Expected genomic dissimilarity

Permutation, a discrete structure, is a sequence over the corresponding alphabet Σ where every element of Σ occurs exactly once. The set of permutations over Σ forms a symmetric group denoted by S
_n
. In evolutionary biology, permutation models a genome. If π = (π
₁
, π
₂
, π
₃
,..., π
_n
) is a permutation over Σ = {0, 1, ... ,n - 1} then π
_i
, and π
_i
+1 form an adjacency if π
_i
+1 = π
_i
, + 1, 0 ≤ i ≤ n-1. Various operations that model genomic mutations are defined over permutations. An operation is defined by a set of generators. Execution of a specific generator is a move. The distance between a pair of permutations α and ß under a given operation ⊕ is the minimum number of moves that are required to transform a into ß. It denotes genomic dissimilarity between a and ß. Computation of distance is intractable for various operations including transposition. We call prefix transposition, suffix transposition and transposition as block-moves. Based on properties of S
_n
related to adjacencies we develop a model that estimates the expected block-move distance between any pair of permutations.