Are pattern-avoidance counting sequences Stieltjes moment sequences?
Conjecture: for every pattern β, the numbers of β-avoiding permutations form a Stieltjes moment sequence. One failing pattern would refute it; Hankel determinants give a direct test.
Cite
@misc{cairn-stieltjes-moment-permutation-classes,
title = {Are pattern-avoidance counting sequences Stieltjes moment sequences?},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/stieltjes-moment-permutation-classes}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
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The problem
The question
A sequence (a_0, a_1, …) is a Stieltjes moment sequence if a_n = ∫ x^n dμ(x) for a positive measure μ supported on [0, ∞). Conjecture: for every permutation β (of length at least 2), the counting sequence of the principal class Av(β) is a Stieltjes moment sequence. More generally: when is the counting sequence of a permutation class a (Stieltjes) moment sequence?
What is known
Confirmed for Av(12…k) and Av(1342); numerical evidence supports all patterns of length 5. Vatter's 2026 survey lists the question as open.
What counts as progress
- Testing positivity of the Hankel determinants det(a_{i+j}) and det(a_{i+j+1}) on the longest known series for all patterns of length 5 and 6; any negative determinant disproves the conjecture for that β.
- Explicit measures (and proofs) for further classes.
How it is checked
A negative Hankel determinant computed in exact integer arithmetic from verified terms is a certificate; data and code must be reproducible (level B).
Source. Posed by Natasha Blitvić and Andrew Elvey Price in the open problem session of the Oberwolfach workshop Mini-Workshop: Permutation Patterns (2024), recorded in Oberwolfach Reports 6/2024, p. 285 (EMS Press, DOI 10.4171/OWR/2024/6), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.