Consecutive patterns in permutations

Abstract

In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in $\text{S}{}_{\mathrm{n}}$. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of certain linear differential equations with polynomial coefficients. Our method is based on the representation of permutations as increasing binary trees and on symbolic methods.