Abstract: Let $spt(n)$ denote the total number of appearances of the smallest parts in all the partitions of $n$. The spt-crank of a vector partition was introduced by Andrews, Garvan and Liang which leads to combinatorial interpretations of the congruences of $spt(n)$ mod $5$ and $7$. Andrews, Dyson and Rhoades proposed the problem of finding a definition of the spt-crank for ordinary partitions. We introduce the structure of double marked partitions and establish a bijection between ordinary partition and double marked partitions. Then we define the spt-crank of a double marked partitions, which can be used to divide the set of partitions counted by $spt(5n+4)$ (or $spt(7n+5)$) into five (or seven) equinumerous classes.
Let $N_S(m,n)$ denote the number of vector partitions of $n$ with spt-crank $m$. In the same paper, Andrews, Dyson and Rhoades conjectured that $\{N_S(m,n)\}_m$ is unimodal for any $n$ and showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. We give a combinatorial proof of this conjecture. This conjecture leads to a bijection $\tau_n$ with interesting property. Finally, using this property, we will give an upper-bound on spt(n).
