k-indivisible noncrossing partitions

International audienceFor 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