Abstract
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.
Original language | English (US) |
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Article number | 113110 |
Journal | Discrete Mathematics |
Volume | 345 |
Issue number | 12 |
DOIs | |
State | Published - Dec 2022 |
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics