Asymptotic inequalities for positive crank and rank moments

Abstract

Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases, and our main result states that while the two families of moment functions are asymptotically equal, the crank moments are always asymptotically larger than the rank moments.Andrews, Chan, and Kim primarily focused on one case, and proved the stronger result that the first crank moment is strictly larger than the first rank moment for all partitions by showing that the difference is equal to a combinatorial statistic on partitions that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function, and we further determine its behavior modulo 2 by relating its parity to Andrews spt-function.

The number of parts in certain residue classes of integer partitions

“…In this case, we employ a different variant of the Circle Method due to Wright [15], which has essentially been rediscovered and used in various contexts by K. Bringmann, K. Mahlburg, and their collaborators, see e.g. [2,3]. We use a very convenient formulation due to Ngo and Rhoades [13].…”

Asymptotics of Higher Order Ospt-Functions for Overpartitions

“…From [6] we know that S r (q) obeys the above expression. For S r (q), we proceed as follows: we perform a Mittag-Leffler expansion on the two functions …”

“…The same argument of [6] can be used to show that S r (q) obeys the same bound as S r away from the dominant pole and therefore SR r (q) obeys the same bound as SC r (q). We see that this error term can be absorbed into the error of the asymptotics near the essential singularity q = 1.…”

Integer partitions, probabilities and quantum modular forms

“…Notable in those are the works dealing with runs and gaps in parts making up a partition [17,18,20,22,48,51,57,78,79]. This paper demonstrates how quantum modular forms and their asymptotic expansions, which involve values of modular L-functions, arise in the resolution of natural integer partition probability problems.…”

“…See the book of Stanley [75] for discussion of sieving in enumerative combinatorics. See the papers of Andrews [4,5], Bringmann and Mahlburg [22], Kim and Lovejoy [54], Kim and Jo [52], Kim et al [55,56], and Grabner et al [48] for more examples of this phenomenon. Most commonly, the "sieving function" is a partial theta function, which is known to be a quantum modular form.…”