Enumerating anchored permutations with bounded gaps

Say that a permutation of 1, 2, ..., n is k-bounded if every pair of consecutive entries in the permutation differs by no more than k. Such a permutation is anchored if the first entry is 1 and the last entry is n. We show that the generating function for the enumeration of k-bounded anchored permutations is always rational, mirroring the known result on (non-anchored) k-bounded permutations due to Avgustinovich and Kitaev. We then explicitly determine the recursive formulas of minimal depth for the number of anchored k-bounded permutations of n for k = 2 and k = 3, resolving a conjecture listed on the Online Encyclopedia of Integer Sequences (entry A249665). We additionally show that, asymptotically, the number of anchored k-bounded permutations of n is O (k(n)) as a function of n for a given k. (C) 2020 Elsevier B.V. All rights reserved.