Higher Order SPT-Functions

TL;DR

This paper explores higher order spt-functions, providing a combinatorial interpretation via Bailey pairs, and establishes new inequalities and congruences for these functions in partition theory.

Contribution

It introduces a combinatorial interpretation of higher order spt-functions using Bailey pairs and derives new inequalities and congruences for these functions.

Findings

Established inequalities between crank and rank moments for all n

Derived several new congruences for higher order spt-functions

Provided a combinatorial interpretation involving weighted sums of partitions

Abstract

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 symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only know previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.