Three Wilf-inequivalent classes with the same growth rate
Av(132456), Av(124356) and Av(123546) share a growth rate, and A and C even share counting sequences — but A and B do not. How different are the sequences of A and B?
Cite
@misc{cairn-principal-classes-length-six-bona,
title = {Three Wilf-inequivalent classes with the same growth rate},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/principal-classes-length-six-bona}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
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The problem
The question
Let A = Av(132456), B = Av(124356) and C = Av(123546). All three have the same growth rate, and A and C have the same counting sequence, but A and B do not. How different are the counting sequences of A and B? Is their ratio asymptotically constant? Is the difference or ratio described by a polynomial factor — and which one?
What counts as progress
- Long initial segments of both counting sequences (reproducible enumeration code).
- A conjectured asymptotic relation fitted to the data, then a proof (a bijection or generating-function identity).
How it is checked
Sequences are checkable by independent enumeration; proofs are reviewed (level B).
Source. Posed by Miklós Bóna 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.