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

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 $c|{\mathcal{P}}_{X^{\prime }}|$ and also $c|{\mathcal{P}}_{Y^{\prime }}|$, where $c$ is some constant related to the original identity, and $X^{\prime }$, respectively $Y^{\prime }$, is a condition on partitions that is a very slight relaxation of condition $X$, respectively $Y$. A second Beck-type identity involves the difference $b^{\prime }(n)$ 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 $r\ge 2$). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.

18 pages