Colored partitions of a convex polygon by noncrossing diagonals

TL;DR

This paper develops a combinatorial framework for counting colored noncrossing partitions of convex polygons into polygons with side counts constrained by modular conditions, providing recurrence relations and explicit formulas.

Contribution

It introduces a novel enumeration method for colored polygon partitions with modular constraints, including recurrence relations and explicit formulas using Bell polynomials.

Findings

Derived recurrence relations for counting partitions.

Provided explicit formulas using partial Bell polynomials.

Enabled efficient incorporation of polygon restrictions.

Abstract

For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions.