MathOverflow

Consider the function $Dist: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ (natural numbers include $0$), by defining $Dist(E,N)$ to be the size of the set
$$ \{ (s_1, s_2, \ldots, s_N ) \in \mathbb{N}^N | \sum_{i=1}^N s_i = E\}.$$
Intuitively, $Dist(E,N)$ counts the ways of distributing $E$ coins to $N$ people, or distributing $E$ amount of energy quanta to $N$ different particles. I am interested in the behavior of $Dist(E,N)$ where $N$ is very large, and $E$ is roughly $100N...

$\newcommand{\R}{\mathbb{R}}$ $\DeclareMathOperator{\law}{Law}$
$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$Let $H$ be a Hilbert space equipped with the orthonormal basis $(e_i)_{i\ge1}$. Let $\xi=\sum_{k=1}^\infty \sigma_ke_k\xi_k$ be a Gaussian random variable on this space; here $\sigma_i\in\R$, and $\xi_i$ are...

The Question is simple, yet I have encountered it multiple times in my mathematical life without finding an obvious answer, so I've decided to post it here.
Say $U, V$ are real vector spaces of dimension $m,n$. What is the largest possible dimension of a subspace $W\subset U \otimes V$ so that $W$ contains no nonzero element of the form $u \otimes v$?...

A set of $k$ generators is called irredundant if it is minimal by inclusion, i.e. any $k-1$ of them generate a proper subgroup. I wonder if there's a sampling procedure for such families of generators for symmetric groups.
If not the irredundancy restriction, the product replacement algorithm would give an approximation to the uniform sampler (I believe a sequence of Nielsen moves can transform any family of $k$ generators to any other such family, so a long enough random...