Papers › Explicit formulas for permutation pattern character polynomials
Explicit formulas for permutation pattern character polynomials
Jonas Iskander
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Given permutations π∈Sₙ and σ∈Sₖ, let N_σ(π) denote the number of occurrences of σ in π. While pattern avoidance and the distribution of pattern occurrences in permutations have been extensively studied, their interactions with the group structure on Sₙ are still poorly understood. Gaetz and Ryba showed that the expected value of χ^(λ[n])(π)N_σ(π) for π∈Sₙ is given by a polynomial a_σ^λ(n). More recently, Gaetz and Pierson derived explicit formulas for a_(idₖ)^λ(n) when |λ|≤2, which led them to conjecture that the polynomials a_(idₖ)^λ(n) are real-rooted and nonnegative for n ≥k. We show that for all partitions λ, the polynomials a_(idₖ)^λ(n) admit explicit closed forms in n and k. These formulas allow us to exhibit counterexamples to Gaetz and Pierson's real-rootedness conjecture as well as to prove special cases of their nonnegativity conjecture. Our results imply that the expected value of f ·N_(idₖ) on Sₙ admits a closed form whenever f is a permutation statistic expressible as a polynomial in the functions mⱼ _(n ≥0) Sₙ →ℤ which count j-cycles in their inputs.
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