Classical and vincular patterns of length three in generalized alternating permutations

Abstract

Let k be an integer at least 2, and let D_{N,k} be the set of permutations of {1,...,N} whose descent set is exactly {k, 2k, ..., k*floor((N-1)/k)}. We enumerate the elements of D_{N,k} avoiding each classical and each vincular pattern of length three. For classical patterns, we give recursive bijections from the 132- and 231-avoiding classes to ordered forests of complete k-ary trees, obtaining the Raney number. The 213- and 312-avoiding classes are obtained from these forest bijections by completing the last block and applying reverse-complement symmetry. The remaining classical case 321 is expressed by RSK. For vincular patterns, we enumerate the six fully consecutive patterns and the twelve patterns with exactly one adjacency. The closed-form results are accompanied by bijective models: the Catalan, Raney, Fuss-Catalan, and RSK cases are natural k-ary or fixed-descent extensions of classical bijections, while the product and poset cases arise from block-insertion and record/tree-poset encodings forced by the adjacency conditions.