Inversion Sequences Avoiding {010, 100, 102, 210} and {011, 201, 210}

David Callan and Toufik Mansour, Inversion Sequences Avoiding Quadruple Length-3 Patterns, Integers 23 (2023), #A78, Section 1, Conjecture 1:

Class 152: {010, 100, 102, 210} ∼_I {011, 201, 210}

Here ∼_I means that, for every n, equally many inversion sequences of length n avoid each set. The authors checked the conjecture up to n = 13 and list the comparison as "still open" in Table 1.

The theorem D5/S3/Combinatorics/InversionSeq/InversionSeqClass152.result establishes the equality for every n; both classes have (1 + Σ_{j<n} binom(2j, j))/2 elements of length n ≥ 1.

Pre-registration issue 11762 records the literature screen: Class 166 of the same conjecture was settled by Asinowski and Polley, while none of the located later papers treats Class 152. This is a bounded negative finding.

1. In the left class a value that occurs after a larger value occurs only once, and every adjacent descent starts at the global maximum; the sequence is cut at its first adjacent descent into a weakly increasing prefix and a suffix of interleaved maxima and singleton values.
2. In the right class 011 forces distinct positive entries; the sequence is cut at its last zero into a Catalan prefix and a unique decomposition into rotation blocks.
3. Weakly increasing inversion sequences are Dyck paths; factorizations of Dyck paths at the first ascent, a marked interior descent and the penultimate descent give the generating functions of both sides, which coincide.

The statement would fail if either decomposition missed a sequence or produced one twice, or if the two generating functions differed in some coefficient.

Every structural lemma and both decompositions were checked on all inversion sequences of length at most 11; the two counts agree through n = 11.

theorem; the statement is Class 152 of Conjecture 1 in Callan and Mansour's paper and is quantified over every n.

The literature screen is limited to the citation and full-text searches, the OEIS entries and the GitHub and repository checks recorded above.