804 research outputs found
The expected number of inversions after n adjacent transpositions
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
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
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