bibkey: rascoe2025a380392
authors: John Tyler Rascoe
year: 2025
title: A380392 — binary matrices counted by their South-East paths of ones
doi: null
url: https://oeis.org/A380392
claim: The entry conjectures that the average number of all-one South-East paths in binary n-square matrices is A001790(n-1)/A101926(n-1).
strata_touched:
  - D5/S3/Arith/Paths/MonotoneOnePaths
license: citation-only
triage: anchor

A380392

Verified locator

Canonical entry URL: https://oeis.org/A380392

Canonical entry URL: https://oeis.org/A380392 The entry has no DOI. Revision 16 of its internal page is dated 2025-02-22. Rascoe’s comment dated 2025-02-21 states the average-path conjecture. The entry defines South-East paths using adjacent cells in the same row or column, with only South and East steps. It supplies no proof of the conjectured average. Its zero-size row is outside the positive-size theorem.

Mathematical statement

For n at least 1, a path from the top left to the bottom right has n-1 east steps and n-1 south steps. The expected number of paths containing only ones, when each binary matrix has equal weight, is C(2n-2,n-1)/2^(2n-1). This is the ratio of the auxiliary numerator and denominator sequences in the conjecture. The formal result proves this binomial formula; it does not define or enumerate the OEIS triangle.

The repository proof encodes the east steps by a fixed-size subset of the step positions. Prefix counts give the two coordinates. Their sum is the time index, so all visited cells are different. Assignments off those cells extend uniquely to matrices supporting the path. Summing first over paths gives C(2n-2,n-1)*2^((n-1)^2) incidences.

Proof scope

The Lean derivation uses a double counting argument and standard finite subset and function cardinality lemmas. It is repository work, not a proof attributed to the OEIS note. No publication priority or exhaustive literature coverage is claimed.