We develop a geometric approach to the study of (s, ms − 1)-core and (s, ms + 1)-core partitions through the associated ms-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub (originally proposed by T. Amdeberhan) on the enumeration of (s, s+1) and (s, ms−1)-core partitions with distinct parts. It also enumerates the (s, ms+1)-cores with distinct parts. Furthermore, we calculate the weight of the (s, ms− 1, ms + 1)-core partition with the largest number of parts. Finally we use 2-core partitions to enumerate self-conjugate core partitions with distinct parts. The central idea is that the ms-abaci of maximal (s, ms ± 1)-cores can be built up from s-abaci of (s, s ± 1)-cores in an elegant way.

A recent refinement of the McKay conjecture is verified for the case of the alternating groups. The argument builds upon the verification of the conjecture for the symmetric groups [P. Fong, The Isaacs-Navarro conjecture for symmetric groups, J. Algebra 250 (1) (2003) 154-161].

Integer partitions which are simultaneously t-cores for distinct values of t have attracted significant interest in recent years. When s and t are relatively prime, Olsson and Stanton have determined the size of the maximal (s, t)-core κ s,t . When k ≥ 2, a conjecture of Amdeberhan on the maximal (2k − 1, 2k, 2k + 1)-core κ 2k−1,2k,2k+1 has also recently been verified by numerous authors.In this work, we analyze the relationship between maximal (2k − 1, 2k + 1)-cores and maximal (2k − 1, 2k, 2k + 1)-cores. In previous work, the first author noted that, for all k ≥ 1, | κ 2k−1,2k+1 | = 4| κ 2k−1,2k,2k+1 | and requested a combinatorial interpretation of this unexpected identity. Here, using the theory of abaci, partition dissection, and elementary results relating triangular numbers and squares, we provide such a combinatorial proof.

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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.