Interlacing Triangles, Schubert Puzzles, and Graph Colorings

We show that interlacing triangular arrays, introduced by Aggarwal–Borodin–Wheeler to study certain probability measures, can be used to compute structure constants for multiplying Schubert classes in the K-theory of Grassmannians, in the cohomology of their cotangent bundles, and in the cohomology of partial flag varieties. Our results are achieved by establishing a splitting lemma, allowing for interlacing triangular arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 with certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson–Tao puzzles. Along the way, we prove one enumerative conjecture of Aggarwal–Borodin–Wheeler and disprove another.