On the asymptotic distribution of cranks and ranks of cubic partitions

Abstract

Cubic partitions are a special kind of bi-partitions whose name is inspired by a connection between a cubic continued fraction and an arithmetic property of this bi-partition. There are two partition statistics, namely rank and crank, for cubic partitions, which explain cubic partition congruences combinatorially. We obtain asymptotics for the number of cubic partitions of rank (resp. crank) m which reveal the distribution of the rank (resp. crank) among cubic partitions. As applications, we derive asymptotic inequalities between cubic partition rank and crank functions, and we remark upon the similarities and the differences between these and the corresponding inequalities for ordinary partitions.

The asymptotic profile of $\chi_y$-genera of Hilbert schemes of points on K3 surfaces

“…To obtain our results we use and develop techniques from analytic number theory. See for example the closely related papers [4,5,6,7,9,14,18,19,27]. We expect that the techniques in this paper might in turn motivate and be relevant for questions in analytic number theory.…”

The asymptotic profile of an eta-theta quotient related to entanglement entropy in string theory

“…In [2], the authors introduce techniques in order to compute the bivariate asymptotic behaviour of coefficients for a Jacobi form in order to answer Dyson's conjecture on the bivariate asymptotic behaviour of the partition crank. This method is used in numerous other papers -for example, in relation to the rank of a partition [8], ranks and cranks of cubic partitions [13], and certain genera of Hilbert schemes [15] (a result that has recently been extended to a complete classification with exact formulae using the Hardy-Ramanujan circle method [9]), along with many other partition-related statistics. Using Wright's circle method [22,23] and following the same approach as [2] we show the following theorem.…”