Related papers: On spt-crank type functions

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…

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 intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical…

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…

We prove a recent conjecture of Berndt and Kim regarding the positivity of the coefficients in the asymptotic expansion of a class of partial theta functions. This generalizes results found in Ramanujan's second notebook, and recent work of…

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…

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit…

In this paper we obtain asymptotic formulas for the positive crank and rank moments for overpartitions. Moreover, we show that crank and rank moments are asymptotically equal while the difference is asymptotically positive. This indicates…

We analyze the mock modular behavior of $\bar{P}_\omega(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta…

In this paper we obtain asymptotic formulas for the Fourier coefficients of an infinite family of crank generating functions. Moreover we use this result to show that the crank obeys certain inequalities. This implies that the crank can not…

In this paper we prove a longstanding conjecture by Freeman Dyson concerning the limiting shape of the crank generating function. We fit this function in a more general family of inverse theta functions which play a key role in physics.

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…

In a recent paper, J. Lovejoy and the second author conjectured that ranks for four types of unimodal like sequences satisfy certain inequalities. In this paper, we prove these conjectures asymptotically. For this, we extend Wright's Circle…

Recently Gordon and McIntosh introduced the third order mock theta function $\xi(q)$ defined by $$ \xi(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of…

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results for…

The two partition functions $p_\omega(n)$ and $p_\nu(n)$ were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions $\omega(q)$ and $\nu(q)$, respectively. Recently, Andrews and Yee analytically…

In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a…

We obtain two-variable Hecke-Rogers identities for three universal mock theta functions. This implies that many of Ramanujan's mock theta functions, including all the third order functions, have a Hecke-Rogers-type double sum…

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous…