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For new exact formulas for these quantities, and their asymptotics, you maybe should see the recent papers http://arxiv.org/abs/1109.1412 http://arxiv.org/abs/1208.3443 (Theorem 1.2) and especially …

Yes, if the tree is rooted and planar (that is, children at each vertex are linearly ordered), you can construct its contour function. See, for example, http://www.math.ens.fr/~legall/Rio-lectures.pdf …

A little off-topic: Gessel-Viennot's lemma was also discovered by Karlin and McGregor (1959, "Coincidence Probabilities") and was used to construct dynamics of noncolliding systems of particles: take …

There is a result by Sergei Kerov (in his book Asymptotic representation theory of the symmetric group and its applications in analysis) which somewhat charaterizes the Macdonald symmetric functions. …

q-Catalan numbers are defined recurrently as C0=1, $C_{N+1}=\sum_{k=0}^N q^k C_k C_{N-k}$. What can be said about the asymptotics of Cn when 0<q<1? P.S. In the case q>1 it is known that as n goes to …

Suppose we are talking about graphs with $n$ labeled vertices. Which graphs are more common: connected or disconnected?

Let $X_1,\ldots ,X_N$ be independent identically distributed random variables (or, more generally, exchangeable random variables, but let's assume independence for simplicity). Then $$ E(X_i\mid X_1+ …

I'm trying to set straight various $q$-deformations of the standard geometric distribution. The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has $$ \mu_1(X=j)=(1-p)p^j,\q …