Related papers: The spt-crank for overpartitions
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…
By considering the $M_2$-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences $\overline{\mbox{spt}}_2(3n)\equiv \overline{\mbox{spt}}_2(3n+1)\equiv 0\pmod{3}$. Here…
We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give…
We continue to investigate spt-type functions that arise from Bailey pairs. In this third paper on the subject, we proceed to introduce additional spt-type functions. We prove simple Ramanujan type congruences for these functions which can…
Let $N(\leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(\leq m,n)\leq…
We introduce several spt-type functions that arise from Bailey pairs. We prove simple Ramanujan type congruences for these functions which can be explained by a spt-crank-type function. The spt-crank-type functions are constructed by adding…
We consider the symmetrized moments of three ranks and cranks, similar to the work of Garvan for the rank and crank of a partition. By using Bailey pairs and elementary rearrangements, we are able to find useful expressions for these…
We investigate spt-crank-type functions arising from Bailey pairs. We recall four spt-type functions corresponding to the Bailey pairs $A1$, $A3$, $A5$, and $A7$ of Slater and given four new spt-type functions corresponding to Bailey pairs…
Dyson famously provided combinatorial explanations for Ramanujan's partition congruences modulo $5$ and $7$ via his rank function, and postulated that an invariant explaining all of Ramanujan's congruences modulo $5$, $7$, and $11$ should…
In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function $\operatorname{spt}_\omega(n)$, which is closely related to Ramanujan's third order mock theta function $\omega(q)$. Garvan and Jennings-Shaffer introduce a crank…
In 2015, Bringmann, Lovejoy and Mahlburg considered certain kinds overpartitions, which can been seen as the overpartition analogue of Schur's partition. The motivation of their work is that the difference between the generating function of…
This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their…
We construct new smallest parts partition functions and smallest parts crank functions by considering variations of Bailey's Lemma and conjugate Bailey pairs. The functions we introduce satisfy simple linear congruences modulo $3$ and $5$.…
Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of…
Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We…
In 1919, Ramanujan discovered his famous congruences for the partition function. Not too long after, Freeman Dyson conjectured a combinatorial statistic existed that explained the three congruences, which he dubbed the \textit{crank}. A…
Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the number of partitions of $n$ with designated summands. A bipartition of $n$ is an ordered pair of partitions $(\pi_1, \pi_2)$ with the sum of all of the parts being…
Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our methods are elementary relying entirely on the three classical…
We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function $\bar{p}(n)$ and two smallest parts functions: $\bar{\operatorname{spt1}}(n)$ for overpartitions and $\operatorname{M2spt}(n)$ for partitions…
Andrews' spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order…