Document Type

Article

Publication Date

5-27-2024

Department

Department of Mathematical Sciences

Abstract

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher’s theorem: for all m ≥ 2 and n ≥ 1, there are at least as many partitions with perimeter n and parts repeating fewer than m times as there are partitions with perimeter n with parts not divisible by m. In this work, we provide a combinatorial proof of their theorem by relating the combinatorics of the partition perimeter to that of compositions. Using this technique, we also show that a composition theorem of Huang implies a refinement of another perimeter theorem of Fu and Tang.

Publisher's Statement

© 2024 Waldron. Publisher’s version of record: https://doi.org/10.5281/zenodo.11353009

Publication Title

Integers

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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Mathematics Commons

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