Noncommutative crossing partitions

TL;DR

This paper introduces noncommutative crossing partitions, a generalization of non-crossing partitions, and explores their lattice structure, properties, and relations to other combinatorial objects, extending classical results and characterizations.

Contribution

It defines a new graded lattice of noncommutative crossing partitions, embeds the Kreweras lattice, and characterizes key maps using this new framework, advancing combinatorial lattice theory.

Findings

The lattice of noncommutative crossing partitions is graded and contains the Kreweras lattice.

Explicit EL-labeling allows calculation of Möbius function and chain counts.

Relations among trees, chains, and tilings are established.

Abstract

We define and study noncommutative crossing partitions which are a generalization of non-crossing partitions. By introducing a new cover relation on binary trees, we show that the partially ordered set of noncommutative crossing partitions is a graded lattice. This new lattice contains the Kreweras lattice, the lattice of non-crossing partitions, as a sublattice. We calculate the M\"obius function, the number of maximal chains and the number of $k$-chains in this new lattice by constructing an explicit $EL$-labeling on the lattice. By use of the $EL$-labeling, we recover the classical results on the Kreweras lattice. We characterize two endomorphism on the Kreweras lattice, the Kreweras complement map and the involution defined by Simion and Ullman, in terms of the maps on the noncommutative crossing partitions. We also establish relations among three combinatorial objects: labeled $k+1$-ary trees, $k$-chains in the lattice, and $k$-Dyck tilings.