The spt-Crank for Ordinary Partitions

William Y.C. Chen, Kathy Q. Ji and Wenston J.T. Zang

  Abstract:   The spt-function spt(n) was introduced by Andrews as the weighted counting of partitions of n with respect to the number of occurrences of the smallest part. Andrews, Garvan and Liang defined the spt-crank of an S-partition which leads to combinatorial interpretations of the congruences of spt(n) mod 5 and 7. Let NS (m,n) denote the net number of S-partitions of n with spt-crank m. Andrews, Garvan and Liang showed that NS (m,n) is nonnegative for all integers mand positive integers n, and they asked the question of finding a combinatorial interpretation of NS (m,n). In this paper, we introduce the structure of doubly marked partitions and define the spt-crank of a doubly marked partition. We show that NS (m,n) can be interpreted as the number of doubly marked partitions of n with spt-crankm. Moreover, we establish a bijection between marked partitions of n and doubly marked partitions of n. A marked partition is defined by Andrews, Dyson and Rhoades as a partition with exactly one of the smallest parts marked. They consider it a challenge to find a definition of the spt-crank of a marked partition so that the set of marked partitions of 5n+ 4 and 7n+ 5 can be divided into five and seven equinumerous classes. The definition of spt-crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions leads to a solution to the problem of Andrews, Dyson and Rhoades.

  AMS Classification:  05A17, 05A19, 11P81, 11P83.

  Keywords:   spt-function, spt-crank, congruence, marked partition, doubly marked partition.

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