A bijection between self-conjugate and ordinary partitions and counting simultaneous cores as its application

TL;DR

This paper establishes a bijection between self-conjugate and ordinary partitions, revealing new combinatorial interpretations for Catalan and Motzkin numbers through self-conjugate simultaneous core partitions.

Contribution

It introduces a novel bijection linking self-conjugate and ordinary partitions and connects hook lengths to these partitions, providing fresh combinatorial insights.

Findings

Bijection between self-conjugate and ordinary partitions

New interpretations of Catalan and Motzkin numbers

Relation between hook lengths and partitions

Abstract

We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.