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 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.