slug: archer-laudone-theta-cube-231-312 bibkey: archer2024fundamental doi: 10.48550/arXiv.2407.06338 url: https://arxiv.org/abs/2407.06338v1 triage: theorem motivation_gids:

• D5/S3/Combinatorics/FundamentalBijection/ThetaCube.result

Avoiders of 231 and 312 Fixed by the Third Iterate of the Fundamental Bijection

Problem

Kassie Archer and Robert P. Laudone, Pattern avoidance and the fundamental bijection, arXiv:2407.06338v1, Section 5, Conjecture 5.6:

For σ ∈ {231, 312}, the generating functions F_σ^k(x) are rational. A few examples of conjectured generating functions for these patterns are found below. F_σ^3(x) = 1/(1 − x − x^2 − 2x^3)

Here F_σ^k(x) = Σ f_n^k(σ) x^n and f_n^k(σ) counts the permutations of [n] avoiding σ and fixed by the k-th iterate of the fundamental bijection θ.

Motivation

The theorem D5/S3/Combinatorics/FundamentalBijection/ThetaCube.result establishes F_σ^3(x) (1 − x − x^2 − 2x^3) = 1 for both σ = 231 and σ = 312, so f_n^3(σ) = 1, 1, 2, 5, 9, 18, 37, 73, 146, 293, … for n = 0, 1, ….

Gap

Pre-registration issue 11298 records the literature screen: none of the papers citing arXiv:2407.06338 treats this conjecture, and the journal version still states it. This is a bounded negative finding.

Route

1. θ commutes with direct sums, and every 231- or 312-avoider is uniquely a direct sum of sum-indecomposable ones, so F_σ^3 = 1/(1 − I_σ^3) where I_σ^3 counts the indecomposable avoiders fixed by θ^3.
2. The inverse of θ cuts a word before each left-to-right maximum and closes each block into a cycle. Following the rows p, θ^{-1}(p), θ^{-2}(p) of a θ^3-fixed indecomposable avoider of size n > 3, the successor equations force two different entries into one position.
3. Hence the indecomposable fixed avoiders are 1, 21 and two blocks of size three, and I_σ^3 = x + x^2 + 2x^3.

Falsifier

The statement would fail if some σ-avoider of size at least four, not a direct sum, were fixed by θ^3.

Evidence

The successor chase was checked on every indecomposable 231- and 312-avoider of size at most 11; the counts agree through n = 11.

Triage

theorem; the formula is stated in Conjecture 5.6 of arXiv:2407.06338 and is quantified over every n.

ASSUMED-UNVERIFIED

The literature screen is limited to the citation list, arXiv searches and repository checks recorded above.