$k$-Indivisible Noncrossing Partitions
Abstract
For a fixed integer , we consider the set of noncrossing partitions, where both the block sizes and the difference between adjacent elements in a block is . We show that these -indivisible noncrossing partitions can be recovered in the setting of subgroups of the symmetric group generated by -cycles, and that the poset of -indivisible noncrossing partitions under refinement order has many beautiful enumerative and structural properties. We encounter -parking functions and some special Cambrian lattices on the way, and show that a special class of lattice paths constitutes a nonnesting analogue.
Cite
@article{arxiv.1904.05573,
title = {$k$-Indivisible Noncrossing Partitions},
author = {Henri Mühle and Philippe Nadeau and Nathan Williams},
journal= {arXiv preprint arXiv:1904.05573},
year = {2021}
}
Comments
23 pages, 7 figures. Updated final version incorporating a result suggested by Christian Krattenthaler on the rank-enumeration of m-divisible k-indivisible noncrossing partitions. This resolves the conjectures from Section 8.2 in the previous versions. Comments welcome