Computing cardinalities of subsets on Sn with k adjacencies

Abstract

Given a permutation \( \pi = (\pi_1, \pi_2, \pi_3, \ldots, \pi_n) \) over the alphabet \(\Sigma = \{0, 1, \ldots, n-1\}\), \(\pi_i\) and \(\pi_{i+1}\) are said to form an adjacency if \(\pi_{i+1} = \pi_i + 1\) where \(1 \leq i \leq n-1\). The set of permutations over \(\Sigma\) is a symmetric group denoted by \(S_n\). \(S_n(k)\) denotes the subset of permutations with exactly \(k\) adjacencies. We study four adjacency types and efficiently compute the cardinalities of \(S_n(k)\). That is, we compute for all \(k\) \(|S_n(k)|\) for each type of adjacency in \(O(n^2)\) time. We define reduction and show that \(S_n(n-k)\) is a multiset consisting exclusively of \(\mu \in \mathbb{Z}^+\) copies of \(S_n(0)\) where \(\mu\) depends on \(n\), \(k\) and the type of adjacency. We derive an expression for \(\mu\) for all types of adjacency.