Convolution Preserves Partial Synchronicity of Log-concave Sequences

TL;DR

This paper investigates how convolution operations affect the synchronicity relations between log-concave sequences, disproving a previous conjecture and introducing a new, preserved partial synchronicity relation.

Contribution

It introduces the concept of partial synchronicity, which is preserved under convolution, contrasting with the previously conjectured weakly synchronicity.

Findings

Counterexample disproves convolution preserves weakly synchronicity.

Partial synchronicity is weaker than synchronicity but stronger than weakly synchronicity.

Convolution preserves the partial synchronicity relation.

Abstract

In a recent proof of the log-concavity of genus polynomials of some families of graphs, Gross et al. defined the weakly synchronicity relation between log-concave sequences, and conjectured that the convolution operation by any log-concave sequence preserves weakly synchronicity. We disprove it by providing a counterexample. Furthermore, we find the so-called partial synchronicity relation between log-concave sequences, which is (i) weaker than the synchronicity, (ii) stronger than the weakly synchronicity, and (iii) preserved by the convolution operation.