2023/02/01 by Hunter Waldron, Waldron, Hunter
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2302.00645
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 \not ≡ 0 \pmodm as partitions with perimeter n and parts repeating fewer than m times. 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.