Higher order spt functions for overpartitions, overpartitions with smallest part even, and partitions without repeated odd parts

TL;DR

This paper explores symmetrized moments of ranks and cranks in overpartitions, deriving inequalities and higher order spt functions using Bailey pairs and elementary methods, advancing understanding of partition statistics.

Contribution

It introduces new expressions for symmetrized moments, proves inequalities between moments, and develops higher order spt functions for specific classes of overpartitions and partitions.

Findings

Crank moments for overpartitions exceed rank moments.

Derived explicit formulas for symmetrized moments using Bailey pairs.

Established higher order spt functions for various overpartition classes.

Abstract

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 moments. We then deduce inequalities between the corresponding ordinary moments. In particular we prove that the crank moment for overpartitions is always larger than the rank moment for overpartitions; with recent asymptotics this was known to hold for sufficiently large values of n for each fixed k. Lastly we provide higher order spt functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and no repeated odds.