k-indivisible noncrossing partitions

Henri Mühle (1) , Philippe Nadeau (2, 3, 4) , Nathan Williams (5)

1 Technische Universität Dresden, Institut für Algebra
2 CTN - Combinatoire, théorie des nombres
3 ICJ - Institut Camille Jordan
4 CNRS - Centre National de la Recherche Scientifique
5 UT Dallas - University of Texas at Dallas [Richardson]

For a fixed integer k, we consider the set of noncrossing partitions, where both the block sizes and the difference between adjacent elements in a block is 1 mod k. We show that these k-indivisible noncrossing partitions can be recovered in the setting of subgroups of the symmetric group generated by (k+1)-cycles, and that the poset of k-indivisible noncrossing partitions under refinement order has many beautiful enumerative and structural properties. We encounter k-parking functions and some special Cambrian lattices on the way, and show that a special class of lattice paths constitutes a nonnesting analogue.

Domaines: Combinatoire [math.CO]

Henri Mühle, Philippe Nadeau, Nathan Williams. k-indivisible noncrossing partitions. Seminaire Lotharingien de Combinatoire, 2020. ⟨hal-02094383⟩