bibkey: franklin2024inversions authors: Atli Fannar Franklín year: 2024 title: “Pattern avoiding permutations enumerated by inversions” doi: 10.48550/arXiv.2410.07467 url: https://arxiv.org/abs/2410.07467v4 claim: “Section 1 conjectures on indecomposable pattern-avoiding permutations counted by inversions, including the count of I_k(321, 1342).” strata_touched:

• D5/S3/Combinatorics/IndecomposableInversion/FranklinInversion license: citation-only triage: anchor

Franklín, pattern avoiding permutations enumerated by inversions

The paper counts pattern-avoiding permutations by their number of inversions. Its central objects are the sets I_k of direct-sum indecomposable permutations, of any length, with exactly k inversions; an indecomposable permutation of length n has at least n − 1 inversions, so each I_k is finite.

Section 1, printed pages 2–3, states two conjectures. The first identifies I_k(132, 4321) with partitions having no part strictly between the smallest and the largest part; it is proved in Claesson, Linusson, Ulfarsson and Verkama, arXiv:2604.01143v1, Section 7.6, Proposition 7.7. The second reads:

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

The printed formula fails at k = 1, where I_1 = {21}. The counts 1, 1, 2, 4, 7, 11, 16, 22, 29 for k = 0, …, 8 equal k(k − 1)/2 + 1; the printed formula equals the count of I_{k+1}(321, 1342). The module D5/S3/Combinatorics/IndecomposableInversion/FranklinInversion proves |I_k(321, 1342)| = k(k − 1)/2 + 1 for every k and refutes the printed formula.

The journal version appeared in Discrete Mathematics & Theoretical Computer Science 27:1, Permutation Patterns 2024 (special issue, 2025); its text was not compared with arXiv v4.

Verified locator

DOI: 10.48550/arXiv.2410.07467

URL: https://arxiv.org/abs/2410.07467v4

• Locator: Section 1, printed pages 2–3, conjecture on I_k(321, 1342).
• Locator: Section 1, printed page 2, conjecture on I_k(132, 4321).
• Locator: Section 1, definition of I_k as the indecomposable permutations with exactly k inversions.