2004/01/03 by Alexander Bérkovich, Alexander Berkovich, Berkovich, Alexander +3
Mathematics · #05A17 #05A19 #11P81 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A17 #msc:05A19 #msc:11P81 #msc:11P83
paper · pdf · doi:10.48550/arxiv.math/0401012
14 pages, 1 figure, 2 tables
arxiv created 2004/01/03 · openalex publication_date 2004/01/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a recent study of sign-balanced, labelled posets Stanley [13], introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0, 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enumerated by p[i](5n + 4) (i = 0, 2) into five equinumerous classes. In this paper we discuss two such statistics. The first one, while new, is intimately related to the Andrews-Garvan [2] crank. The second one is in terms of the 5-core crank, introduced by Garvan, Kim and Stanton [9]. Finally, we discuss some new formulas for partitions that are 5-cores.