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The spt-Crank for Ordinary Partitions
The spt-function was introduced by Andrews as the weighted counting
of partitions of with respect to the number of occurrences of the smallest
part. Andrews, Garvan and Liang defined the spt-crank of an -partition which
leads to combinatorial interpretations of the congruences of mod 5 and
7. Let denote the net number of -partitions of with spt-crank
. Andrews, Garvan and Liang showed that is nonnegative for all
integers and positive integers , and they asked the question of finding
a combinatorial interpretation of . In this paper, we introduce the
structure of doubly marked partitions and define the spt-crank of a doubly
marked partition. We show that can be interpreted as the number of
doubly marked partitions of with spt-crank . Moreover, we establish a
bijection between marked partitions of and doubly marked partitions of .
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 and 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.Comment: 22 pages, 6 figure
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