Combinatorics of Integer Partitions With Prescribed Perimeter

TL;DR

This paper explores the combinatorial properties of integer partitions with fixed perimeter, revealing distributional symmetries and inequalities related to partition statistics, with proofs provided through analytical and combinatorial methods.

Contribution

It refines existing theorems on partition distributions and generalizes statistics to establish new distribution inequalities for partitions with fixed perimeter.

Findings

Number of even parts and repeated parts are equally distributed over fixed-perimeter partitions.

Generalization of partition statistics to part-difference and residue classes.

Analytical and combinatorial proofs of distribution symmetries and inequalities.

Abstract

We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.