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The Descending Parameter Family

Abstract

The decreasing word has an explicit fundamental image and characterizes parameters ending with one.

Theorem 1.1 (The decreasing family and its later orbit).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateVFamilies.V_family

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateVFamilies.V_family (✓ 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.

For h at least two, the fundamental image of the decreasing word begins with h divided by two plus one when h is odd, then consists of pairs x, h plus one minus x for increasing x from the integer part of h divided by two plus one plus the parity of h through h. Its successor word is increasing. Among parameters beginning with h and ending with one, P avoids 132 through depth two exactly for the decreasing parameter. For h at least four, the second fundamental image of that decreasing parameter contains 132.

References