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Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equ …

Interestingly, the probability that $t$ occurs for a standard die $n=6$ as $t\to\infty$ is $\frac27$, with exponential convergence towards that value. This is a standard generating function problem. T …

The number of ordered pairs of commuting functions is A181162. I agree with those counts up to n=7. There is little in OEIS that helps to answer the asymptotics question. Incidentally, the probabil …

I think the expected time for stumbling across the solution is roughly proportional to the number of configurations. The process you describe is walking randomly on a Cayley graph. The limiting dist …

As marcoromito wrote, this is an elementary calculation. However, I thought I would record a nice approximation that I stumbled across. Whether it is new, I have no idea. ADDED: The following sente …

Just expand $ln(X+\alpha)$ as a Taylor series about $X=np$ and then do the binomial sum term by term. Use tail bounds on the binomial distribution to show that the error terms are meaningful. Without …

Take the case of choosing edges independently with probability $p=n^{-2+\epsilon}$. As you say, it won't make much difference compared to choosing $n^{1+\epsilon}$ edges. Assume $\epsilon<\frac12$. …

James correctly identified percolation theory as the place where something like this is studied seriously. But let's do an elementary calculation. Each possible path consists of $4n-1$ squares and i …

Consider the model where a random graph is made by adding one edge at a time chosen uniformly at random from edges not yet present. Łuczak, Pittel and Wierman (1994) showed that there is a function $f …

For $3\le d\le n-4$ the group size is almost always 1. The next most likely group size is 2, which most probably occurs due to a transposition (I don't know where this is proved formally). There is no …

It's unlikely that there is an exact answer simpler than the inclusion-exclusion sum. One can also use much the same calculations to show that the number of matched pairs adjacent to each other is asy …

EDITED: As pointed out by Anthony and John, my 2am solution was anything but. In summary, the conjecture is TRUE for $C$ smaller than approximately 0.6880137 and false for larger $C$. The exact value …

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$ …

There are $$ \frac{\prod_{i=0}^{k-1}(n-i)^2}{k} $$ possible directed cycles with $2k$ vertices. Each such cycle occurs with probability $n^{-2k}$, so the exact expectation of the number of cycles is …

For the case $\lambda_1=\cdots=\lambda_n=1$, the answer is $1/n$ by symmetry. Not using symmetry, $Y = \sum_{j\ne i} X_j^2$ has the distribution $\chi_{n-1}^2$. Now look up the $\chi^2$ density and f …