Compositions constrained by graph Laplacian minors

TL;DR

This paper explores compositions constrained by graph Laplacian minors, providing generating functions for trees and cycles, and linking these to algebraic properties of reflexive simplices, thus advancing combinatorial and algebraic understanding.

Contribution

It offers a complete description of generating functions for Laplacian-minor constrained compositions on trees and cycles, extending prior work and answering open questions.

Findings

Derived multivariate generating functions for tree-constrained compositions.

Solved an open question on super convex compositions constrained by Laplacian minors.

Connected compositions constrained by leafed cycle Laplacians to reflexive simplices.

Abstract

Motivated by examples of symmetrically constrained compositions, super convex partitions, and super convex compositions, we initiate the study of partitions and compositions constrained by graph Laplacian minors. We provide a complete description of the multivariate generating functions for such compositions in the case of trees. We answer a question due to Corteel, Savage, and Wilf regarding super convex compositions, which we describe as compositions constrained by Laplacian minors for cycles; we extend this solution to the study of compositions constrained by Laplacian minors of leafed cycles. Connections are established and conjectured between compositions constrained by Laplacian minors of leafed cycles of prime length and algebraic/combinatorial properties of reflexive simplices.