Nonnesting Permutations Avoiding 1231, 1312, 2231 and 3221

Sergi Elizalde and Amya Luo, Pattern avoidance in nonnesting permutations, arXiv:2412.00336v6, Section 4, Table 4, row {1231, 1312, 2231, 3221}: the conjectured ordinary generating function of c_n({1231, 1312, 2231, 3221}) is (1 - 3x + 2x^2)/((1 - 3x)(1 - x - x^2)), with the remark

All the conjectures have been checked for n up to 8.

Here a nonnesting permutation of size n is a permutation of {1, 1, 2, 2, …, n, n} avoiding 1221 and 2112, and pattern containment keeps equal letters equal.

The theorem D5/S3/Combinatorics/Nonnesting/NonnestingFour.result establishes the identity of formal power series C(x)(1 - 3x)(1 - x - x^2) = 1 - 3x + 2x^2 for C(x) = Σ c_n x^n, so c_n = 1, 1, 4, 11, 33, 98, 293, 877, 2628, 7879, … and c_n = 4c_{n-1} - 2c_{n-2} - 3c_{n-3} for n ≥ 3.

Pre-registration issue 11301 records the literature screen: none of the papers citing arXiv:2412.00336 treats this row, and the repository had no result for it. This is a bounded negative finding.

1. A doubled word avoids 1221 and 2112 exactly when its first and second occurrences appear in the same letter order.
2. Every word of the class splits uniquely at its value cuts into cut-free words, and direct sums of words of the class stay in the class; so C = 1/(1 - D) for the generating function D of cut-free words.
3. The four patterns force the first-occurrence order into consecutive blocks k, 1, 2, …, k - 1; a cut-free word of size n ≥ 2 is either n n v with v of increasing first order, or one of two chains, so there are 2^{n-2} + 2 of them, and D = x + x^2/(1 - 2x) + 2x^2/(1 - x).

The statement would fail if some n had a count different from the coefficient of the rational function. It depends on reading containment with equal letters kept equal.

Exhaustive enumeration through n = 11 agrees with every structural lemma and with the counts, including c_{10} = 23629 and c_{11} = 70874.

theorem; the conjecture is stated in Table 4 of arXiv:2412.00336 and is quantified over every n.

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