Séminaire Lotharingien de Combinatoire, 93B.26 (2025), 12 pp.
Christian Gaetz and Yibo Gao
Bijections between Interlacing Triangles, Schubert Puzzles, and Graph Colorings
Abstract.
We show that \emph{interlacing triangular arrays}, introduced by Aggarwal-Borodin-Wheeler to study certain probability measures, can be used to compute various kinds of Schubert structure constants. We do this by establishing a splitting lemma, allowing for arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 and certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson-Tao puzzles. This proves one enumerative conjecture of Aggarwal-Borodin-Wheeler and disproves another.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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