From Motzkin to Catalan permutations

AbstractFor every integer j⩾1, we define a class of permutations in terms of certain forbidden subsequences. For j=1, the corresponding permutations are counted by the Motzkin numbers, and for j=∞ (defined in the text), they are counted by the Catalan numbers. Each value of j>1 gives rise to a counting sequence that lies between the Motzkin and the Catalan numbers. We compute the generating function associated to these permutations according to several parameters. For every j⩾1, we show that only this generating function is algebraic according to the length of the permutations