TY - JOUR

T1 - Refinements of Beck-type partition identities

AU - Amdeberhan, T.

AU - Andrews, G. E.

AU - Ballantine, C.

N1 - Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/12

Y1 - 2022/12

N2 - Franklin's identity generalizes Euler's identity and states that the number of partitions of n with j different parts divisible by r equals the number of partitions of n with j different parts repeated at least r times. In this article, we give a refinement of Franklin's identity when j=1. We prove Franklin's identity when j=1, r=2 for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter M over the number of parts in all partitions into distinct parts with perimeter M equals the number of partitions with perimeter M whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.

AB - Franklin's identity generalizes Euler's identity and states that the number of partitions of n with j different parts divisible by r equals the number of partitions of n with j different parts repeated at least r times. In this article, we give a refinement of Franklin's identity when j=1. We prove Franklin's identity when j=1, r=2 for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter M over the number of parts in all partitions into distinct parts with perimeter M equals the number of partitions with perimeter M whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.

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U2 - 10.1016/j.disc.2022.113110

DO - 10.1016/j.disc.2022.113110

M3 - Article

AN - SCOPUS:85135707658

SN - 0012-365X

VL - 345

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 12

M1 - 113110

ER -