bibkey: thompson2026a398720
authors: G. Gordon Thompson
year: 2026
title: A398720 — weight distribution of square single parity check product codes
doi: null
url: https://oeis.org/A398720
claim: The entry conjectures that odd square binary matrices with even row and column weights and total weight n(n-1) are counted by n factorial.
strata_touched:
  - D5/S1/Words/ParityCode/OddTopWeight
license: citation-only
triage: anchor

A398720

Verified locator

url: https://oeis.org/A398720

doi: null

Revision 66 of https://oeis.org/A398720/internal is dated 2026-08-20 and credits G. Gordon Thompson, 2026-08-07. Its formula field states: “Conjecture: T(n, n*(n-1)/2) = n! for n odd.”

Meaning of the indices

The comment defines T(n,k) as the number of n by n binary matrices with exactly 2k ones and an even number of ones in each row and each column. This is the convention formalized here. The name field instead uses the code parameters [(n+1)^2,n^2,4]; that shift is not used to index the matrices. The example rows start at n=0, and rows n=1,3,5 end in 1,6,120 respectively. The case n=1 consists of the single zero matrix.

Repository derivation

For odd n, a row of even weight has at most n-1 ones. If the total weight is n(n-1), every row must attain that bound and hence has exactly one zero. Transposition gives the same conclusion for columns. Choosing the zero in each row defines an injection from the row indices to the column indices, and finiteness makes it a permutation. Conversely the complement of a permutation matrix has n-1 ones in every row and column. These constructions are inverse, so Mathlib’s permutation cardinality theorem gives n!.

Proof scope

The proof derives unique zero positions from parity and total weight, using Mathlib’s finite-sum equality and permutation cardinality APIs. No publication priority or exhaustive literature coverage is claimed.