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 $c|\mathcal P_{X'}|$ 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.