 q2 (111), q1 q1 q q ϕq ((312)) = ϕq ◦ L( q 32 ) = Γq ( q 32 ) = (312) + q1 (211) + q2 (212) + q1 q22 (111). Moreover: q ϕq ◦ L( q 1 q 32 ) = ϕq ((123) + (213) + (312)) = (123) + (213) + (312) + (q1 + q2 )(112) + q1 (211) + q1 (122) + (q1 + q2 )(212) + (q13 + q12 q2 + q1 q22 )(111) q = Γq ( q 1 q 32 ) = (123) + (213) + (312) + (112) + q1 (211) + q1 (122) + (212) + q1 (111). Identifying these two expressions, the coefficient of (112) gives q1 + q2 = 1; the coefficient of (111) gives: q1 = q13 + q12 q2 + q2 q12 ⇐⇒ q1 (q12 + q1 q2 + q22 − 1) = 0 ⇐⇒ q1 ((q1 + q2 )2 − q1 q2 − 1) = 0 ⇐⇒ q12 q2 = 0. So (q1 , q2 ) = (1, 0) or (0, 1). Moreover: q q ϕq ((121)) = ϕq ◦ L( q 21, 3 ) = Γq ( q 21, 3 ) = (121), q 3 q 1, 3 ϕq ((212)) = ϕq ◦ L( q 1, 2 ) = Γq ( q 2 ) = (212), THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 29 and: (121)+(212) = ϕq ((121)+(212)) = ϕq ◦L( q 1, 3 q 2 ) = Γq ( q 1, 3 q 2 ) = (121)+(212)+q3(111). The coefficient of (111) gives q3 = 0. So q = (1, 0, 0) or (0, 1, 0). ⇐=. This comes from proposition 19.  4.4. Links with special posets. If T ∈ T is T0 , then all its linear extensions are permutations. Seeing FQSym as a Hopf subalgebra of (WQSym, , ∆), by restriction we obtain a Hopf algebra morphism L : HSP −→ FQSym, which is the morphism L defined in section 1.2. If T ∈ T is not T0 , then no generalized T-partition of T is a permutation, so ̟ ◦ Γq (T ) = 0. Hence, we have a commutative diagram of Hopf algebras: HT θ0 Γq // WQSym ̟  HSP L  // FQSym If T ∈ T is T0 , then a T-partition of T is a P-partition of the poset associated to T , in Stanley’s sense [16], and we obtain the commutative diagram: FQSym 99 ss s s θ(1,0,0) s ss  ss // WQSym HSP L Γ 4.5. The order on packed words. Let us precise the properties of the order on packed words of definition 17. We shall use the following notion, which, in some sense, is the generalization of the notion of ascents of permutations. Definition 22. Let f be a packed word of length n, and let i ∈ [n]. We shall say that i ∈ M (f ) if: • f (i) < max(f ). • For all j ∈ [n], f (j) = f (i) =⇒ j ≤ i. • For all j ∈ [n], f (j) = f (i) + 1 =⇒ j > i. Example. If f = (412133), then M (f ) = {3}. Remark. If f is a permutation, then i ∈ M (f ) if, and only if, f −1 (f (i) + 1) > i. The aim of this section is to prove the following theorem: Theorem 23. (1) Let f, g be two packed words. Then f ≤ g if, and only if, Std(f ) = Std(g) and M (f ) ⊆ M (g). (2) For all n ≥ 1, there is an isomorphism of posets:  G  ({packed words of length n}, ≤) −→ ({subsets of M (σ) }, ⊆) Φ: σ∈Sn  f −→ M (f ) ⊆ M (Std(f )). 30 LOÏC FOISSY AND CLAUDIA MALVENUTO Hence, for all n ≥ 1, the poset of packed words of length n is a disjoint union of posets, indexed by Sn , the part indexed by σ being isomorphic to the poset of subsets of M (σ), partially ordered by the inclusion. Lemma 24. (1) For any packed word g, g ≤ Std(g). (2) Let f, g be packed words. If f ≤ g, then Std(f ) = Std(g). Proof. We put σ = Std(f ), τ = Std(g) and, for all p ∈ [max(g)], g −1 ({p}) = Cp . 1. Let i, j ∈ [n]. We assume that i ∈ Cp and j ∈ Cq . If τ (i) ≤ τ (j), by definition of the standardization, p ≤ q, so g(i) = p ≤ q = g(j). If τ (i) = τ (j), as τ is a permutation, i = j, and g(i) = g(j). If τ (i) < τ (j) and j > i, then p ≤ q. As τ is increasing on Cp by definition of the standardization, p = q is impossible. So p < q, and g(i) = p < q = g(j). We obtain g ≤ τ . 2. As f ≤ g, f is constant on Cp for all p. We put f (Cp ) = {cp }. If i ∈ Cp and j ∈ Cq , with p < q, then g(i) = p ≤ q = g(j), so f (i) ≤ f (j): cp ≤ cq . If cp = cq and p < q, let i ∈ Cp and j ∈ Cq . If j < i, as g(i) = p < q = g(j) and f ≤ g, cp = f (i) < f (j) = cq : contradiction. Hence, if cp = cq , for all i ∈ Cp , for all j ∈ Cq , i < j, which is shortly denoted by Cp < Cq . As f is constant on Cp , σ is increasing on Cp . If p < q and cp 6= cq , then cp < cq . By definition of the standardization, for all i ∈ Cp , j ∈ Cq , σ(i) < σ(j). If p < q and cp = cq , then for all i ∈ Cp , j ∈ Cq , i < j. As f is constant on Cp ⊔ Cq , σ(i) < σ(j). Finally: • σ is increasing on Cp for all p. • If p < q, i ∈ Cp and j ∈ Cq , σ(i) < σ(j). So σ = Std(g).  Lemma 25. Let σ ∈ Sn , n ≥ 1. The following map is bijective:  {f packed word | Std(f ) = σ} −→ {I | I ⊆ M (σ)} φσ : f −→ M (f ). Proof. Let f be a packed word such that Std(f ) = σ. We put f −1 ({p}) = Cp for all p ∈ [max(f )]. Let i ∈ M (f ). Assume that i ∈ Cp . Then p < max(f ), i is the greatest element of Cp , and, if j is the smallest element of Cp+1 , i < j. By definition of the standardization, σ(j) = σ(i) + 1. As j > i, i ∈ M (σ), so M (f ) ⊆ M (σ), and φσ is well-defined. We define a map ψσ : {I | I ⊆ M (σ)} −→ {f packed word | Std(f ) = σ} in the following way. If I ⊆ M (σ), we define f (σ −1 (i)) by induction: f (σ −1 (1)) = 1, and, for all i ∈ [n − 1]: • If σ −1 (i) ∈ I or if σ −1 (i) ∈ / M (σ), then f (σ −1 (i + 1)) = f (σ −1 (i)) + 1. −1 −1 • If σ (i) ∈ M (σ) \ I, f (σ (i + 1)) = f (σ −1 (i)). Clearly, f is a packed word. Let us prove that Std(f ) = σ. For all p ∈ [max(f )], we put f −1 ({p}) = Cp . By definition of f , for all p, there exist ip ≤ jp such that Cp = σ −1 ({ip , . . . , jp }), and σ −1 (ip ), . . . , σ −1 (jp − 1) ∈ M (σ), which implies: σ −1 (ip ) < σ −1 (ip + 1) < . . . < σ −1 (jp − 1) < σ −1 (jp ). We obtain that σ −1 is increasing on ip , . . . , jp , so σ is increasing on Cp . Moreover, if p < q, i ∈ Cp and j ∈ Cq , by definition of f , putting σ −1 (i) = k and σ −i (j) = l, k < l. Consequently, σ(i) = k < l = σ(j). We obtain that Std(f ) = σ. We can put THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 31 ψσ (I) = f , and then ψσ is a well-defined map. Let I ⊆ M (σ), and f = ψσ (I). For all p ∈ [max(f )], we put f −1 ({p}) = Cp . If i ∈ M (f ), then i is the greatest element of Cp with p = f (i) < max(f ), and if j is the smallest element of Cp+1 , then i < j. As σ = Std(f ), σ(j) = σ(i) + 1, so i ∈ M (σ), and M (f ) ⊆ M (σ). By definition of f , i ∈ I or i ∈ / M (σ), so i ∈ I: M (f ) ⊆ I. Let i ∈ I. We put k = σ(i). Then f (σ −1 (k + 1)) = f (σ −1 (k)) + 1. By definition of f , i is the greatest element of Cp , with p = f (i) and j = σ −1 (k + 1) is the smallest element of Cp+1 . As σ −1 (k) = i ∈ M (σ), σ −1 (k + 1) = j > σ −1 (k) = i: i ∈ M (f ). We obtain M (f ) = I, that is to say φσ ◦ ψσ (I) = I. Let f be a packed word such that Std(f ) = σ, I = M (f ) and g = ψσ (I). For all p ∈ [max(f )], we put f −1 ({p}) = Cp . Let us prove that f (σ −1 (i)) = g(σ −1 (i)) for all i by induction. If i = 1, as σ = Std(f ), f (σ −1 (1)) = 1 = g(σ −1 (1)). Let us assume that f (σ −1 (i)) = g(σ −1 (i)). We obtain three different cases. (1) If σ −1 (i) ∈ I, then σ −1 (i) is the greatest element of Cp , with p = f (σ −1 (i)), and if j is the smallest element of Cp+1 , then i < j. As Std(f ) = σ, j = σ −1 (σ(σ −1 (i))+1) = σ −1 (i+1), and f (σ −1 (i+1)) = p+1 = f (σ −1 (i))+1 = g(σ −1 (i + 1)). (2) If σ −1 (i) ∈ / M (σ), then σ −1 (i + 1) < σ −1 (i). As σ = Std(f ), necessarily −1 σ (i) is the greatest element of Cp and σ −1 (i + 1) is the smallest element of Cp+1 . We obtain f (σ −1 (i + 1)) = p + 1 = f (σ −1 (i)) + 1 = g(σ −1 (i + 1)). (3) If σ −1 (i) ∈ M (σ) \ I, then σ −1 (i) < σ −1 (i + 1). As σ = Std(f ) and i∈ / I, σ −1 (i) and σ −1 (i + 1) are in the same Cp , so f (σ −1 (i + 1)) = p = f (σ −1 (i)) = g(σ −1 (i + 1)). As a conclusion, g = f , so ψσ ◦ φσ (f ) = f .  Proof. (theorem 23). 1. =⇒. If f ≤ g, by lemma 24, Std(f ) = Std(g). We denote by σ this permutation. If I = M (f ) and J = M (g), then f = ψσ (I) and g = ψσ (J). For all p ∈ [max(f )], we put f −1 ({p}) = Cp . For all q ∈ [max(g)], we put g −1 ({q}) = Cq′ . Let k ∈ I. We put σ(k) = i. By construction of ψσ (I), k = σ −1 (i) is the greatest letter of Cp for p = f (k), and if l = σ −1 (i + 1) is the smallest letter of Cp+1 , then k < l. Consequently, if k ′ ∈ Cp , l′ ∈ Cp+1 , then k ′ ≤ k < l ≤ l′ . If g(k ′ ) ≥ g(l′ ), as f ≤ g, we should have f (k ′ ) > f (l′ ): this is a contradiction, as f (k ′ ) = p and f (l′ ) = p + 1. So g(k ′ ) < g(l′ ). Moreover, f is constant on Cq′ for all q, as f ≤ g. If k ∈ Cq′ , then Cq′ ⊆ Cp with p = f (k). Moreover, l ∈ Cp+1 , so l ∈ / Cq′ . As ′ −1 ′ Std(g) = σ, l = σ (i + 1) ∈ Cq+1 , which implies Cq+1 ⊆ Cp+1 . So for all k ′ ∈ Cq , l′ ∈ Cq+1 , k ′ < l′ : k ∈ M (g) = J, and I ⊆ J. 1. ⇐=. We put I = M (f ), J = M (g), such that f = ψσ (I) and g = ψσ (J), with σ = Std(f ) = Std(g). • As I ⊆ J, the change of values of f in the definition of ψσ (I) are also change of values of g in the definition of ψσ (J); consequently, if g(i) = g(j), then f (i) = f (j). • If g(k) ≤ g(l), we put σ(k) = i and σ(l) = j. By construction of ψσ (J), i < j. By construction of ψσ (I), f (k) = f (σ −1 (i)) =≤ f (σ −1 (j)) = f (l). 32 LOÏC FOISSY AND CLAUDIA MALVENUTO • If g(k) < g(l) and k > l, we put σ(k) = i and σ(l) = j. Then the interval {i, . . . , j−1} contains an element which does not belong to M (σ) (otherwise, it would contain only elements of M (σ), and then k ≤ l). By definition of ψσ (I), f (k) < f (l). Finally, f ≤ g. 2. For all σ ∈ Sn , φσ is bijective: this implies that Φ is bijective. By the first point, Φ is an isomorphism of posets.  Remark. In particular, if σ is a permutation, ϕ(1,0,0) (σ) is the sum of all packed words f such that P ack(f ) = σ. This implies that the restriction of ϕ(1,0,0) to FQSym is the map ϕ defined in section 1.2. Corollary 26. For all n ≥ 1, for all σ ∈ Sn : ♯{w packed word of length n | Std(w) = σ} = 2♯M(σ) . For all n ≥ 1: ♯{packed words of length n} = X 2♯M(σ) . σ∈Sn References 1. Gérard Duchamp, Florent Hivert, and Jean-Yves Thibon, Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput. 12 (2002), no. 5, 671–717. 2. Marcel Erné and Kurt Stege, Counting finite posets and topologies, Order 8 (1991), no. 3, 247–265. 3. Loïc Foissy, Algebraic structures on double and plane posets, J. Algebraic Combin. 37 (2013), no. 1, 39–66, arXiv:1101.5231. 4. Loïc Foissy, Claudia Mavenuto, and Frédéric Patras, b∞ -algebras, their enveloping algebras, and finite spaces, arXiv:1403.7488. 5. Loïc Foissy, Frédéric Patras, and Jean-Yves Thibon, Deformations of shuffles and quasishuffles, arXiv:1311.1464. 6. Ira M. Gessel, Multipartite P -partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. 7. Jean-Louis Loday and María Ronco, On the structure of cofree Hopf algebras, J. Reine Angew. Math. 592 (2006), 123–155, arXiv:math/0405330. 8. Claudia Malvenuto, P -partitions and the plactic congruence, Graphs Combin. 9 (1993), no. 1, 63–73. 9. Claudia Malvenuto and Christophe Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967–982. , Plethysm and conjugation of quasi-symmetric functions, Discrete Mathematics 193 10. (1998), no. 3, 225–233, Selected papers in honor of Adriano Garsia (Taormina, 1994). 11. , A self paired Hopf algebra on double posets and a Littlewood-Richardson rule, J. Combin. Theory Ser. A 118 (2011), no. 4, 1322–1333, arXiv:0905.3508. 12. Rémi Maurice, A polynomial realization of the Hopf algebra of uniform block permutations, Adv. Appl. Math. 51 (2013), no. 2, 285–308. 13. J.-C. Novelli and J.-Y. Thibon, Construction of dendriform trialgebras, C. R. Acad. Sci. Paris 342 (2006), no. 6, 365–446, arXiv:math/0605061. , Polynomial realization of some trialgebras, FPSAC’06 (San Diego) (2006), 14. arXiv:math/0605061v1. 15. N. J. A Sloane, On-line encyclopedia of integer sequences, http://oeis.org/. 16. Richard P. Stanley, Ordered structures and partitions, American Mathematical Society, Providence, R.I., 1972, Memoirs of the American Mathematical Society, No. 119. 17. R. E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966), 325–340. THE HOPF ALGEBRA OF FINITE TOPOLOGIES AND T -PARTITIONS 33 18. Andrei Zelevinsky, A generalization of the Littlewood-Richardson rule and the RobinsonSchensted-Knuth correspondence, J. Algebra 69 (1981), 82–94. Fédération de Recherche Mathématique du Nord Pas de Calais FR 2956, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Université du Littoral Côte d’opale, Centre Universitaire de la Mi-Voix, 50, rue Ferdinand Buisson, CS 80699, 62228 Calais Cedex, France, email: [email protected] Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy, email: [email protected]