On colored set partitions of type BnB_n

Generalizing Reiner's notion of set partitions of type $B_n$, we define colored $B_n$-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored $B_n$-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored $B_n$-partition. We find an asymptotic expression of the total number of colored $B_n$-partitions up to an error of $O(n^{-1/2}\log^{7/2}{n})$, and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored $B_n$-partitions.Comment: 10 page

The Real-Rootedness and Log-concavities of Coordinator Polynomials of Weyl Group Lattices

It is well-known that the coordinator polynomials of the classical root lattice of type $A_n$ and those of type $C_n$ are real-rooted. They can be obtained, either by the Aissen-Schoenberg-Whitney theorem, or from their recurrence relations. In this paper, we develop a trigonometric substitution approach which can be used to establish the real-rootedness of coordinator polynomials of type $D_n$. We also find the coordinator polynomials of type $B_n$ are not real-rooted in general. As a conclusion, we obtain that all coordinator polynomials of Weyl group lattices are log-concave.Comment: 8 page

Root geometry of polynomial sequences III: Type (1,1)(1,1) with positive coefficients

In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the overall real-rootedness of all the polynomials, in terms of the polynomial coefficients of the recurrence. Moreover, in the real-rooted case, we find the set of limits of zeros, which turns out to be the union of a closed interval and one or two isolated points; when non-real-rooted polynomial exists, we present a sufficient condition under which every polynomial with $n$ large has a real zero.Comment: 14page, 3 figure

Surface embedding of non-bipartite kk-extendable graphs

We find the minimum number $k=\mu'(\Sigma)$ for any surface $\Sigma$, such that every $\Sigma$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that they are $3$-extendable. This confirms the existence of an infinite number of $3$-extendable non-bipartite graphs which can be embedded in the Klein bottle.Comment: 17 pages, 5 figure

A Note on {k,n−k}\{k,n-k\}-Factors of Regular Graphs

Let $r$ be an odd integer, and $k$ an even integer. In this note, we present $r$-regular graphs which have no $\{k,r-k\}$-factors for all $1\le k\le {r\over2}-1$. This gives a negative answer to a problem posed by Akbari and Kano recently

The maximum number of perfect matchings of semi-regular graphs

Let $n\ge 34$ be an even integer, and $D_n=2\lceil n/4 \rceil-1$. In this paper, we prove that every $\{D_n,\,D_n+1\}$-graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings. This result is sharp in the sense that (i) there exists a $\{D_n,\,D_n+1\}$-graph containing exactly $\lceil n/4 \rceil$ disjoint perfect matchings, and that (ii) there exists a $\{D_n-1,\,D_n\}$-graph without perfect matchings for each $n$. As a consequence, for any integer $D\ge D_n$, every $\{D,\,D+1\}$-graph of order $n$ contains $\lceil (D+1)/2 \rceil$ disjoint perfect matchings. This extends Csaba et~al.'s breathe-taking result that every $D$-regular graph of sufficiently large order is $1$-factorizable, generalizes Zhang and Zhu's result that every $D_n$-regular graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings, and improves Hou's result that for all $k\ge n/2$, every $\{k,\,k+1\}$-graph of order $n$ contains $(\lfloor n/3\rfloor+1+k-n/2)$ disjoint perfect matchings.Comment: 30 pages, 9 figure

Piecewise interlacing zeros of polynomials

We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly speaking, one constructs a collection of disjoint intervals such that one may show by induction that consecutive polynomials have interlacing zeros over each of the intervals. We confirm the real-rootedness of some polynomials satisfying a recurrence with linear polynomial coefficients. This extends Gross et al.'s work where one of the polynomial coefficients is a constant.Comment: 18 pages, 6 figure

The Tutte's condition in terms of graph factors

Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$. We obtain that $G$ satisfies the Tutte's condition $$o(G-S)\le \sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$},$$ with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\colon V(G)\to 2^\N$ such that $H(v)\in \{J_f(v),\,J_f^+(v)\}$ for each $v\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a characterization for graphs of odd order satisfying the Tutte's condition with respect to a function.Comment: 5 page

Common zeros of polynomials satisfying a recurrence of order two

We give a characterization of common zeros of a sequence of univariate polynomials $W_n(z)$ defined by a recurrence of order two with polynomial coefficients, and with $W_0(z)=1$. Real common zeros for such polynomials with real coefficients are studied further. This paper contributes to the study of root distribution of recursive polynomial sequences.Comment: 12 page

Counting subwords in flattened permutations

In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on $\mathcal{S}_n$ in the flattened sense. For the other four patterns in question, we develop a unified approach to obtain explicit formulas for the comparable distributions. We find that the formulas so obtained for 123- and 321-subwords can be expressed in terms of the Chebyshev polynomials of the second kind, while those for peaks and valleys are more related to the Eulerian polynomials. We also provide a bijection showing the equidistribution of descents in flattened permutations of a given length with big descents in permutations of the same length in the usual sense.Comment: 21 page