Complexes of pattern-avoiding injective words

Abstract

The complex of injective words is a cell complex that arises in a number of different areas. It has applications to proving homological stability and to the study of group cohomology, and it is closely related to the random-to-random Markov chain. This complex was first studied by Farmer, who proved it has the homology of a wedge of top-dimensional spheres. Later, Björner and Wachs established its shellability, and Reiner and Webb uncovered its $S_n$-module structure, observing in the process that the rank of its top homology group is the $n$th derangement number. We introduce natural subcomplexes of the complex of injective words by fixing a permutation pattern $σ$ and considering only those injective words in the alphabet $\{1,2,\dots,n\} $ that avoid $σ$. We prove that such pattern-avoiding complexes are shellable if $σ$ begins or ends with its largest or smallest letter, and we construct homology bases for the complexes avoiding such patterns. For patterns of length 3, all of which have this property, we show that the rank of the top homology of the resulting complex is a Riordan number. All but four patterns of length 4 also have this property, and for two of the remaining four patterns, we establish shellability using a different method. We also introduce a technique to use enumerative combinatorics to prove shellability, and we apply it to the complex of separable injective words, thereby deducing shellability in this case. Along the way, we give a combinatorial formula for all of the $h$-numbers in the full complex of injective words as well as for each of the subcomplexes which we prove are shellable. Going in the other direction, we use shellability of complexes of pattern-avoiding injective words to deduce new refined counting formulas for pattern-avoiding permutations.