Beck-type identities: new combinatorial proofs and a theorem for parts congruent to $t$ mod $r$

Abstract

Let ${\mathcal{O}}_r(n)$ be the set of $r$-regular partitions of $n$, ${\mathcal{D}}_r(n)$ the set of partitions of $n$ with parts repeated at most $r-1$ times, ${\mathcal{O}}_{1,r}(n)$ the set of partitions with exactly one part (possibly repeated) divisible by $r$, and let ${\mathcal{D}}_{1,r}(n)$ be the set of partitions in which exactly one part appears at least $r$ times. If $E_{r,t}(n)$ is the excess in the number of parts congruent to $t(\mathrm{mod}r)$ in all partitions in ${\mathcal{O}}_r(n)$ over the number of different parts appearing at least $t$ times in all partitions in ${\mathcal{D}}_r(n)$, then $E_{r,t}(n)=|{\mathcal{O}}_{1,r}(n)|=|{\mathcal{D}}_{1,r}(n)|$. We prove this analytically and combinatorially using a bijection due to Xiong and Keith. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in ${\mathcal{O}}_r(n)$ over the number of parts in all partitions in ${\mathcal{D}}_r(n)$ equals $(r-1)|{\mathcal{O}}_{1,r}(n)|$ and also $(r-1)|{\mathcal{D}}_{1,r}(n)|$. Our work provides a new combinatorial proof of this result that does not use Glaisher's bijection. We also give a new combinatorial proof based of the Xiong-Keith bijection for a second Beck-Type identity that has been proved previously using Glaisher's bijection.

12 pages

References in corpus (1)

• Generalizing a partition theorem of Andrews