Periodic P-Partitions

TL;DR

This paper introduces periodic (P, ω)-partitions, proves they satisfy a matrix difference equation, and shows they can be enumerated with linear recurrences, providing insights into their asymptotic growth.

Contribution

It generalizes previous recurrence results to a broader class of partitions called periodic (P, ω)-partitions, with new enumeration and asymptotic analysis.

Findings

Periodic (P, ω)-partitions satisfy a homogeneous first-order matrix difference equation.

Enumeration of these partitions can be achieved through constant coefficient linear recurrences.

The paper establishes the asymptotic growth rate of the number of such partitions.

Abstract

In this paper, we introduce a class of $(P, \omega)$-partitions that we call periodic $(P, \omega)$-partitions, then prove that such $(P, \omega)$-partitions satisfy a homogeneous first-order matrix difference equation. After defining an appropriate counting problem for the above $(P, \omega)$-partitions, we show that as a consequence of this equation, periodic $(P, \omega)$-partitions can be enumerated with constant coefficient linear recurrence relations. By analysing the above matrix difference equation, we also prove a result for the asymptotic growth rate for the number of periodic $(P, \omega)$-partitions. The results of this paper generalizes and strengthens the constant coefficient linear recurrence results proved by Sun and by L\'opez, Mart\'inez, P\'erez, P\'erez, and Basova for enumerating standard Young tableaux on shifted strips with constant width.