Related papers: Partitions with unique largest part and their gene…

A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.

For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After…

Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are…

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

We classify the subsets of a group by their sizes, formalize the basic methods of partitions and apply them to partition a group to subsets of prescribed sizes.

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…

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…

The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and smallest part is a rational function. This result is similar to the closely related case of…

Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate…

We give a complete classification of the unique path partitions and study congruence properties of the function which enumerates such partitions.

In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a…

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 study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part…

An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so…

We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating…

For $n$ and $k$ integers we introduce the notion of some partition of $n$ being able to generate another partition of $n$. We solve the problem of finding the minimum size partition for which the set of partitions this partition can…

The main aim of this paper is twofold: (1) Suggesting a statistical mechanical approach to the calculation of the generating function of restricted integer partition functions which count the number of partitions --- a way of writing an…

This article introduces recursive relations allowing the calculation of the number of partitions with constraints on the minimum and/or on the maximum fragment size.

In this paper, we extend the work of Andrews, Beck and Hopkins by considering partitions and compositions with bounded gaps between each pair of consecutive parts. We show that both their generating functions and two matrices determined by…

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we…