The number of self-conjugate core partitions

TL;DR

This paper investigates the properties and counts of self-conjugate t-core partitions, proposing conjectures, partial proofs, and applications in group theory, with implications for asymptotic and combinatorial analysis.

Contribution

It introduces formulas for counting self-conjugate t-core partitions and explores monotonicity conjectures, extending understanding in partition theory and representation theory.

Findings

Formulas for counting self-conjugate t-core partitions

Partial results supporting monotonicity conjectures

Applications to symmetric and alternating group block theory

Abstract

A conjecture on the monotonicity of t-core partitions in an article of Stanton [Open positivity conjectures for integer partitions, Trends Math., 2:19-25, 1999] has been the catalyst for much recent research on t-core partitions. We conjecture Stanton-like monotonicity results comparing self-conjugate (t+2)- and t-core partitions of n. We obtain partial results toward these conjectures for values of t that are large with respect to n, and an application to the block theory of the symmetric and alternating groups. To this end we prove formulas for the number of self-conjugate t-core partitions of n as a function of the number of self-conjugate partitions of smaller n. Additionally, we discuss the positivity of self-conjugate 6-core partitions and introduce areas for future research in representation theory, asymptotic analysis, unimodality, and numerical identities and inequalities.