Beck-type identities for Euler pairs of order $r$

TL;DR

This paper extends Beck-type identities to Euler pairs of any order, providing new identities relating partition counts and parts, with analytic and bijective proofs.

Contribution

It generalizes Beck-type identities to all Euler pairs of order r, introducing many new identities and proof methods.

Findings

Extended Beck-type identities to all Euler pairs of order r.

Derived numerous new identities relating partition differences and counts.

Provided both analytic and bijective proofs for the identities.

Abstract

Partition identities are often statements asserting that the set $\mathcal P_X$ of partitions of $n$ subject to condition $X$ is equinumerous to the set $\mathcal P_Y$ of partitions of $n$ subject to condition $Y$. A Beck-type identity is a companion identity to $|\mathcal P_X|=|\mathcal P_Y|$ asserting that the difference $b(n)$ between the number of parts in all partitions in $\mathcal P_X$ and the number of parts in all partitions in $\mathcal P_Y$ equals a $c|\mathcal P_{X'}|$ and also $c|\mathcal P_{Y'}|$, where $c$ is some constant related to the original identity, and $X'$, respectively $Y'$, is a condition on partitions that is a very slight relaxation of condition $X$, respectively $Y$. A second Beck-type identity involves the difference $b'(n)$ between the total number of different parts in all partitions in $\mathcal P_X$ and the total number of different parts in all partitions in $\mathcal P_Y$. We extend these results to Beck-type identities accompanying all identities given by Euler pairs of order $r$ (for any $r\geq 2$). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.