An inequality between finite analogues of rank and crank moments

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan’s conjecture. In this paper, we give a proof of this conjecture.

Arithmetic properties of l-regular overpartitions

Recently, Andrews introduced the partition function [Formula: see text] as the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we call the overpartitions enumerated by the function [Formula: see text] [Formula: see text]-regular overpartitions. For [Formula: see text] and [Formula: see text], we obtain some explicit results on the generating function dissections. We also derive some congruences for [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] which imply the congruences for [Formula: see text] proved by Andrews. By introducing a rank of vector partitions, we give a combinatorial interpretation of the congruences of Andrews for [Formula: see text] and [Formula: see text].

RAMANUJAN’S SPT-CRANK FOR MARKED OVERPARTITIONS

In 1916, Ramanujan’s showed the spt-crank for marked overpartitions. The corresponding special functions , and are found in Ramanujan’s notebooks, part 111. In 2009, Bingmann, Lovejoy and Osburn defined the generating functions for , and . In 2012, Andrews, Garvan, and Liang defined the in terms of partition pairs. In this article the number of smallest parts in the overpartitions of n with smallest part not overlined, not overlined and odd, not overlined and even are discussed, and the vector partitions and - partitions with 4 components, each a partition with certain restrictions are also discussed. The generating functions , , , , are shown with the corresponding results in terms of modulo 3, where the generating functions , are collected from Ramanujan’s notebooks, part 111. This paper shows how to prove the Theorem 1 in terms of ,Theorem 2 in terms of and Theorem 3 in terms of respectively with the numerical examples, and shows how to prove the Theorems 4,5 and 6 with the help of in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer are able to defined the for marked overpartitions. This paper also shows another results with the help of 6 -partition pairs of 3, help of 20 -partition pairs of 5 and help of 15 -partition pairs of 8 respectively.