Let $P(x)$ be a polynomial of degree $m$, with nonnegative and non-decreasing coefficients. We settle the conjecture that for any positive real number $d$, the coefficients of $P(x+d)$ form a unimodal sequence, of which the special case $d$ being a positive integer has already been asserted in a previous work. Further, we explore the location of modes of $P(x+d)$ and present some sufficient conditions on $m$ and $d$ for which $P(x+d)$ has the unique mode $\lceil{m-d\over d+1}\rceil$.
A polynomial $A(q)=\sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0\le a_1\le \cdots \le a_k\ge a_{k+1} \ge \cdots \ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= \frac{1}{[n+m]} \left[ m+n \atop n\right]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $m\le 5$ in a straightforward way.
Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive lattice L(alpha) consisting of all lower order ideals of F(alpha). These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that L(alpha) is rank unimodal. We show that if one of the…
Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required…
Recently, much attention has been given to various inequalities among partition functions. For example, Nicolas, {and later DeSavlvo--Pak,} proved that $p(n)$ is eventually log-concave, and Ji--Zang showed that the cranks are eventually unimodal. This has led to a flurry of recent activity generalizing such results in different directions. At the same time, Stanton recently made deep conjectures on the positivity of certain polynomials associated to ranks and cranks of partitions, with the ultimate goal of pointing the way to ``deeper'' structure refining ranks and cranks. These have been show…
In this article, we present a short, non-exhaustive study of an important and well-known property of combinatorial sequences - unimodality. We shall have a look at a sample of classical results on unimodality and related properties, and then proceed to understand the unimodality of the Gaussian polynomial in more detail. We will look at an outline of O'Hara's proof of the unimodality of the Gaussian polynomial. In order to grasp the challenge of the problem of obtaining an injective proof of the unimodality of the Gaussian polynomial (which is still an open question), we make several attempts …
A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of domination sets of each cardinality in $G$, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode $ \lceil \frac{n}{2}\rceil $.
Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture
一个图$G$是良好覆盖的，如果它的所有极大稳定集都具有相同的大小，记为alpha（G）（M）. D. Plummer，1970）。如果对任何$k$，多项式I（G;x）的第$k$个系数等于图$G$中基数$k$的稳定集的个数，则称它为$G$的独立多项式（Gutman and Harary，1983）. J. I.布朗，K。Dilcher和R. J. Nowakowski（2000）证明了对于任何覆盖良好的图$G$，I（G;x）是单峰的（即，存在一个索引$k$，使得从第一个到第$k$ -th的系数序列的一部分是非减的，而系数的另一部分是非增的）。T. S.迈克尔和W。N. Traves（2002）证明了这个断言对于alpha（G）< 4是正确的，而对于集合{4，5，6，7}中的alpha（G），他们提供了反例。本文证明了对任意整数$alpha$ > 7，存在一个（不）连通的良好覆盖图$G$，满足$alpha$ = alpha（G），且其独立多项式不是单峰的.此外，我们给出了α（G）< 7的图$G$具有单峰独立多项式的若干充分条件.
Let $Q_n(z)$ be the polynomials associated with the Nekrasov-Okounkov formula $$\sum_{n\geq 1} Q_n(z) q^n := \prod_{m = 1}^\infty (1 - q^m)^{-z - 1}.$$ In this paper we partially answer a conjecture of Heim and Neuhauser, which asks if $Q_n(z)$ is unimodal, or stronger, log-concave for all $n \geq 1$. Through a new recursive formula, we show that if $A_{n,k}$ is the coefficient of $z^k$ in $Q_n(z)$, then $A_{n,k}$ is log-concave in $k$ for $k \ll n^{1/6}/\log n$ and monotonically decreasing for $k \gg \sqrt{n}\log n$. We also propose a conjecture that can potentially close the gap.
The symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)={n+k\brack k}-q^{n}{n+k-2\brack k-2}$ was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k \geq 2$ and $n$ even by using the representation theory for Lie algebras. Based on Sylvester's proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality o…
Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order $n$ is unimodal with the maximum value occurring at $\lfloor\frac{n}{2}\rfloor$. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.
In algebraic, topological, and geometric combinatorics inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternat…
Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$, and let $M(m,n)$ denote the number of partitions of $n$ with crank $m$. Chan and Mao proved that for any nonnegative integers $m$ and $n$, $N(m,n)\geq N(m+2,n)$ and for any nonnegative integers $m$ and $n$ such that $n\geq12$, $n\neq m+2$, $N(m,n)\geq N(m,n-1)$. Recently, Ji and Zang showed that for $n\geq 44$ and $1\leq m\leq n-1$, $M(m-1,n)\geq M(m,n)$ and for $n\geq 14$ and $0\leq m\leq n-2$, $M(m,n)\geq M(m,n-1)$. In this paper, we analogue the result of Ji and Zang to overpartitions. Note that Bringmann, Lovejoy and Osburn…
