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Active Positions in Direct Sums Avoiding 2431 and 3241

Abstract

Maximum-insertion positions in a direct sum avoiding 2431 and 3241 are determined componentwise.

Theorem 1.1 (Componentwise insertion and crossing inequalities).

Lean statement: D5/S3/Combinatorics/FishburnTenThirteen/FishburnTenThirteenBSums.crossing_sum_sites

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FishburnTenThirteen/FishburnTenThirteenBSums.crossing_sum_sites (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Commentary.

Let u and v be Fishburn permutations of lengths m and n avoiding 2431 and 3241, and suppose their direct sum avoids the same patterns and is Fishburn. For every position s from zero through m, insertion of m + n + 1 at s in the direct sum preserves these conditions exactly when insertion of m + 1 at s in u does so. For every position s from zero through n, insertion at m + s in the direct sum is permitted exactly when insertion of n + 1 at s in v is permitted. Moreover, if the entry at a position h in the direct sum is m + n and a permitted insertion position s is strictly after h, every entry strictly between h and s is less than every entry at or after s. Positions are numbered from zero.

References