slug: franklin-indecomposable-321-1342-count bibkey: franklin2024inversions doi: 10.48550/arXiv.2410.07467 url: https://arxiv.org/abs/2410.07467v4 triage: theorem motivation_gids:

• D5/S3/Combinatorics/IndecomposableInversion/FranklinInversion.result
• D5/S3/Combinatorics/IndecomposableInversion/FranklinInversion.avoiders_ncard

The Number of Indecomposable 321- and 1342-Avoiders with k Inversions

Problem

Atli Fannar Franklín, Pattern avoiding permutations enumerated by inversions, arXiv:2410.07467v4, Section 1, printed pages 2–3:

Another case is I_k(321, 1342), which we conjecture to have k(k + 1)/2 + 1 elements.

Here I_k is the set of direct-sum indecomposable permutations, of any length, with exactly k inversions, and I_k(321, 1342) is its subset avoiding the classical patterns 321 and 1342.

Motivation

The theorem D5/S3/Combinatorics/IndecomposableInversion/FranklinInversion.avoiders_ncard proves |I_k(321, 1342)| = k(k − 1)/2 + 1 for every k ≥ 0, and FranklinInversion.result derives from it that the printed formula is false.

Gap

Pre-registration issue 12442 records the literature screen: the paper citing arXiv:2410.07467 that settles the companion conjecture on I_k(132, 4321) (arXiv:2604.01143) does not treat I_k(321, 1342), and the GitHub and repository searches located no treatment. This is a bounded negative finding.

Route

1. Every member of I_k(321, 1342) is either a star of length k + 1 or a five-block permutation B(r, t, d, h) with explicit parameter ranges; the two families are disjoint and the parametrization is injective.
2. Stars have k = n − 1 inversions, and B(r, t, d, h) has rt + d + h inversions.
3. Euclidean division matches the five-block members with k inversions to the pairs 1 ≤ t < q ≤ k, giving k(k − 1)/2 of them, plus one star.

Falsifier

The statement would fail if a member of I_k(321, 1342) had neither form, if two parameter tuples gave the same permutation, or if an inversion formula were wrong.

Evidence

The classification, the inversion formulas and the counts were checked on all 321-avoiders with at most 14 inversions; the counts for k = 0, …, 14 are 1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92. An independent referee implementation confirmed every lemma through k = 12.

Triage

theorem; the printed conjecture of Section 1 of arXiv:2410.07467 is refuted, and the corrected count is proved for every k.

• Proved (formalized): |I_k(321, 1342)| = k(k − 1)/2 + 1 for every k ≥ 0; the printed formula is the count of I_{k+1}(321, 1342), an index shift.
• Open: the exact enumeration of the indecomposable permutations of length n avoiding 321 and 1342 (the source reports experimental growth like n³/6), and the asymptotic bound the source derives from the printed formula, which has to be re-derived from the corrected count.

ASSUMED-UNVERIFIED

The literature screen is limited to the citing papers, the arXiv and GitHub searches and the repository checks recorded above; the journal version was not compared with arXiv v4.