Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and k-ARY trees

Abstract

In this paper we compute the number of reduced decompositions of certain permutations σ ∈ S n as a product of transpositions (1, 2), (1, 3), …, (1, n). We present several combinatorial correspondences between these decompositions and combinatorial objects such as Catalan paths and k-ary trees.