Ordered set partition posets

Abstract

A set partition is said to be ordered if the blocks of the partition are listed in a specific order. The ordered set partitions of $\{1,\ldots,n\}$, with a unique minimal element adjoined, form a lattice $\Om_n$ with respect to refinement. The lattice $\Om_n$ is well known to be the face lattice of the permutohedron. In this paper we study the combinatorics and topology of two subposets of $\Om_n$ with restricted block sizes, either all divisible by some fixed $d\ge2$, or all congruent to $1$ modulo $d$. For the $d$-divisible case we derive an explicit recursive atom ordering for the lattice, as well as formulas for the action of the symmetric group on the Whitney homology and the rank-selected homology, and also for the multiplicity of the trivial representation. In the 1 mod $d$ case we show that the poset has a curious interval structure related to the $k$-Catalan numbers. Our investigations lead to enumerative invariants in both cases. Open problems and avenues for future research are scattered throughout.