In this paper we present grammatical interpretations of the alternating Eulerian polynomials of t... more In this paper we present grammatical interpretations of the alternating Eulerian polynomials of types A and B. As applications, we derive several properties of the type B alternating Eulerian polynomials, including combinatorial expansions, recurrence relations and generating functions. We establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials of permutations in the symmetric group, which implies that the type B alternating Eulerian polynomials have gamma-vectors alternate in sign.
In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomi... more In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.
Let $\vec{r}=(r_i)_{i=1}^n$ be a sequence of real numbers of length $n$ with sum $s$. Let $s_0=0$... more Let $\vec{r}=(r_i)_{i=1}^n$ be a sequence of real numbers of length $n$ with sum $s$. Let $s_0=0$ and $s_i=r_1+\ldots +r_i$ for every $i\in\{1,2,\ldots,n\}$. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums $s_i$. Define $p(\vec{r})$ to be the number of positive sum $s_i$ among $s_1,\ldots,s_n$ and $m(\vec{r})$ to be the smallest index $i$ with $s_i=\max\limits_{0\leq k\leq n}s_k$. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper, let $\vec{r}_i=(r_i,\ldots,r_n,r_1,\ldots,r_{i-1})$ be the $i$-th cyclic permutation of $\vec{r}$. For $s>0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$; for $s\leq 0$, we giv...
In this paper we develop a general method to enumerate the congruences of finite summations $\sum... more In this paper we develop a general method to enumerate the congruences of finite summations $\sum_{k=0}^{p-1} \frac{a_k}{m^k} \!\pmod{p}$ and $\sum_{k=0}^{p-1-h} \frac{a_k a_{k+h}}{B^k} \!\pmod{p}$ for the the infinite sequence $\{a_n\}_{n\ge 0}$ with generating functions $(1+x f(x))^\frac{N}{2}$, where $f(x)$ is an integer polynomial and $N$ is an odd integer with $|N|< p$. We also enumerate the congruences of some similar finite summations involving generating functions $\frac{1-\alpha x -\sqrt{1-2(\alpha+\beta)x + Bx^2}}{\beta x}$ and $\frac{1-\alpha x-\sqrt{1-2\alpha x+(\alpha^2-4\beta)x^2}}{2\beta x^2}$.
In this paper we provide constructive proofs that the following three statistics are equidistribu... more In this paper we provide constructive proofs that the following three statistics are equidistributed: the number of ascent plateaus of Stirling permutations of order $n$, a weighted variant of the number of excedances in permutations of length $n$ and the number of blocks with even maximal elements in perfect matchings of the set $\{1,2,3,\ldots,2n\}$.
Mathematical Proceedings of the Cambridge Philosophical Society
Let χ(t) = a 0 t n – a 1 t n−1 + ⋯ + (−1) r a r t n−r be the chromatic polynomial of a graph, the... more Let χ(t) = a 0 t n – a 1 t n−1 + ⋯ + (−1) r a r t n−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is \begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*} where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney’s sign-alternating theorem, Meredith’s upper bound theorem, and Dowling and Wilson’s lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.
Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-i... more Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-inversion sequence of length $n$ is a sequence ${\bf e} = (e_1, e_2, \ldots, e_n)$ of nonnegative integers such that $0 \leq e_i < s_i$ for $1\leq i\leq n$. When $s_i=(i-1)k+1$ for any $i\geq 1$, we call the ${\bf s}$-inversion sequences the $k$-inversion sequences. In this paper, we provide a bijective proof that the ascent number over $k$-inversion sequences of length $n$ is equidistributed with a weighted variant of the ascent number of permutations of order $n$, which leads to an affirmative answer of a question of Savage (2016). A key ingredient of the proof is a bijection between $k$-inversion sequences of length $n$ and $2\times n$ arrays with particular restrictions. Moreover, we present a bijective proof of the fact that the ascent plateau number over $k$-Stirling permutations of order $n$ is equidistributed with the ascent number over $k$-inversion sequences of length $n$.
It is observed (among other things) that a theorem on bilinear and bilateral generating functions... more It is observed (among other things) that a theorem on bilinear and bilateral generating functions, which was given recently in the predecessor of this Journal, does not hold true as stated and proved earlier. Several possible remedies and generalizations, which indeed are relevant to the present investigation of various other results on bilinear and bilateral generating functions, are also considered.
The circular descent of a permutation σ is a set {σ(i) | σ(i) > σ(i + 1)}. In this paper, we focu... more The circular descent of a permutation σ is a set {σ(i) | σ(i) > σ(i + 1)}. In this paper, we focus on the enumerations of permutations by the circular descent set. Let cdes n (S) be the number of permutations of length n which have the circular descent set S. We derive the explicit formula for cdes n (S). We describe a class of generating binary trees T k with weights. We find that the number of permutations in the set CDES n (S) corresponds to the weights of T k. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.
The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a con... more The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume G is a graph and f ∶ V (G) → N is a mapping. For a nonnegative integer m, let f (m) be the extension of f to the graph G K m for which f (m) (v) = V (G) for each vertex v of K m. Let m c (G, f) be the minimum m such that G K m is not f (m)-choosable and m p (G, f) be the minimum m such that G K m is not f (m)-paintable. We study the parameter m c (K n , f) and m p (K n , f) for arbitrary mappings f. For ⃗ x = (x 1 , x 2 ,. .. , x n), an ⃗ x-dominated path ending at (a, b) is a monotonic path P of the a × b grid from (0, 0) to (a, b) such that each vertex (i, j) on P satisfies i ≤ x j+1. Let ψ(⃗ x) be the number of ⃗ x-dominated paths ending at (x n , n). By this definition, the Catalan number C n equals ψ((0, 1,. .. , n−1)). This paper proves that if G = K n has vertices v 1 , v 2 ,. .. , v n and f (v 1) ≤ f (v 2) ≤. .. ≤ f (v n), then m c (G, f) = m p (G, f) = ψ(⃗ x(f)), where ⃗ x(f) = (x 1 , x 2 ,. .. , x n) and x i = f (v i)−i for i = 1, 2,. .. , n. Therefore, if f (v i) = n, then m c (K n , f) = m p (K n , f) equals the Catalan number C n. We also show that if G = G 1 ∪ G 2 ∪. .. ∪ G p is the disjoint union of graphs G 1 , G 2 ,. .. , G p and f = f 1 ∪ f 2 ∪. .. ∪ f p , then m c (G, f) = ∏ p i=1 m c (G i , f i) and m p (G, f) = ∏ p i=1 m p (G i , f i). This generalizes a result in [
We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of e... more We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of edges of G whose deletion results in a subgraph with a unique perfect matching M , denoted by af (G, M). The anti-forcing number of a graph proposed by Vukičević and Trinajstić in Kekulé structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite graph G with a perfect matching M , we obtain a minimax result: af (G, M) equals the maximal number of M-alternating cycles of G where any two either are disjoint or intersect only at edges in M. For a hexagonal system H, we show that the maximum anti-forcing number of H equals the Fries number of H. As a consequence, we have that the Fries number of H is between the Clar number of H and twice. Further, some extremal graphs are discussed.
Let L(T, λ) = n k=0 (−1) k c k (T)λ n−k be the characteristic polynomial of its Laplacian matrix ... more Let L(T, λ) = n k=0 (−1) k c k (T)λ n−k be the characteristic polynomial of its Laplacian matrix of a tree T. This paper studied some properties of the generating function of the coefficients sequence (c 0 , • • • , c n) which are related with the matching polynomials of division tree of T. These results, in turn, are used to characterize all extremal trees having the minimum Laplacian coefficient generation function and the minimum incidence energy of trees with described maximum degree, respectively.
If a set can be partitioned into disjoint subsets which have the same cardinality, then we call t... more If a set can be partitioned into disjoint subsets which have the same cardinality, then we call the set has a uniform partition. The classic Chung-Feller theorem says that a free Dyck path has a uniform partition, and one of the subsets is a Dyck path. In this paper, from classic Chung-Feller theorem and its generalizations, we survey the research on uniform partitions for combinatorial objects.
In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote... more In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote the set of ordered preference sets of length $n$ with at least $k$ flaws and $\mathcal{S}_{n,k}=\{(x_1,...,x_{n-k})\mid x_1+x_2+... +x_{n-k}=n+k, x_i\in\mathbb{N}\}$. We obtain a bijection from the sets $\mathcal{OP}_{n,\geq k}$ to $\mathcal{S}_{n,k}$. Let $\mathcal{OP}_{n,k}$ denote the set of ordered preference sets of length $n$ with exactly $k$ flaws. An $(n,k)$-\emph{flaw path} is a lattice path starting at $(0,0)$ and ending at $(2n,0)$ with only two kinds of steps--rise step: $U=(1,1)$ and fall step: $D=(1,-1)$ lying on the line $y = -k$ and touching this line. Let $\mathcal{D}_{n,k}$ denote the set of $(n, k)$-flaw paths. Also we establish a bijection between the sets $\mathcal{OP}_{n,k}$ and $\mathcal{D}_{n,k}$. Let $op_{n,\geq k,\leq l}^m$ $(op_{n, k, =l}^m)$ denote the number of preference sets $\alpha=(a_1,...,a_n)$ with at least $k$ (exact) flaws and leading term $m$ satisf...
Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0... more Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the notation ${\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\bf r}\geq {\bf r}'$ means that $r_i\geq r'_i$ for every $i$. Let $\mathscr{R}(\Delta)$ be the set of vectors ${\bf r}$ such that ${\bf r}>0$ and ${\bf r}\Delta\geq 0$. In this paper, $(\Delta,{\bf r})$-parking functions are defined for any ${\bf r}\in\mathscr{R}(\Delta)$. It is proved that the set of $(\Delta,{\bf r})$-parking functions is independent of ${\bf r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$. For this reason, $(\Delta,{\bf r})$-parking functions are simply called $\Delta$-parking functions. It is shown that the number of $\Delta$-parking functions is less than or equal to the determinant of $\Delta$. Moreover, the definition of $(\Delta,{\bf r})$-recurrent c...
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with ... more The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be ...
In this paper we present grammatical interpretations of the alternating Eulerian polynomials of t... more In this paper we present grammatical interpretations of the alternating Eulerian polynomials of types A and B. As applications, we derive several properties of the type B alternating Eulerian polynomials, including combinatorial expansions, recurrence relations and generating functions. We establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials of permutations in the symmetric group, which implies that the type B alternating Eulerian polynomials have gamma-vectors alternate in sign.
In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomi... more In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.
Let $\vec{r}=(r_i)_{i=1}^n$ be a sequence of real numbers of length $n$ with sum $s$. Let $s_0=0$... more Let $\vec{r}=(r_i)_{i=1}^n$ be a sequence of real numbers of length $n$ with sum $s$. Let $s_0=0$ and $s_i=r_1+\ldots +r_i$ for every $i\in\{1,2,\ldots,n\}$. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums $s_i$. Define $p(\vec{r})$ to be the number of positive sum $s_i$ among $s_1,\ldots,s_n$ and $m(\vec{r})$ to be the smallest index $i$ with $s_i=\max\limits_{0\leq k\leq n}s_k$. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper, let $\vec{r}_i=(r_i,\ldots,r_n,r_1,\ldots,r_{i-1})$ be the $i$-th cyclic permutation of $\vec{r}$. For $s>0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$; for $s\leq 0$, we giv...
In this paper we develop a general method to enumerate the congruences of finite summations $\sum... more In this paper we develop a general method to enumerate the congruences of finite summations $\sum_{k=0}^{p-1} \frac{a_k}{m^k} \!\pmod{p}$ and $\sum_{k=0}^{p-1-h} \frac{a_k a_{k+h}}{B^k} \!\pmod{p}$ for the the infinite sequence $\{a_n\}_{n\ge 0}$ with generating functions $(1+x f(x))^\frac{N}{2}$, where $f(x)$ is an integer polynomial and $N$ is an odd integer with $|N|< p$. We also enumerate the congruences of some similar finite summations involving generating functions $\frac{1-\alpha x -\sqrt{1-2(\alpha+\beta)x + Bx^2}}{\beta x}$ and $\frac{1-\alpha x-\sqrt{1-2\alpha x+(\alpha^2-4\beta)x^2}}{2\beta x^2}$.
In this paper we provide constructive proofs that the following three statistics are equidistribu... more In this paper we provide constructive proofs that the following three statistics are equidistributed: the number of ascent plateaus of Stirling permutations of order $n$, a weighted variant of the number of excedances in permutations of length $n$ and the number of blocks with even maximal elements in perfect matchings of the set $\{1,2,3,\ldots,2n\}$.
Mathematical Proceedings of the Cambridge Philosophical Society
Let χ(t) = a 0 t n – a 1 t n−1 + ⋯ + (−1) r a r t n−r be the chromatic polynomial of a graph, the... more Let χ(t) = a 0 t n – a 1 t n−1 + ⋯ + (−1) r a r t n−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is \begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*} where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney’s sign-alternating theorem, Meredith’s upper bound theorem, and Dowling and Wilson’s lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.
Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-i... more Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-inversion sequence of length $n$ is a sequence ${\bf e} = (e_1, e_2, \ldots, e_n)$ of nonnegative integers such that $0 \leq e_i < s_i$ for $1\leq i\leq n$. When $s_i=(i-1)k+1$ for any $i\geq 1$, we call the ${\bf s}$-inversion sequences the $k$-inversion sequences. In this paper, we provide a bijective proof that the ascent number over $k$-inversion sequences of length $n$ is equidistributed with a weighted variant of the ascent number of permutations of order $n$, which leads to an affirmative answer of a question of Savage (2016). A key ingredient of the proof is a bijection between $k$-inversion sequences of length $n$ and $2\times n$ arrays with particular restrictions. Moreover, we present a bijective proof of the fact that the ascent plateau number over $k$-Stirling permutations of order $n$ is equidistributed with the ascent number over $k$-inversion sequences of length $n$.
It is observed (among other things) that a theorem on bilinear and bilateral generating functions... more It is observed (among other things) that a theorem on bilinear and bilateral generating functions, which was given recently in the predecessor of this Journal, does not hold true as stated and proved earlier. Several possible remedies and generalizations, which indeed are relevant to the present investigation of various other results on bilinear and bilateral generating functions, are also considered.
The circular descent of a permutation σ is a set {σ(i) | σ(i) > σ(i + 1)}. In this paper, we focu... more The circular descent of a permutation σ is a set {σ(i) | σ(i) > σ(i + 1)}. In this paper, we focus on the enumerations of permutations by the circular descent set. Let cdes n (S) be the number of permutations of length n which have the circular descent set S. We derive the explicit formula for cdes n (S). We describe a class of generating binary trees T k with weights. We find that the number of permutations in the set CDES n (S) corresponds to the weights of T k. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.
The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a con... more The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume G is a graph and f ∶ V (G) → N is a mapping. For a nonnegative integer m, let f (m) be the extension of f to the graph G K m for which f (m) (v) = V (G) for each vertex v of K m. Let m c (G, f) be the minimum m such that G K m is not f (m)-choosable and m p (G, f) be the minimum m such that G K m is not f (m)-paintable. We study the parameter m c (K n , f) and m p (K n , f) for arbitrary mappings f. For ⃗ x = (x 1 , x 2 ,. .. , x n), an ⃗ x-dominated path ending at (a, b) is a monotonic path P of the a × b grid from (0, 0) to (a, b) such that each vertex (i, j) on P satisfies i ≤ x j+1. Let ψ(⃗ x) be the number of ⃗ x-dominated paths ending at (x n , n). By this definition, the Catalan number C n equals ψ((0, 1,. .. , n−1)). This paper proves that if G = K n has vertices v 1 , v 2 ,. .. , v n and f (v 1) ≤ f (v 2) ≤. .. ≤ f (v n), then m c (G, f) = m p (G, f) = ψ(⃗ x(f)), where ⃗ x(f) = (x 1 , x 2 ,. .. , x n) and x i = f (v i)−i for i = 1, 2,. .. , n. Therefore, if f (v i) = n, then m c (K n , f) = m p (K n , f) equals the Catalan number C n. We also show that if G = G 1 ∪ G 2 ∪. .. ∪ G p is the disjoint union of graphs G 1 , G 2 ,. .. , G p and f = f 1 ∪ f 2 ∪. .. ∪ f p , then m c (G, f) = ∏ p i=1 m c (G i , f i) and m p (G, f) = ∏ p i=1 m p (G i , f i). This generalizes a result in [
We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of e... more We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of edges of G whose deletion results in a subgraph with a unique perfect matching M , denoted by af (G, M). The anti-forcing number of a graph proposed by Vukičević and Trinajstić in Kekulé structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite graph G with a perfect matching M , we obtain a minimax result: af (G, M) equals the maximal number of M-alternating cycles of G where any two either are disjoint or intersect only at edges in M. For a hexagonal system H, we show that the maximum anti-forcing number of H equals the Fries number of H. As a consequence, we have that the Fries number of H is between the Clar number of H and twice. Further, some extremal graphs are discussed.
Let L(T, λ) = n k=0 (−1) k c k (T)λ n−k be the characteristic polynomial of its Laplacian matrix ... more Let L(T, λ) = n k=0 (−1) k c k (T)λ n−k be the characteristic polynomial of its Laplacian matrix of a tree T. This paper studied some properties of the generating function of the coefficients sequence (c 0 , • • • , c n) which are related with the matching polynomials of division tree of T. These results, in turn, are used to characterize all extremal trees having the minimum Laplacian coefficient generation function and the minimum incidence energy of trees with described maximum degree, respectively.
If a set can be partitioned into disjoint subsets which have the same cardinality, then we call t... more If a set can be partitioned into disjoint subsets which have the same cardinality, then we call the set has a uniform partition. The classic Chung-Feller theorem says that a free Dyck path has a uniform partition, and one of the subsets is a Dyck path. In this paper, from classic Chung-Feller theorem and its generalizations, we survey the research on uniform partitions for combinatorial objects.
In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote... more In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote the set of ordered preference sets of length $n$ with at least $k$ flaws and $\mathcal{S}_{n,k}=\{(x_1,...,x_{n-k})\mid x_1+x_2+... +x_{n-k}=n+k, x_i\in\mathbb{N}\}$. We obtain a bijection from the sets $\mathcal{OP}_{n,\geq k}$ to $\mathcal{S}_{n,k}$. Let $\mathcal{OP}_{n,k}$ denote the set of ordered preference sets of length $n$ with exactly $k$ flaws. An $(n,k)$-\emph{flaw path} is a lattice path starting at $(0,0)$ and ending at $(2n,0)$ with only two kinds of steps--rise step: $U=(1,1)$ and fall step: $D=(1,-1)$ lying on the line $y = -k$ and touching this line. Let $\mathcal{D}_{n,k}$ denote the set of $(n, k)$-flaw paths. Also we establish a bijection between the sets $\mathcal{OP}_{n,k}$ and $\mathcal{D}_{n,k}$. Let $op_{n,\geq k,\leq l}^m$ $(op_{n, k, =l}^m)$ denote the number of preference sets $\alpha=(a_1,...,a_n)$ with at least $k$ (exact) flaws and leading term $m$ satisf...
Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0... more Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the notation ${\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\bf r}\geq {\bf r}'$ means that $r_i\geq r'_i$ for every $i$. Let $\mathscr{R}(\Delta)$ be the set of vectors ${\bf r}$ such that ${\bf r}>0$ and ${\bf r}\Delta\geq 0$. In this paper, $(\Delta,{\bf r})$-parking functions are defined for any ${\bf r}\in\mathscr{R}(\Delta)$. It is proved that the set of $(\Delta,{\bf r})$-parking functions is independent of ${\bf r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$. For this reason, $(\Delta,{\bf r})$-parking functions are simply called $\Delta$-parking functions. It is shown that the number of $\Delta$-parking functions is less than or equal to the determinant of $\Delta$. Moreover, the definition of $(\Delta,{\bf r})$-recurrent c...
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with ... more The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be ...
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Papers by Yeong-Nan Yeh