Related papers: Reciprocals of Partition Polynomials

About four centuries ago, Johann Faulhaber developed formulas for the power sum $1^n + 2^n + \cdots + m^n$ in terms of $m(m+1)/2$. The resulting polynomials are called the Faulhaber polynomials. We first give a short survey of Faulhaber's…

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…

We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to…

Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with…

In the present note we obtain new results on two conjectures by Csordas et al. regarding the interlacing property of zeros of special polynomials. These polynomials came from the Jacobi tau methods for the Sturm-Liouville eigenvalue…

Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann,…

Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical…

For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing…

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…

In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim…

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands…

We establish a recursive relation for the bipartition number $p_2(n)$ which might be regarded as an analogue of Euler's recursive relation for the partition number $p(n)$. Two proofs of the main result are proved in this article. The first…

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating…

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums…

In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn…

In this paper, we discuss an alternative approach to determine an asymptotic equivalent of the partial sum of the reciprocals of prime numbers. This well-known result, related to Merten's second theorem, is usually derived through methods…

Let $p_1,p_2,\dots,p_n, a_1,a_2,\dots,a_n \in \N$, $x_1,x_2,\dots,x_n \in \R$, and denote the $k$th periodized Bernoulli polynomial by $\B_k(x)$. We study expressions of the form \[ \sum_{h \bmod{a_k}} \ \prod_{\substack{i=1\\ i\not=k}}^{n}…

In [2], while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, the authors found that their boundary words, encoded by the Freeman chain coding on a four letters alphabet, have…

We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…