A bijection proof of Andrews-Merca integer partition theorem

TL;DR

This paper provides a bijective proof of Andrews and Merca's partition theorem, extending their results to all moduli and introducing new partition statistics with equidistribution properties.

Contribution

It introduces a novel bijection on partitions that generalizes and extends previous generating function proofs to all moduli, revealing new equidistribution results.

Findings

Established a bijection that proves Andrews-Merca theorem.

Extended results to all integer moduli m ≥ 2.

Identified six partition statistics with equidistribution properties.

Abstract

Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on generating functions. Motivated by Andrews and Merca's results, we define six statistics related to the partitions of $n$ and show that the two triples of the six statistics are equidistributed. From this equidistributed result, we derive modulo $m$ extensions of Andrews and Merca's results for all integers $m\ge 2$. The proof of the main result is based on a general bijection on the set of partitions of $n$.