Combinatorial Proofs for Two-Color Partition Identities Involving Overpartitions and Minimal Excludants

Chen Dandan; Zhao Mengjie; Zou Ziyin

Combinatorial Proofs for Two-Color Partition Identities Involving Overpartitions and Minimal Excludants

Combinatorics 2026-06-27 v1

Authors: Chen Dandan , Zhao Mengjie , Zou Ziyin

Abstract

Andrews and El Bachraoui investigated two-color partitions in which even parts may occur only in blue and obtained several identities relating parity refinements of these partitions to overpartitions and minimal excludants. Some of their proofs are analytic and rely on $q$ -series manipulations. In this paper, we construct explicit weight-preserving bijections and parity-reversing involutions that provide combinatorial proofs of these identities. We also give a new combinatorial proof of related overpartition refinements involving $\overline{p}(n)$ and $\overline{p}_o(n)$ .

Keywords

two-color partitions overpartitions minimal excludants combinatorial proof bijection

Cite

@article{arxiv.2607.02556,
  title  = {Combinatorial Proofs for Two-Color Partition Identities Involving Overpartitions and Minimal Excludants},
  author = {Chen Dandan and Zhao Mengjie and Zou Ziyin},
  journal= {arXiv preprint arXiv:2607.02556},
  year   = {2026}
}