bibkey: blitvickammounsteingrimsson2026patterns
authors: Natasha Blitvić, Mohammed Slim Kammoun, and Einar Steingrímsson
year: 2026
title: "Permutations, Pattern Covers, Posets and Positivity: Part I: Patterns and Posets"
doi: 10.48550/arXiv.2610.29846
url: https://arxiv.org/abs/2610.29846v1
claim: "Conjecture 7.1: prescribed interleaved minimal prefix and suffix attain the maximum packing count for m≤K/2."
strata_touched:
  - D5/S3/Combinatorics/Permutation/PackingMaximumRefutation
license: citation-only
triage: anchor

Prescribed permutation packing

The primary source is arXiv:2610.29846v1, submitted 21 October 2026. Definition 2.2 defines order isomorphism by agreement of every strict comparison. Definition 2.6 and equation (2.1) define a permutation of length $n=K+(j-1)(K-m)$ that has a copy of the $K$-pattern at each start $1+t(K-m)$, for $0\le t<j$. Remark 2.7 explicitly permits additional occurrences. This is a prescribed packing count, rather than avoidance or a count with exactly $j$ total occurrences.

Definition 3.3 and its following paragraph require order-compatible $m$-prefix and $m$-suffix. The important convention in Section 3 uses increasing prefix and suffix, with the first prefix value smaller, after count-preserving symmetries. The Lean comparison domain includes every compatible pattern directly. No normalization theorem is needed to admit the counterexample, which already satisfies the convention.

Section 7 item 4 introduces the maximum count as a function of $(K,m,j)$. Conjecture 7.1 states:

For overlap $m\le K/2$, the maximum is obtained when prefix and suffix are interleaved and minimal, that is, for patterns with prefix $135\dots (2m-1)$ and suffix $246\dots (2m)$.

This asserts attainment by a member of the prescribed family. It does not assert uniqueness or fix the order of the remaining middle values. The equality case $K=2m$ has an empty middle.

The repository calculation gives $C(13472568,4,2)=9>8=C(13572468,4,2)$. At this equality boundary, the prefix and suffix determine every prescribed pattern, so the comparison refutes the entire maximum-attainment assertion. The exact counts and source-permutation correspondence are the repository result; the general packing definitions and the packing/linear-extension interpretation (Lemma 5.1 and Theorem 5.3) are the source authors’ results.

The finite mechanism is two independent three-way choices for the competitor, compared with three independent binary choices for the prescribed pattern. The actual maximum, the strict-overlap range $2m<K$, and Section 7 item 8’s separate cutoff question are not settled.

The bounded prior check covered the primary version, selected symmetry and enumeration results, the FPSAC precursor, Elizalde arXiv:1203.1636v1 (overlap-one results), and Iriarte arXiv:1405.4880v2 (orientations of a fixed graph). None supplied a located whole answer in that checked scope. The graph-orientation theorem does not directly compare these changing pattern posets. Ineffective general web queries establish no absence. Worldwide priority, exhaustive literature absence, and provider memory are unverified; no such claim is made. Ordinary counting ingredients receive no separate novelty claim.

Locator

• DOI: 10.48550/arXiv.2610.29846.
• https://arxiv.org/abs/2610.29846v1
• https://arxiv.org/html/2610.29846v1#S7.Thmtheorem1
• Definitions 2.1–2.2, 2.5–2.6; equation (2.1); Remark 2.7.
• Definition 3.3 and the important convention in Section 3.
• Lemma 5.1 and Theorem 5.3 for the original chain interpretation.

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