Counting even and odd restricted permutations

Abstract

Let p be a permutation of the set N<inf>n</inf> = {1; 2· n}. We introduce techniques for counting N(n; k; r; I;π), the number of even or odd restricted permutations of N<inf>n</inf> satisfying the conditions-k≤ p(i)-iπ r (for arbitrary natural numbers k and r) and p(i)-i ∉ I (for some set I) andπ = 0 for even permutations andπ = 1 for odd permutations.