Cliff operads: a hierarchy of operads on words

TL;DR

This paper introduces a new hierarchy of combinatorial operads based on words called $ ext{delta}$-cliffs, revealing their complex structure, relations, and applications to various combinatorial objects.

Contribution

It constructs novel operads on $ ext{delta}$-cliffs, analyzes their properties, and applies them to permutation spaces and path structures, extending the theory of combinatorial operads.

Findings

Operads are infinitely generated with nonquadratic, nonhomogeneous relations.

Operads can be constructed on various combinatorial objects like permutations and paths.

Koszul duals of some operads generalize duplicial and triplicial operads.

Abstract

A new hierarchy of operads over the linear spans of $\delta$-cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of $\delta$-cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite to the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, $m$-increasing trees, $c$-rectangular paths, and $m$-Dyck paths with operad structures. The operads on $c$-rectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads.