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If you just care about asymptotics, then you can use central limit theorem or LDP techniques to get an exponential estimate. Here is how: let $x_1, \ldots, x_M$ be iid uniform random variables in $[0, …

I don't think constant is the only tempered growth solution to your 2d recurrence. Essentially the recurrence needs a 1-dimensional subspace in $\mathbb{Z}^2$ of boundary conditions. Then your recurre …

The answer is given in terms of inclusion exclusion principle, much as the solution for coupon collector's problem. Let $p_t$ be the probability that at time $t$ there are still two vertices with no e …

Consider the complete graph on $n$ vertices. Each step, one chooses one of the $\binom{n}{2}$ edges iid uniformly at random. Say a sequence of choice is successful if there is some permutation of the …

This question is concerned about the paper "The 3-XORSAT threshold" by O. Dubois, J.Mandler. Here is the link: http://dx.doi.org/10.1016/S1631-073X(02)02563-3 Basically one would like to know when is …

My question is most precisely stated in the title. As an example, if we consider base 10, and k=4, then I am asking, is it possible to have a sequence of length 10^4 + 3, such that each 4 digit number …

The process described is also known as the stick-breaking process for sampling the cycle type of a uniformly chosen permutation in $S_n$. So we have the set $[n]$ and we want to choose a uniformly …

Recall that a permutohedron is a graph on the set of permutations $S_n$ with an edge between $\sigma$ and $\tau$ if they differ by one adjacent transposition: $\tau = (i,i+1) \circ \sigma$ for some $i …

Got stuck on this one for months. Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k c …

I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the symmetri …

Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence configuration …

Markov chains on symmetric groups converging to various distributions other than uniform provide a fertile ground for the marriage between modern combinatorics such as Macdonald polynomials and hard a …

To deal with root systems of type B C D, one needs to understand symmetric Laurent polynomials $\Lambda$. I am wondering if the naive definition of power sum symmetric Laurent polynomials form a basis …

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's a …

For $k :=|\lambda| \ge r$, the statement that all $s_\lambda(x_1, \ldots, x_r)$ vanish is equivalent to all the elementary polynomials $e_j(x_1, \ldots, x_r) := \sum_{i_1 < \ldots < i_j} x_{i_1} \ldot …