Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

bibkey: egge2022pattern authors: Eric S. Egge year: 2022 title: “Pattern-Avoiding Fishburn Permutations and Ascent Sequences” doi: 10.48550/arXiv.2208.01484 url: https://arxiv.org/abs/2208.01484v1 claim: “Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.” strata_touched:

  • D5/S3/Combinatorics/Fishburn/FishburnTenFour
  • D5/S3/Combinatorics/Fishburn/FishburnTenFive
  • D5/S3/Combinatorics/FishburnTenSeven/FishburnTenSeven
  • D5/S3/Combinatorics/Fishburn/FishburnTenNine
  • D5/S3/Combinatorics/Fishburn/FishburnTenTen
  • D5/S3/Combinatorics/Fishburn/FishburnTenEleven
  • D5/S3/Combinatorics/Fishburn/FishburnTenTwelve
  • D5/S3/Combinatorics/FishburnTenThirteen/FishburnTenThirteen license: citation-only triage: anchor

Egge, pattern-avoiding Fishburn permutations and ascent sequences

The paper enumerates Fishburn permutations avoiding classical patterns of lengths three and four, relates them to pattern-avoiding ascent sequences, and closes with a list of conjectured enumerations and Wilf equivalences.

Verified locator

DOI: 10.48550/arXiv.2208.01484

URL: https://arxiv.org/abs/2208.01484v1

  • Locator: Section 1, a permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B) and S_n(B) denote the Fishburn permutations and all permutations of length n avoiding the classical patterns in B.
  • Locator: Section 10, Conjecture 10.4: |F_n(1324, 2143)| = |F_n(1423, 2143)| = |F_n(1423, 3124)| = (n − 1)2^{n−2} + 1 for n ≥ 1.
  • Locator: Section 10, Conjecture 10.5: |F_n(1324, 1423)| = |F_n(1324, 3124)| = F_{2n−2} for n ≥ 1, with F_0 = F_1 = 1.
  • Locator: Section 10, Conjecture 10.7: |F_n(1324, 2143, 1423)| = |F_n(1324, 2143, 3124)| = |F_n(1324, 1423, 3124)| = 2^n − n for n ≥ 1.
  • Locator: Section 10, Conjecture 10.9: |F_n(2143, 3124)| = |S_n(231, 4123)| for n ≥ 1.
  • Locator: Section 10, Conjecture 10.10: |F_n(2143, 1423, 3124)| = |S_n(321, 2143, 3124)| = |S_n(231, 4132, 2134)| for n ≥ 0, verified there for n ≤ 17.
  • Locator: Section 10, Conjecture 10.11: |F_n(1243, 2134)| = |S_n(123, 3241)| for n ≥ 0, verified there for n ≤ 15.
  • Locator: Section 10, Conjecture 10.12: |F_n(1243, 3124)| = |S_n(231, 4123)| for n ≥ 0, verified there for n ≤ 15.
  • Locator: Section 10, Conjecture 10.13: |F_n(2413, 2431)| = |F_n(2431, 3241)| = Σ_{k=1}^{n} C(n − 1, k − 1) C_{n−k} for n ≥ 1, verified there for n ≤ 14; C_m is the Catalan number.

Reading of the statement

For n = 1, …, 8 the counts are 1, 2, 5, 13, 33, 81, 193, 449 (Conjecture 10.4), 1, 2, 5, 13, 34, 89, 233, 610 (Conjecture 10.5), 1, 2, 5, 13, 33, 82, 202, 497 (Conjectures 10.9 and 10.12) and 1, 2, 5, 13, 32, 74, 163, 347 (Conjecture 10.11), 1, 2, 5, 12, 25, 46, 77, 120 (Conjecture 10.10), 1, 2, 5, 12, 27, 58, 121, 248 (Conjecture 10.7) and 1, 2, 5, 15, 51, 188, 731, 2950 (Conjecture 10.13).

Bounded prior-resolution evidence

Read on 2026-10-01: of the six papers listed by Semantic Scholar as citing arXiv:2208.01484, only Du and Zhang (arXiv:2302.13767; Discrete Math. 347 (2024) 113952) settle conjectures of the paper, namely Conjectures 10.14 and 10.17. None of the located later papers treats Conjectures 10.4, 10.5, 10.9, 10.11 10.12 or 10.13. This is a bounded negative finding.