Non-Koszulness of operads and positivity of Poincar\'e series

TL;DR

This paper proves that a specific operad related to mock partially associative n-ary algebras is not Koszul, using hypergeometric summation techniques, and shows that its non-Koszulness cannot be detected by negative coefficients in its Poincaré series inverse.

Contribution

It confirms the non-Koszulness of the operad of mock partially associative n-ary algebras and introduces a novel approach using Zeilberger's algorithm to analyze Poincaré series.

Findings

The operad of mock partially associative n-ary algebras is not Koszul.

Negative coefficients in the inverse Poincaré series do not indicate non-Koszulness.

Zeilberger's algorithm can be used to study properties of operad series.

Abstract

We prove that the operad of mock partially associative $n$-ary algebras is not Koszul, as conjectured by the second and the third author in 2009, and utilise the Zeilberger's algorithm for hypergeometric summation to demonstrate that non-Koszulness of that operad cannot be established by hunting for negative coefficients in the inverse of its Poincar\'e series.