All Questions

Is there a known expression for, or a nontrivial upper bound on, the number of permutations in $S_k$ with longest increasing subsequence of length at most $n$? Let $l(\sigma)$ denote the length of the ...

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices. I am interested in the following sequence which showed up in a calculation I was doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...