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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 N S (m, n) denote the net number of S-partitions of n with spt-crank m. Andrews, Garvan and Liang showed that N S (m, n) is nonnegative for all integers m and positive integers n, and they asked the question of finding a combinatorial interpretation of N S (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 N S (m, n) can be interpreted as the number of doubly marked partitions of n with spt-crank m. 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.

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We use the Algorithm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan's 1 ψ 1 summation formula.

The rank of partitions play an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the D-rank and M 2 -rank of an overpartition were introduced by Lovejoy, respectively. Let N(m, n) and N2(m, n) denote the number of overpartitions of n with D-rank m and M 2 -rank m, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of N (m, n) and N2(m, n). In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for any integer m and nonnegative integer n, N2(m, n) ≤ N2(m, n + 1); and for (m, n) = (0, 4) with n = |m| + 2, we have N(m, n) ≤ N (m, n + 1). Furthermore, when m increases, we prove that N (m, n) ≥ N(m + 2, n) and N2(m, n) ≥ N2(m + 2, n) for any m, n ≥ 0, which is an analogue of Chan and Mao's result for partitions.