Lili Mu

• Jiangsu Normal University

The edge cover polynomial of a graph G is the function E(G,x)=∑i≥1e(G,i)xi, where e(G,i) is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph Kn, based on which we establish that the average edge cover polynomial of order...

Let R=R(d(t),h(t)) be a Riordan array, where d(t)=∑n≥0dntn and h(t)=∑n≥0hntn. We show that if the matrix[d0h000⋯d1h1h00d2h2h1h0⋮⋮⋱] is totally positive, then so is the Riordan array R.

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\ d_2 & h_2 & h_1 & h_0 &\\ \vdots&\vdots&&&\ddots \end{array}\right] \end{equation*} is totally positive, then...

We introduce the notion of order-chain polytopes, which generalizes both order polytopes and chain polytopes arising from finite partially ordered sets. Since in general order-chain polytopes cannot be integral, the problem when order-chain polytopes are integral will be studied. Furthermore, we discuss the question whether every integral order-cha...

Let be the generating functions of Catalan-like numbers . We show that the corresponding Hankel determinants satisfy a three-term recurrence relation. As applications, we express the Hankel determinants of some classical combinatorial counting coefficients, including the Catalan numbers, the central binomial coefficients, the large and little Schrö...

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...

Let R=[rn,k]n,k≥0 be a Riordan array. Define the row polynomials Rn(q)=∑k=0ⁿrn,kqk and the row polynomial matrix R(q)=[rn,k(q)]n,k≥0 by rn,k(q)=∑j=kⁿrn,jqj−k. Then R(q) is also a Riordan array with the Rn(q) located on the leftmost column of R(q). In this paper we investigate combinatorial properties of the matrix R(q) and the sequence (Rn(q))n≥0,...

Define an infinite lower triangular matrix D(e, h) = [dn,k]n,k≥0 by the recurrence d0,0 = d1,0 = d1,1 = 1, dn,k = dn−1,k−1 + edn−1,k + hdn−2,k−1, where e, h are both nonnegative and dn,k = 0 unless n ≥ k ≥ 0. We call D(e, h) the Delannoy-like triangle. The entries dn,k count lattice paths from (0, 0) to (n − k, k) using the steps (0, 1), (1, 0) and...

Let $(a_0,a_1,a_2,\ldots)$ be the sequence of Catalan-like numbers. We evaluate the Hankel determinants of the shifted sequence $(0,a_0,a_1,a_2,\ldots)$. As an application, we settle Barry's three conjectures about Hankel determinant evaluations of certain sequences in a unified approach. We also provide some Somos sequences by means of Hankel dete...

Let $[n]=\{1,2,\ldots,n\}$ and $\mathscr{B}_n=\{A: A\subseteq [n]\}$. A family $\mathscr{A}\subseteq \mathscr{B}_n$ is a Sperner family if $A\nsubseteq B$ and $B\nsubseteq A$ for distinct $A,B\in\mathscr{A}$. Sperner's theorem states that the density of the largest Sperner family in $\mathscr{B}_n$ is $\binom{n}{\left\lceil{n/2}\right\rceil}/2^n$....

Let [n] = {1,2…,n} and Bn = {A : A ⊆ [n]}. A family A ⊆ Bn is a Sperner family if A ⊈ B and B ⊈ A for distinct A;B ϵ A. Sperner’s theorem states that the density of the largest Sperner family in Bn is (Formula presented). The objective of this note is to show that the same holds if Bn is replaced by compressed ideals over [n].

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schr\"oder numbers, a...

Let $G$ be a simple graph on the vertex set $\{v_1,\dots,v_n\}$ with edge set $E$. Let $K$ be a field. The graphical arrangement $\mathcal{A}_G$ in $K^n$ is the arrangement $x_i-x_j=0, v_iv_j \in E$. An arrangement $\mathcal{A}$ is supersolvable if the intersection lattice $L(c(\mathcal{A}))$ of the cone $c(\mathcal{A})$ contains a maximal chain of...

A family P of subsets of the set {1, 2,. .. , n} is called a Sperner family if A B for distinct A, B ∈ P, and a convex family if A, B ∈ P and A ⊆ C ⊆ B imply that C ∈ P. It is conjectured that given a convex family P one can find a Sperner family A in P satisfying the inequality |A |/|P| ≥ n n/2 /2 n , which is a generalization of a classical resul...