The degree of parity bias among fixed-perimeter partitions into distinct parts

Abstract

The perimeter of a partition is its largest part plus its number of parts minus one, the largest hook length of its Ferrers diagram; Straub proved that exactly Fₙ partitions into distinct parts have perimeter n. Gray, Payne, Swisher and Watson asked how those Fₙ partitions split by parity, conjectured that more of them carry an excess of odd parts than an excess of even parts for every n ≥ 9, and verified this to n=150; Mahanta has since proved it. We study the refinement that neither paper defines: the count rd(n,m) of distinct-part partitions of perimeter n whose odd parts outnumber their even parts by exactly m, and the excess D(n,m)=rd(n,m)-rd(n,-m). We record four facts about D, each verified by exhaustive computation over an explicit range of perimeters and none of them proved in general: an exact reflection D(n+1,m)=-D(n,m) whenever m ≡ n (mod 2), which pins the value and not merely the sign; a sign law signD(n,m)=(-1)ⁿ⁻ᵐ that holds exactly up to a vanishing threshold m ≤ ⌊ (n+2)/3⌋, a threshold with no counterpart in the unrestricted case; single-sum closed forms collapsing the natural Θ(n²)-term double sums for rdₒ-rdₑ and for rdₜ to Θ(n) terms; and the identification of the first two columns of the excess table with the generalized Catalan numbers A004148 and with A166297, two OEIS entries that record no partition-perimeter content. The sign law exhibits rdₒ(n)-rdₑ(n) as an alternating sum of those columns; the unrestricted analogue alternates in the same way, but there the alternating sum is evaluated exactly by an involution, and for distinct parts we know of no such evaluation. We also verify the conjecture for every 9 ≤ n ≤ 1500, ten times the published range, and prove in general the counting comparison that the published proof asserts as an elementary calculation. All statements are machine-checked in Lean 4.

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