On the unimodality of convolutions of sequences of binomial coefficients

TL;DR

This paper establishes precise conditions for the unimodality of convolutions of binomial coefficient sequences, linking them to rank sequences of certain tree posets, and counts non-unimodal cases for fixed vertices.

Contribution

It provides necessary and sufficient criteria for unimodality of convoluted binomial sequences and characterizes non-unimodal tree posets with a fixed number of vertices.

Findings

Conditions for unimodality of convoluted binomial sequences

Characterization of non-unimodal tree posets

Enumeration of non-unimodal cases for fixed vertices

Abstract

We provide necessary and sufficient conditions on the unimodality of a convolution of two sequences of binomial coefficients preceded by a finite number of ones. These convolution sequences arise as as rank sequences of posets of vertex-induced subtrees for a particular class of trees. The number of such trees whose poset of vertex-induced subgraphs containing the root is not rank unimodal is determined for a fixed number of vertices $i$.