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Low Entries in the Forced Prefix

Abstract

The prescribed boundary data for two consecutive inverse images determine additional small entries.

Theorem 1.1 (Four forced small entries).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaCube231LargePrefix.forced_low_prefix

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaCube231LargePrefix.forced_low_prefix (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Kassie Archer, Robert P. Laudone (2024). Pattern avoidance and the fundamental bijection. DOI: 10.48550/arXiv.2407.06338. URL: https://arxiv.org/abs/2407.06338v1.

Commentary.

Let p and q be permutations of size n at least eleven with p avoiding 231, q the inverse image of p, and the second inverse image of q equal to p. Suppose p starts with n, one, n minus two, two and ends with n minus one, while q has n minus two, n minus one, n, one at positions zero, n minus four, n minus two, n minus one. Then q has four and three at positions one and three, and p has four and three at positions four and five.

References