Interlacing triangles, Schubert puzzles, and graph colorings

TL;DR

This paper establishes a deep link between interlacing triangular arrays (originating in LLT/puzzle contexts) and Schubert calculus across multiple cohomology theories. It provides a splitting lemma that reduces high-rank interlacing structures to lower-rank components and builds a chain of bijections: interlacing arrays to 1/2/3-puzzles, puzzles to triangular-grid colorings, and extensions to square-grid edge-labelings. These tools yield explicit combinatorial formulas for structure constants in $H^*(Gr(d,n))$, $K(Gr(d,n))$, and localized $H^*_{\mathbb{C}^\times}(T^*Gr(d,n))$, and for pullbacks in partial flag varieties, with geometric interpretations tied to puzzle counts. The work confirms a rank-3 equinumerosity conjecture with colorings, refutes the analogous rank-4 conjecture, and provides a unified framework for multi-fold products without iterative binary product rules. Overall, the results bridge combinatorics, puzzle theory, and Schubert calculus, offering new counting interpretations for a range of geometric structure constants.

Abstract

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.

Figures (7)

• Figure 1: An interlacing triangular array $T$ of rank $3$ and height $4$.
• Figure 2: Interlacing triangular arrays with top row $\lambda^{(1)}=3434,\lambda^{(2)}=2323,\lambda^{(3)}=1212,\lambda^{(4)}=4141$.
• Figure 3: Interlacing triangular arrays with top row $\lambda^{(1)}=3434,\lambda^{(2)}=2323,\lambda^{(3)}=1212,\lambda^{(4)}=4411$.
• Figure 4: A $1/2/3$-puzzle $P$ with boundary conditions $(1213, 1332, 1232)$.
• Figure 5: The bijection $\mathscr{P}$ from \ref{['thm:counting-m3']}.

Theorems & Definitions (53)

• Conjecture 1.1: Conj. A.3 of Aggarwal--Borodin--Wheeler Aggarwal-Borodin-Wheeler
• Theorem 1.2
• Conjecture 1.3: Conj. A.5 of Aggarwal--Borodin--Wheeler Aggarwal-Borodin-Wheeler
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Example 1.9
• Definition 2.1: Aggarwal--Borodin--Wheeler Aggarwal-Borodin-Wheeler