Self-conjugate [formula omitted]-core partitions and free Motzkin paths

Abstract

Simultaneous core partitions have been extensively exploited after Anderson’s work on the enumeration of (s,t)-core partitions. Ford, Mai and Sze established a bijection between self-conjugate (s,t)-core partitions and lattice paths in the ⌊s2⌋×⌊t2⌋ rectangle consisting of north and east steps, thereby showing that the number of such partitions is given by ⌊s2⌋+⌊t2⌋⌊s2⌋ for relatively prime integers s and t. In this paper, we explore self-conjugate (s,s+d,s+2d)-core partitions in the spirit of the work of Ford, Mai and Sze. We provide a lattice path interpretation of self-conjugate (s,s+d,s+2d)-core partitions in terms of free Motzkin paths and obtain the enumeration of such core partitions.