Enumeration of crossings in two-step puzzles

TL;DR

This paper derives a boundary-dependent formula for counting specific configurations in two-step puzzles, which are tilings used to compute Schubert structure constants in algebraic geometry.

Contribution

It introduces a new formula for counting occurrences in two-step puzzles based solely on boundary conditions, advancing combinatorial methods in algebraic geometry.

Findings

Derived a formula for counting configurations in two-step puzzles

Connected puzzle configurations to Schubert structure constants

Provided a proof based on color map tilings

Abstract

We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.