Related papers: Flagged $(\mathcal{P},\rho)$-partitions
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…
Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals…
The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…
We study a graph partitioning problem motivated by the simulation of the physical movement of multi-body systems on an atomistic level, where the forces are calculated from a quantum mechanical description of the electrons. Several advanced…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…
Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\pi=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\pi$ is connected for…
Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…
These are extended notes for a series of lectures on Hall algebras given at the CRM Barcelona in February 2015. The basic idea of the theory of Hall algebras is that the collection of flags in an exact category encodes an associative…
In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k}…
For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…
To factorize and to decompose the graphs of representative functions on the free monoid X * (generated by the alphabet X ) with values in the ring A containing Q, we examine various products of series (as concatenation, shuffle and its…
One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After "normalization," a sum of products has a natural graphical representation, called a normal…
We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the…
For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…
We prove that with high probability $G(n,p)$ with $p \geq n^{-4/11 + o(1)}$ admits a fractional triangle decomposition (FTD), i.e., a nonnegative weighting of its triangles such that for each edge, the total weight of the triangles…
We generalize the seminal polynomial partitioning theorems of Guth and Katz to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is…
A periodic parallelogram polyomino is a parallelogram polyomino such that we glue the first and the last column. In this work we extend a bijection between ordered trees and parallelogram polyominoes in order to compute the generating…
The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f=g(h) over finite fields…
Let $p$ be a prime and let $S$ be a non-empty subset of $\mathbb{F}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$…