Abstract
In this paper, we study the distribution of the number of consecutive pattern matches of the five up-down permutations of length four, $1324$, $2314$, $2413$, $1432$, and $3412$, in the set of up-down permutations. We show that for any such $\tau$, the generating function for the distribution of the number of consecutive pattern matches of $\tau$ in the set of up-down permutations can be expressed in terms of what we call the generalized maximum packing polynomials of $\tau$. We then provide some systematic methods to compute the generalized maximum packing polynomials for such $\tau$.
Highlights
If σ = σ1 . . . σn is a permutation in the symmetric group Sn, we letDes(σ) = {i : σi > σi+1} and Ris(σ) = {i : σi < σi+1}.Let N = {0, 1, . . . , } denote the natural numbers, P = {1, 2, . . .} denote the set of positive integers, E = {0, 2, 4, . . .} denote the set of even numbers in N, and [n] = {1, 2, . . . , n} for n ∈ P
We study the distribution of the number of consecutive pattern matches of the five up-down permutations of length four, 1324, 2314, 2413, 1432, and 3412, in the set of up-down permutations
We show that for any such τ, the generating function for the distribution of the number of consecutive pattern matches of τ in the set of up-down permutations can be expressed in terms of what we call the generalized maximum packing polynomials of τ
Summary
These definitions are extended to sets of permutations Υ ⊆ Sj. For example, we say that σ has a Υ-match starting at position i in σ if red(σiσi+1 . We let GMP2n+1,τ(i) denote the set of σ ∈ S2n+1 which are generalized maximum packings for τ (i). We shall call GM Pn,τ(i)(x) the generalized maximum packing polynomial for τ (i) of length n. 2n n−2 where for any formal power series f (x) = n 0 fnxn, we write f (x)|xn the coefficient of xn in f Using these facts, we can compute the generating functions for the number of up-down permutations with no τ (1)-matches or with exactly one τ (1)-match.
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