The expected number of inversions after n adjacent transpositions

Mireille Bousquet-Mélou (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.

Mots clés: Permutations; Markov chains; inversions

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

Mireille Bousquet-Mélou. The expected number of inversions after n adjacent transpositions. Discrete Mathematics and Theoretical Computer Science, 2010, 12 (2), pp.65--88. ⟨hal-00412143v2⟩