Consecutive Patterns: From Permutations to Column-Convex Polyominoes and Back

Abstract

We expose the ties between the consecutive pattern enumeration problems associated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column-convex polyominoes, and of words to be applied directly to the enumeration of permutations by consecutive patterns. We deduce a host of new consecutive pattern results, including a solution to the $(2m+1)$-alternating pattern problem on permutations posed by Kitaev.