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arXiv · 2210.11858

Tight Lower Bound for Pattern Avoidance and Symmetric Functions

Avichai Marmor
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Abstract

For a set of permutations (patterns) $\Pi$ in $S_k$, consider the set of permutations in $S_n$ that avoid all patterns in $\Pi$. In current algebraic combinatorics, a significant problem is to identify pattern sets $\Pi$ for which the corresponding quasisymmetric function is symmetric for all $n$. Recently, Bloom and Sagan proved that unless $\Pi \subseteq \{12\dots k, k \dots 21\}$, the size of such $\Pi$ must be at least $3$ for any $k \ge 4$. They also posed a general lower bound conjecture. In this work, we resolve this conjecture and give a tight lower bound, namely, the minimal size of such $\Pi$ is exactly $k - 1$. The proof relies on a novel generalization of Bose's theorem in extremal combinatorics, utilizing the multilinear polynomial approach introduced by Alon, Babai, and Suzuki.

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Avichai Marmor. 2022-10-21. Tight Lower Bound for Pattern Avoidance and Symmetric Functions. https://arxiv.org/abs/2210.11858

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