Positive $m$-divisible non-crossing partitions and their Kreweras maps

TL;DR

This paper explores positive m-divisible non-crossing partitions and their Kreweras maps, providing combinatorial models, geometric realizations, enumeration of invariants, and establishing cyclic sieving phenomena across classical and exceptional types.

Contribution

It introduces new combinatorial and geometric models for positive m-divisible non-crossing partitions and their Kreweras maps, extending understanding to exceptional types and cyclic sieving.

Findings

Combinatorial realizations of partitions in classical types

Geometric models as pseudo-rotations on circles and annuli

Enumeration of invariant partitions under Kreweras map powers

Abstract

We study positive $m$-divisible non-crossing partitions and their positive Kreweras maps. In classical types, we describe their combinatorial realisations as certain non-crossing set partitions. We also realise these positive Kreweras maps as pseudo-rotations on a circle, respectively on an annulus. We enumerate positive $m$-divisible non-crossing partitions in classical types that are invariant under powers of the positive Kreweras maps with respect to several parameters. In order to cope with the exceptional types, we develop a different combinatorial model in general type describing positive $m$-divisible non-crossing partitions that are invariant under powers of the positive Kreweras maps. We finally show that altogether these results establish several cyclic sieving phenomena.