Related papers: Towards Heim and Neuhauser's Unimodality Conjectur…

In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If $f(z) = \sum_n a_nz^n$ is an infinite series with $a_n \geq 1$ and $a_0 + \cdots + a_n = O(n + 1)$ for all $n$, we prove that a…

We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…

Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the…

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…

We prove strict unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of S_n representations.

Let $P_{n,k}$ be the number of permutations $\pi$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $\pi$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define…

Let $A(\ell,n,k)$ denote the number of $\ell$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We provide a new proof of an explicit formula for $A(\ell,n,k)$…

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…

Pak and Panova recently proved that the $q$-binomial coefficient ${m+n \choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n \geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of…

For univalent and normalized functions $f$ the logarithmic coefficients $\gamma_n(f)$ are determined by the formula $\log(f(z)/z)=\sum_{n=1}^{\infty}2\gamma_n(f)z^n$. In the paper \cite{Pon} the authors posed the conjecture that a locally…

In his study of Ramanujan-Sato type series for $1/\pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials…

The Nekrasov--Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The…

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and…

Bergeron--Ceballos--K\"ustner introduced the $q$-Fibonomial coefficients \( \qfibonom{m+n}{n}\), gave a combinatorial interpretation of the $q$-Fibonomial coefficients via a weighted path-domino tiling model, and conjectured that these…

Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we…

The classical Eulerian Numbers $A_{n,k}$ are known to be log-concave. Let $P_{n,k}$ and $Q_{n,k}$ be the number of even and odd permutations with $k$ excedances. In this paper, we show that $P_{n,k}$ and $Q_{n,k}$ are log-concave. For this,…

We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers.

Let q be a prime power and N_n count degree-n monic irreducible polynomials in F_q[x]. We show that prod_{n=1}^{\infty} (1 -z^n)^N_n = 1 - qz for small |z|.

Let $m$ and $k \geq 2$ be positive integers. We show that polynomial $P = (1+x)^m(1+x^k)$ is strongly unimodal (frequently known as {\it log concave\/}) if and only if $m \geq k^2 -3$; this is also the criterion for $P$ to be merely…

Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the…