Simultaneous core multipartitions

TL;DR

This paper introduces the concept of simultaneous core multipartitions, generalizes existing theory, and provides conditions for finiteness and enumeration results for specific cases, advancing combinatorial partition theory.

Contribution

It defines (s|c)-core multipartitions, establishes finiteness criteria, and offers exact enumeration for certain cases, extending the study of core partitions.

Findings

Necessary and sufficient conditions for finiteness of simultaneous core multipartitions.

Exact enumeration results for special cases of simultaneous core bipartitions.

Generalization of core partition theory to multipartitions.

Abstract

We initiate the study of simultaneous core multipartitions, generalising simultaneous core partitions, which have been studied extensively in the recent literature. Given a multipartition datum (s|c), which consists of a non-negative integer s and an l-tuple c of integers, we introduce the notion of an (s|c)-core multipartition. Given an arbitrary set of multicore data, we give necessary and sufficient conditions for the corresponding set of simultaneous core multipartitions to be finite. We then study the special case of simultaneous core bipartitions, giving exact enumerative results in some special subcases.