146 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
The vertical profile of embedded trees
Consider a rooted binary tree with n nodes. Assign with the root the abscissa
0, and with the left (resp. right) child of a node of abscissa i the abscissa
i-1 (resp. i+1). We prove that the number of binary trees of size n having
exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is with n_{l-1}=n_{r+1}=0. The
sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the
tree. The vertical profile of a uniform random tree of size n is known to
converge, in a certain sense and after normalization, to a random mesure called
the integrated superbrownian excursion, which motivates our interest in the
profile. We prove similar looking formulas for other families of trees whose
nodes are embedded in Z. We also refine these formulas by taking into account
the number of nodes at abscissa j whose parent lies at abscissa i, and/or the
number of vertices at abscissa i having a prescribed number of children at
abscissa j, for all i and j. Our proofs are bijective.Comment: 47 page
Polynomial equations with one catalytic variable, algebraic series, and map enumeration
Let be a formal power series in with polynomial
coefficients in . Let be formal power series in ,
independent of . Assume all these series are characterized by a polynomial
equation We prove that, under a mild
hypothesis on the form of this equation, these series are algebraic,
and we give a strategy to compute a polynomial equation for each of them. This
strategy generalizes the so-called kernel method, and quadratic method, which
apply respectively to equations that are linear and quadratic in .
Applications include the solution of numerous map enumeration problems, among
which the hard-particle model on general planar maps
- …