Enumeration of the Arrow Pattern (32; 1 → 3)

Robin D.P. Zhou and Xinyang Yu, Arrow-Wilf equivalences and enumerative results for short arrow patterns, arXiv:2609.29392v1, abstract:

Together with earlier work of Archer and Laudone, this leaves only (32;1→3) unresolved for |ν|≤2, which we pose as an open problem.

Here S_n(32; 1 → 3) consists of permutations of 1, …, n avoiding the arrow pattern in the paper's sense: the selected values have the required order in the one-line word, and the prescribed arrow holds in π̂ = θ⁻¹(π).

Write a_n = |S_n(32; 1 → 3)|, with a_0 = 1, and F(x) = Σ_{n≥0} a_n x^n. The theorem D5/S3/Combinatorics/ArrowThirtyTwoOneThree.result establishes

1 + (3x − 2)F + (1 − x)(1 − 2x)F² + x³F³ = 0.

Among integer formal power series satisfying this cubic, F is uniquely determined by F(0) = 1 and [x]F = 1. The initial counts for n = 0, …, 8 are 1, 1, 2, 5, 15, 51, 190, 757, 3171.

Pre-registration issue 11059 records the literature screen. The arXiv record has no later paper on arrow patterns, OEIS has no entry for 1, 2, 5, 15, 51, 190, 757, 3171, and the repository had no result for this enumeration before the theorem above. These are bounded negative findings, not a claim about all mathematical literature.

1. An avoider is characterized by its inverse Foata edges: whenever a < c < π̂(a), the value c must precede π̂(a) in the one-line word. For words of distinct values, a classical 132 occurrence exists exactly when an adjacent 13-2 occurrence exists.
2. Split an avoider at its largest value. The letters of its final cycle form a selected set; its last letter is the largest selected letter, and the preceding order avoids 132. Missing selected values separate the prefix into ordered value intervals. Conversely, interval avoiders and a 132-avoiding final-cycle order reconstruct a unique avoider.
3. A final cycle with k ≥ 1 other letters contributes C_{k−1} times the coefficient of x^{n−1−k} in F^{k+1}; a singleton final cycle contributes a_{n−1}. The resulting recurrence, together with C(z) = 1 + zC(z)², gives F = 1 + xF + (xF)² C(xF), hence the cubic above.
4. If two integer formal series satisfy the cubic and both have constant and linear coefficients equal to one, the difference of their cubic equations factors by their series difference. The remaining factor has first nonzero coefficient −1 at degree one, so it cannot annihilate a nonzero series difference. Thus the distinguished branch is unique.

The enumeration statement would fail if some n had a different avoidance count from the coefficients of the distinguished cubic branch. Fixing only F(0) = 1 does not characterize that branch: the cubic also has a formal solution beginning 1 + 2x + 3x² + x³ − 16x⁴ + …. The conclusion depends on the paper's arrow-containment convention and the inverse Foata cycle map; changing either changes the class being counted.

The count recurrence is obtained from the final-cycle bijection, and the cubic follows as a formal power-series identity over the integers. Independent exhaustive enumeration through n = 11 agrees with the edge criterion, reconstruction, refined counts, and cubic coefficients.

theorem; the open question is stated in the abstract of arXiv:2609.29392v1. The result is quantified over every size and is not a bounded enumeration.

The literature screen is limited to the arXiv, OEIS, and repository checks recorded above. It does not establish a worldwide priority claim.