### The expected number of inversions after n adjacent transpositions

*Mireille Bousquet-Melou*

#### Abstract

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

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