Yeong-nan Yeh

• Research Associate at Academia Sinica

Based on a determinantal formula for the higher derivative of a quotient of two functions, we first present the determinantal expressions of Eulerian polynomials and Andre polynomials. In particular, we discover that the Euler number (number of alternating permutations) can be expressed as a lower Hessenberg determinant. We then investigate the det...

One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010), 2921--2966), we provide a curious $t$-symmetric decomposition for the generating polynomial of the joint distribution o...

In the context of Stirling polynomials, Gessel and Stanley introduced the definition of Stirling permutation, which has attracted extensive attention over the past decades. Recently, we introduced Stirling permutation code and provided numerous equidistribution results as applications. The purpose of the present work is to further analyse Stirling...

In algebraic combinatorics and formal calculation, context-free grammar is defined by a formal derivative based on a set of substitution rules. In this paper, we investigate this issue from three related viewpoints. Firstly, we introduce a differential operator method. As one of the applications, we deduce a new grammar for the Narayana polynomials...

In their study of a quartic integral, Boros and Moll introduced a special case of Jacobi polynomials, which are now known as the Boros-Moll polynomials. In this paper, we study a symmetric decomposition of Boros-Moll polynomials. We discover that both of the polynomials in the symmetric decomposition are alternatingly gamma-positive polynomials.

Motivated by the work of Visontai and Dey-Sivasubramanian on the gamma-positivity of some polynomials, we find the commutative property of a pair of Eulerian operators. As an application, we show the bi-gamma-positivity of the descent polynomials on permutations of the multiset $\{1^{a_1},2^{a_2},\ldots,n^{a_n}\}$, where $0\leqslant a_i\leqslant 2$...

The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial interpretations of the second-order Eulerian polynomials. Recently, there is a growing interest in the propert...

Assume G is a graph and S is a set of permutations of positive integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e=(u,v) of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,...,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say S is...

We first present grammatical interpretations for the alternating Eulerian polynomials of types A and B. As applications, we then derive several properties of the type B alternating Eulerian polynomials, including recurrence relations, generating function and unimodality. And then, we establish an interesting connection between alternating Eulerian...

In this paper, we introduce the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems. This generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including up-down run polynomials for symmetric groups, alternati...

Gamma-positivity appears frequently in finite geometries, combinatorics and number theory. Motivated by the recent work of Sagan and Tirrell (Adv. Math., 374 (2020), 107387), we study the relationships between gamma-positivity and alternating gamma-positivity. As applications, we derive several alternatingly gamma-positive polynomials related to Na...

The Jacobian elliptic function $\sn(u,k)$ is the inverse of the elliptic integral of the first kind and $\cn(u,k)=\sqrt{1-\sn^2(u,k)}$. In this paper, we study coefficient polynomials in the Taylor series expansions of $\sn(u,k)$ and $\cn(u,k)$. We first provide a combinatorial expansion for a family of bivariate peak polynomials, which count permu...

Inspired by the recent work of Chen and Fu on the e-positivity of trivariate second-order Eulerian polynomials, we show the e-positivity of a family of multivariate k-th order Eulerian polynomials. A relationship between the coefficients of this e-positive expansion and second-order Eulerian numbers is established. Moreover, we present a grammatica...

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 Eul...

The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or no...

In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent polynomials for hyperoctahedral group, flag ascent-plateau polynomials for Stirling permutations, up-down run polynomi...

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 po...

This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics. Properties of the enumerative polynomials for permutations, signed permutations and derangements, including generating functions and gamma-positivity are studied, which generalize and unify earlie...

In this paper, we give a type B analogue of the 1/k-Eulerian polynomials. Properties of this kind of 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 obtain the corresponding resu...

Let s = (s 1 , s 2 ,. .. , s n ,. . .) be a sequence of positive integers. An s-inversion sequence of length n is a sequence e = (e 1 , e 2 ,. .. , e n) of nonnegative integers such that 0 e i < s i for 1 i n. When s i = (i − 1)k + 1 for any i 1, we call the s-inversion sequences the k-inversion sequences. In this paper, we provide a bijective proo...

A polynomial p(z) of degree d is alternatingly increasing if and only if it can be decomposed into a sum p(z)=a(z)+zb(z), where a(z) and b(z) are symmetric and unimodal polynomials with dega(z)=d and degb(z)<d. We say that p(z) is bi-gamma-positive if a(z) and b(z) are both gamma positive. In this paper, we present a unified elementary proof of the...

In this paper, we study gamma-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present a unified grammatical proof of the gamma-positivity of Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide part...

Let [Rn,k]n,k≥0 be an array of nonnegative numbers satisfying the recurrence relation Rn,k=(a1n+a2k+a3)Rn−1,k+(b1n+b2k+b3)Rn−1,k−1+(c1n+c2k+c3)Rn−1,k−2 with R0,0=1 and Rn,k=0 unless 0≤k≤n. In this paper, we first prove that the array [Rn,k]n,k≥0 can be generated by some context-free Grammars, which gives a unified proof of many known results. Furth...

In this paper, we first consider an alternate formulation of the David-Barton identity which relates the alternating run polynomials to Eulerian polynomials. By using this alternate formulation, we see that for any γ-positive polynomial, there exists a David-Barton type identity. We then consider the joint distribution of cycle runs and cycles over...

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial a...

In this paper, several variants of the ascent-plateau statistic are introduced, including flag ascent-plateau, double ascent and descent-plateau. We first study the flag ascent-plateau statistic on Stirling permutations by using context-free grammars. We then present a unified refinement of the ascent polynomials and the ascent-plateau polynomials....

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. As a continuation of the work of Andrews and Littlejohn (Proc. Amer. Math. Soc., 137 (2009), 2581-2590), we provide a combinatorial code for Legendre-Stirling set partitions. As an application...

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (1979) and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi-pairs of differential equations. We present a general met...

The Jacobian elliptic function $\sn(u,k)$ is the inverse of the elliptic integral of the first kind and $\cn(u,k)=\sqrt{1-\sn^2(u,k)}$. In this paper, we study coefficient polynomials in the Taylor series expansions of $\sn(u,k)$ and $\cn(u,k)$. We first provide a combinatorial expansion for a family of bivariate peak polynomials, which count permu...

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreo...

In this paper, we study gamma-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present grammatical proofs of the gamma-positivity of the Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide partial g...

In this paper, several variants of the ascent-plateau statistic are introduced, including flag ascent-plateau, double ascent and descent-plateau. We first study the flag ascent-plateau statistic on Stirling permutations by using context-free grammars. We then present a unified refinement of the ascent polynomials and the ascent-plateau polynomials....

Binomial-Eulerian polynomials were introduced by Postnikov, Reiner and Williams. In this paper, properties of the binomial-Eulerian polynomials, including recurrence relations and generating functions are studied. We present three constructive proofs of the recurrence relations for binomial-Eulerian polynomials.

Let (an)n≥0 be a sequence of the Catalan-like numbers. We evaluate Hankel determinants det[λai+j+μai+j+1]0≤i,j≤n and det[λai+j+1+μai+j+2]0≤i,j≤n for arbitrary coefficients λ and μ. Our results unify many known results of Hankel determinant evaluations for classic combinatorial counting coefficients, including the Catalan, Motzkin and Schröder numbe...

In this paper we introduce the definition of marked permutations. We first present a bijection between Stirling permutations and marked permutations. We then present an involution on Stirling derangements. Furthermore, we present a symmetric bivariate enumerative polynomials on $r$-colored marked permutations. Finally, we give an explanation of $r$...

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

In this paper, we first present combinatorial proofs of a kind of expansions of the Eulerian polynomials of types A and B, and then we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles.

In this paper, we first present combinatorial proofs of a kind of expansions of the Eulerian polynomials of types A and B, and then we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles.

Let $\mathcal{L}(T,\lambda)=\sum_{k=0}^n(-1)^{k}c_{k}(T)\lambda^{n-k}$ 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 $(c_0, \cdots, c_n)$ which are related with the matching polynomials of division tree of $T$. These results, in turn...

In this paper, we introduce the definitions of simsun succession, simsun cycle succession and simsun pattern. In particular, the ordinary simsun permutations are permutations avoiding simsun pattern 321. We study the descent and peak statistics on permutations avoiding simsun successions. We give a combinatorial interpretation of the q-Eulerian pol...

In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations. Properties of the generating polynomials, including the recurrence relation, generating function and real-rootedness are studied. Moreover, we give a constructive proof of a connection between the number of simsun permutations of lengt...

In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations. Properties of the generating polynomials, including the recurrence relation, generating function and real-rootedness are studied. Moreover, we introduce and study simsun permutations of the second kind.

In this paper, let Δ be a nonsingular M-matrix. A generalization of G-parking functions, which is called Δ-parking functions, is studied. An explicit characterization for Δ-parking functions is given. It is shown that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M-matrix Δ. It is proved that t...

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the LDU decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatoria...

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...

In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. As an application, we obtain an interlacing property for the zeros of alternating Eulerian polynomials.

In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.

In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.

In this paper, a bivariate generating function \(CF(x,y) = \frac{{f(x) - yf(xy)}} {{1 - y}}\) is investigated, where f(x) = Σn⩾0f n x n is a generating function satisfying the functional equation f(x) = 1 + Σ j=1r Σ i=j−1m a ij x i f(x)j . In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths ar...

A partition of a given set is said to be uniform if all the partition classes have the same cardinality. In this paper, we will introduce the concepts of rooted -lattice path and rooted cyclic permutation and prove some fundamental theorems concerning the actions of rooted cyclic permutations on rooted lattice -paths. The main results obtained have...

In this paper, we present a generalized Toeplitz determinant solution for the generalized Schur flow and propose a mixed form of the two known relativistic Toda chains together with its generalized Toeplitz determinant solution. In addition, we also give a Hankel type determinant solution for a nonisospectral Toda lattice. All these results are obt...

In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these numbers.

In this paper we investigate the connections between Stirling permutations, cycle structures of permutations and perfect matchings. The main tool of our investigations is MY-sequences. In particular, we discover that the Eulerian polynomials have a simple combinatorial interpretation in terms of some statistics on MY-sequences.

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 e...

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 Vuki\v{c}evi\'{c} and Trinajsti\'c in Kekul\'e structures of molecular graphs is in fact the mi...

In this paper we develop a general method to the congruences of finite summations Sigma(p-1)(k=0)a(k)/m(k) (mod p) and Sigma(p-1-h)(k=0) a(k)a(k)+h/B-k (mod p) for the infinite sequence {a(n)}n >= 0 with generating functions (1+x f (x))(N/2), where f (x) is an integer polynomial and N is an odd integer with vertical bar N vertical bar < p. We also...

The determinant of the adjacentcy matrix, the algebraic structure count and the Kekulé structure count of circulenes are shown to conform to simple expressions containing Fibonacci numbers.

The Hosoya polynomial (also called Wiener polynomial) of a graph G with the vertex set V (G) is defined as H(G, x) = � {u,v}⊆V (G) xdG(u,v) on variable x, where the sum is over all unordered pairs {u, v} of distinct vertices in G, dG(u, v) is the distance of two vertices u, v in G. In 2004, Yang and Yeh evaluated H(G, x) for certain graphs of chemi...

The classical Chung-Feller theorem (2) tells us that the number of Dyck paths of length n with m ∞aws is the n-th Catalan number and independent on m. In this paper, we consider the reflnements of Dyck paths with ∞aws by four parameters, namely peak, valley, double descent and double ascent. Let pn;m;k be the number of all the Dyck paths of semi-le...

In this paper, we find a new phenomenon on chromatic polynomials of graphs. Let $\chi_G(t)=a_0t^n-a_1t^{n-1}+...(-1)^ra_rt^{n-r}$ be the chromatic polynomial of a simple graph $G$. For any $q,k\in \Bbb{Z}$ with $0\le k\le \min\{r, q+r+1\}$, we show that the partial binomial sum $\sum_{i=0}^{k}{q\choose i}a_{k-i}$ of $a_i$ is bounded above by ${m+q\...

In this paper, we consider the asymptotic behavior of the number of spanning trees and the Kirchhoff index of iterated line graphs and iterated para-line graphs (or clique-inserted graphs) of a regular graph GG. We show that the asymptotic behavior of these indices (except the Kirchhoff index of the iterated para-line graphs) is independent of the...

Let G be a connected and simple graph with vertex set {1, 2, …, n + 1} and TG(x, y) the Tutte polynomial of G. In this paper, we give combinatorial interpretations for TG(1, −1). In particular, we give the definitions of even spanning tree and left spanning tree. We prove TG(1, −1) is the number of even-left spanning trees of G. We associate a perm...

It is well known that there are three types of dimers belonging to the three different orientations in a honeycomb lattice, and in each type all dimers are mutually parallel. Based on a previous result, we can compute the partition function of the dimer problem of the plane (free boundary) honeycomb lattices with three different activities by using...

The classical Chung-Feller Theorem offers an elegant perspective for enumeratin the Catalan number cn =1/n+1(2n/n) One of the various proofs is by the uniform-partit(on)method. The method shows that the set of the free Dyck n-paths, which have (2n/n) in total, is uniformly partitioned into n + 1 blocks, and the ordinary Dyc(n)-paths form one of the...

In this note we consider unimodality problems of sequences of multinomial coefficients and symmetric functions. The results presented here generalize our early results for binomial coefficients. We also give an answer to a question of Sagan about strong q-concavity of certain sequences of symmetric functions, which can unify many known results for...

The classical Chung–Feller theorem tells us that the number of (n,m)-Dyck paths is the nth Catalan number and independent of m. In this paper, we consider refinements of (n,m)-Dyck paths by using four parameters, namely the peak, valley, double descent and double ascent. Let pn,m,k be the total number of (n,m)-Dyck paths with k peaks. First, we der...

In this paper, we focus on a "local property" of permutations: value-peak. A permutationhas a value-peak �(i) if �(i − 1) < �(i) > �(i + 1) for some i ∈ (2,n − 1). Define V P(�) as the set of value-peaks of the permutation �. For any S ⊆ (3,n), define V Pn(S) such that V P(�) = S. Let Pn = {S | V Pn(S) 6= ∅}. we make the set Pn into a poset Pn by d...

Let ⃗r = (ri) n i=1 be a sequence of real numbers of length n with sum s. Let s0 = 0 and si = r1 +... + ri for every i ∈ {1,2,...,n}. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums si. Define p(⃗r) to be the number of positive sum si among s1,...,sn and m(⃗r) to be the s...

In this paper, we give a new expression for the Tutte polynomial of a general connected graph G in terms of statistics of G-parking functions. In particular, given a G-parking function f, let cbG(f) be the number of critical-bridge vertices of f and denote wG(f)=|E(G)|−|V(G)|−∑v∈V(G)f(v). We prove that TG(x,y)=∑f∈PGxcbG(f)ywG(f), where PG is the se...

Let G be a graph with n vertices and ν(G) be the matching number of G. Let η(G) denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G)=n-2ν(G). Tan and Liu [X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Alg. Appl. 408 (2005) 212–220] proved that the nullity set of all uni...

In the vicinity of boundaries the bulk universality class of critical phenomena splits into several boundary universality classes, depending upon whether the tendency to order in the boundary is smaller or larger than in the bulk. For Ising universality class there are five different boundary universality classes: periodic, antiperiodic, free, fixe...

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 $m$. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths. Mohanty's book [5] devotes an entire section to exploring Chung-Feller theorem. Many Chung-Feller theorems are consequ...

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of t...

Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we give the definition of the bridge vertex of a $G$-parking function and obtain an expression of the Tutte polynomi...

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...

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regul...

The conceptions of $G$-parking functions and $G$-multiparking functions were introduced in [15] and [12] respectively. In this paper, let $G$ be a connected graph with vertex set $\{1,2,...,n\}$ and $m\in V(G)$. We give the definition of $(G,m)$-multiparking function. This definition unifies the conceptions of $G$-parking function and $G$-multipark...

We obtain explicit expressions of the number of close-packed dimers and entropy for three types of lattices (the so-called 8.8.6, 8.8.4, and hexagonal lattices) with cylindrical boundary condition and the entropy of the 8.8.6 lattice with toroidal boundary condition. Our results and the one on 8.8.4 and hexagonal lattices with toroidal boundary con...

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak and valley heights to avoid congruence classes modulo k. We study the shift equivalence on sequences...

In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8.

We present a new class of LYM orders, which generalizes Lih's result and is a common generalization of Griggs' result and a result of West, Harper and Daykin.

The circular descent of a permutation $\sigma$ is a set $\{\sigma(i)\mid \sigma(i)>\sigma(i+1)\}$. 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...

The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid \sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. Let $\mathcal{P}_n$ be the set of all the subset $S\subseteq [n]$ such that there exists a permutation $\sigma$ which has the circular set $S$. We can make the set $\mathcal{P}_n$ into a poset $\mathscr{P}_n$ by defining $S\preceq T$ if...

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 $\mathcal{S}_{n,k}$....

The pure-dimer problem was solved in exact closed form for many lattice graphs. Although some numerical solutions of the monomer–dimer problem were obtained, no exact solutions of the monomer–dimer problem were available (except in one dimension). Let G be an arbitrary graph with N vertices. Construct a new graph R(G) from G by adding a new verex e...

In this paper we prove that two quantities relating to the length of permutations defined on trees are independent of the structures of trees. We also find that these results are closely related to the results obtained by Graham and Pollak [R.L. Graham, H.O. Pollak, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971) 2495–2...

The “pentachains” studied in this paper are graphs formed of concatenated 5-cycles. Explicit formulas are obtained for the Schultz and modified Schultz indices of these graphs, as well as for generalizations of these indices. In the process we give a more refined version of the procedure that earlier was reported for the ordinary Wiener index.

The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q-analogue of this index, termed the Wiener polynomial by Hosoya but also known...

A triangle {a(n,k)}0⩽k⩽n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n...

Let R and F be two disjoint edge sets in an n-dimensional hypercube Q n . We give two constructing methods to build a Hamiltonian cycle or path that includes all the edges of R but excludes all of F. Besides, considering every vertex of Q n incident to at most n−2 edges of F, we show that a Hamiltonian cycle exists if (A)|R|+2|F|≤2n−3 when |R|≥2,...

The sum of distances between all vertices pairs in a connected graph is known as the Wiener Index. It is the earliest of the indices that correlates well with many physico- chemical properties of organic compounds and as such has been well-studied over the last quarter of a century. A q-analogue of this index, termed the Wiener Polynomial by Hosoya...