On pattern avoiding indecomposable permutations

Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating 12 ••• k-avoiding indecomposable permutations for k ≥ 3. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies

Some statistics on the hypercubes of Catalan permutations

For a permutation σ of length 3, we define the oriented graph Qn(σ). The graph Qn(σ) is obtained by imposing edge constraints on the classical oriented hypercube Qn, such that each path going from 0^n to 1^n in Qn(σ) bijectively encodes a permutation of size n avoiding the pattern σ. The orientation of the edges in Qn(σ) naturally induces an order relation ≼_σ among its nodes. First, we characterize ≼_σ. Next, we study several enumerative statistics on Qn(σ), including the number of intervals, the number of intervals of fixed length k, and the number of paths (or permutations) intersecting a given node