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This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture by George Andrews and that those generating functions satisfy…
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…
We study the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank, and, using combinatorial and analytical techniques, show that this generating function is closely connected with…
We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…
Recently, Andrews and Paule studied Schmidt type partitions using MacMahon's Partition Analysis and obtained various interesting results. In this paper, we focus on the combinatorics of Schmidt type partition theorems and characterize them…
Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's…
There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r+1. Motivated by work on rational numbers in the Calkin-Wilf tree, we classify partitions…
Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the number of partitions of $n$ with designated summands. In a recent work, Lin studied a partition function $PD_{t}(n)$ which counts the number of tagged parts over…
The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence…
In this paper, cylindric partitions into profiles $c=(1,1)$ and $c=(2,0)$ are considered. The generating functions into unrestricted cylindric partitions and cylindric partitions into distinct parts with these profiles are constructed. The…
Given an integer $n$, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in $\{1,2,\dots, n-1\}$. Starting from an abelian subring, we…
We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $\Pi^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c},…
The algebra of symmetric functions contains several interesting families of symmetric functions indexed by integer partitions or skew partitions. Given a sequence $\{u_n\}$ of symmetric functions taken from one of these families such that…
We study sequences of partitions of the unit interval into subintervals, starting from the trivial partition, in which each partition is obtained from the one before by splitting its subintervals in two, according to a given rule, and then…
Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core…
Within the replica framework we study analytically the instance space of the number partitioning problem. This classic integer programming problem consists of partitioning a sequence of N positive real numbers $\{a_1, a_2,..., a_N}$ (the…
This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…
Improving upon previous work on the subject, we use Wright's Circle Method to derive an asymptotic formula for the number of parts in all partitions of an integer that are in any given arithmetic progression.
Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…