Related papers: A new order on integer partitions

We propose a $q$-analog of classical plethystic conjectures due to Foulkes. In our conjectures, a divided difference of plethysms of Hall-Littlewood polynomials $H_n(\mathbf{x};q)$ replaces the analogous difference of plethysms of complete…

Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these…

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…

We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a…

We investigate the set of partial partitions of a finite set, ordered by inclusion. With this ordering the set of partial partitions can be studied as an abstract simplicial complex. We use the theory of shellable nonpure complexes to find…

A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in "Symmetric polynomials and symmetric mean inequalities". Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a…

We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…

We present Euler-type recurrence relations for some partition functions. Some of our results provide new recurrences for the number of unrestricted partitions of $n$, denote by $p(n)$. Others establish recurrences for partition functions…

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…

In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$…

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a…

In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture.

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 determine the partitions $\lambda$ for which the corresponding induced module (or Schur module in the language of Buchsbaum et. al., [1]) $\nabla(\lambda)$ is injective in the category of polynomial modules for a general linear group…

Plethysm of two Schur functions can be expressed as a linear combination of Schur functions, and monomial symmetric functions. In this paper, we express the coefficients combinatorially in the case of monomial symmetric functions. And by…

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on…

Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions $s_\nu$ and $s_\mu$. In this article we define an explicit polynomial representation corresponding to $s_\nu \circ s_\mu$ with basis indexed by certain…

We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kur\c{s}ung\"oz. We also present some related polynomial and $q$-series…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we…