Boustrophedon transform

Abstract: A generalization of the Seidel-Entringer-Arnold method for calculating the alternating permutation numbers (or secant-tangent numbers) leads to a new operation on integer sequences, the Boustrophedon transform.

Abstract: In this paper, we investigate impulse response sequences over the integers by presenting their generating functions and expressions. We also establish some of the corresponding identities. In addition, we give the relationship between an impulse response sequence and all linear recurring sequences satisfying the same linear recurrence relation, which can be used to transfer the identities among different sequences. Finally, we discuss some applications of impulse response sequences to the structure of Stirling numbers of the second kind, the Wythoff array, and the Boustrophedon transform.

Abstract: We give a short proof that the f-vector of the descent polytope $${{\,\mathrm{DP}\,}}_{\mathbf {v}}$$ is componentwise maximized when the word $$\mathbf {v}$$ is alternating. Our proof uses an f-vector analog of the boustrophedon transform.

Abstract: For a subsetS, let the descent statistic β(S) be the number of permutations that have descent setS. We study inequalities between the descent statistics of subsets. Each subset (and its complement) is encoded by a list containing the lengths of the runs. We define two preorders that compare different lists based on the descent statistic. Using these preorders, we obtain a complete order on lists of the form (k i ,P,k n−i , whereP is a palindrome, whose first entry is larger thank. We prove a conjecture due to Gessel, which determines the list that maximizes the descent statistic, among lists of a given size and given length. We also have a generalization of the boustrophedon transform of Millar, Sloane and Young.