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.
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Open problems about permutation patterns
Counting permutations that avoid a pattern such as 1324, growth rates of permutation classes, and the enumeration sequences behind them. Terms of a sequence can be computed and checked independently.
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.
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Finitely based permutation classes with growth rate below 4
Conjecture (Brignall): every finitely based permutation class with growth rate less than 4 has a rational generating function.
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Permuton limits of classes avoiding α ⊖ 1
Conjecture (Bevan–Troyka): for every nonempty α, a uniformly random large permutation avoiding α ⊖ 1 looks like the diagonal (identity) permuton.
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Prove that some small permutation classes are not D-finite
Classes such as Av(1243, 1324, 1432) are believed to have non-D-finite generating functions. An explicit q-series for one of them is known — can it prove non-D-finiteness?
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Strict monotonicity of growth rates of permutation classes
If a pattern π properly contains ρ, is the growth rate of Av(π) strictly larger than that of Av(ρ)? A first test case: gr(Av(π)) < gr(Av(π⊕1, 1⊕π)) < gr(Av(1⊕π)).
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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?
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Unavoidable k-shuffles in long permutations
Does every sufficiently long permutation contain a run of consecutive entries that splits into k copies of the same pattern? Known thresholds: n0(2) = 6, n0(3) = 12.
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