A Recursive Relation for Bipartition Numbers

TL;DR

This paper introduces a recursive relation for bipartition numbers, analogous to Euler's relation for partition numbers, with proofs via generating functions and combinatorial objects from representation theory.

Contribution

It establishes a new recursive formula for bipartition numbers and provides two distinct proofs, one combinatorial and one algebraic, expanding understanding of bipartition enumeration.

Findings

Derived a recursive relation for bipartition numbers p_2(n)

Provided two proofs: generating function and combinatorial objects

Enhanced theoretical understanding of bipartition enumeration

Abstract

We establish a recursive relation for the bipartition number $p_2(n)$ which might be regarded as an analogue of Euler's recursive relation for the partition number $p(n)$. Two proofs of the main result are proved in this article. The first one is using the generating function, and the second one is using combinatoric objects (called ``symbols'') created by Lusztig for studying representation theory of finite classical groups.