1324-avoiders with a fixed number of inversions
Is the number of 1324-avoiding permutations of length n with exactly k inversions non-decreasing in n? A yes would bound the growth rate of Av(1324) by about 13.002.
Cite
@misc{cairn-av1324-inversion-monotonicity,
title = {1324-avoiders with a fixed number of inversions},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/av1324-inversion-monotonicity}},
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
Let av_n^k(1324) be the number of permutations of length n with exactly k inversions that avoid the pattern 1324. Conjecture (Claesson–Jelínek–Steingrímsson): av_n^k(1324) ≤ av_{n+1}^k(1324) for all n and k.
Why it matters
The growth rate of Av(1324) is one of the best-known unknown constants in permutation patterns; current bounds are about 10.27 ≤ gr(Av(1324)) ≤ 13.5 (lower bound since raised slightly). The conjecture would give gr(Av(1324)) ≤ e^{π√(2/3)} ≈ 13.002.
What is known
Linusson and Verkama enumerated these permutations for n ≥ (k + 7)/2, which settles that range. Later work proves the analogous monotonicity for some sets of patterns containing 1324 and extends the verified range; the full conjecture is open.
What counts as progress
- Exact counts av_n^k(1324) in new (n, k) regions, with efficient, reproducible counting code.
- A counterexample (n, k) with av_n^k > av_{n+1}^k — a finite, checkable fact.
- An injection proving the conjecture in a larger range, or in full.
How it is checked
Counts are reproducible computations (level B); a counterexample is checked by independent enumeration.
Source. Posed by Anders Claesson in the open problem session of the Oberwolfach workshop Mini-Workshop: Permutation Patterns (2024), recorded in Oberwolfach Reports 6/2024, p. 284 (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.