The number of self-conjugate core partitions

Abstract

A conjecture on the monotonicity of t-core partitions in an article of Stanton [Dennis Stanton, Open positivity conjectures for integer partitions, Trends Math. 2 (1999) 19-25] has been the catalyst for much recent research on t-core partitions. We conjecture Stantonlike 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 selfconjugate 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.

On t-Core Towers and t-Defects of Partitions

“…e.g. [1,4,13,15,16,20,21,26]), which has important implications for the representation theory of the symmetric group. Specificallly, the t-cores of size n correspond to t-defect-zero blocks of the corresponding irreducible representations of S n [19].…”

Class Numbers and Self-Conjugate 7-Cores

“…Proof of Theorem 1. If t is a positive odd integer, then the generating function for sc t (n) (see ( 2) of [8]) is ( 4)…”

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

“…A number of properties of self-conjugate core partitions have been found and proved. (See, [5,6]) Similarly, monotonicity conjecture on sc t (n) has been suggested in [11] and asymptotics are provided in [1].…”