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If $B_{n,t}$ is the number of partitions of a set of size $n$, with $t$ parts marked (hopefully as desired, though I find the description unclear), then $$ \sum_{n=0}^\infty \sum_{t=0}^\infty \frac{B_ …

If $\mathcal{O}$ consists of all subsets of size 2, then you are counting graphs without isolated vertices, see A006129. My feeling is that in general this problem will be #P-hard, but don't ask me f …

An explicit formula for this was published about 30 years ago, but it was wrong. As the matter stands, there is no explicit formula. The values up to m=15 are here. The value for m=16 is known too, …

If I understand the way things are counted, the maximum possible value of $\mathrm{al}(M)$ is $\binom{n/2}{2}$. Since this is less than half $\binom{n}{2}$, there is no way to put a matching having n …

If $d\to\infty$ sufficiently much faster than $n\to\infty$, you can get an estimate of $M(n,d)$ using the central limit theorem. The uniform distribution on $\{0,1,\ldots,n\}$ has variance $n(n+2)/12$ …

Let $\alpha\approx 0.0493932733732$ be the positive zero of $621\alpha^3+1242\alpha^2+585\alpha-32$. Then the limit you want is probably $$ \rho = \frac{729(2+3\alpha)^2}{64(1+\alpha)^2} \left(\fra …

If independent variables $X,Y$ are distributed Binom$(n,p)$, Binom$(m,p)$, respectively, then $q_1$ is the probability that $X>Y$. If $mp,np$ are large and the line $X=Y$ is not too far from the poin …

One of the recent volumes of Knuth's "Art of Computer Programming" (maybe volume 4), has these sequences and some things like a generating function. As far as I know, the asymptotic behaviour is not k …

If you have a polynomial or sufficiently convergent power series $f(x)$, and you sum it over $x$ being each of the $k$-th roots of unity, then you get $k$ times the sum of the coefficients of the powe …

Seems like (but needs checking that) $$ \sum \frac{1}{\ell!} s(\ell,k) z^\ell t^k = \exp(t \sinh(z)). $$ That could probably be used to find other formulas, recurrences, etc. ADDED: http://oeis.org/A …

$$ 1 + \frac{(n+1)^{n-1}-1}{n!}.$$ For the record, I'll mention how I found this formula. First I wrote a Maple procedure for it (about 5 Maple statements). Then I noticed it seemed to be integer$(n) …

(Corrected and expanded, again!) As mentioned in the comments, the number of third permutations depends on the relationship between the first two. Asymptotically, the number of third permutations is $ …

There is no simple formula except for very small $k$ or $N-k$. The most general asymptotic formula, though it seems to have not appeared in print yet, is by Liebenau and Wormald and the references th …

If $P(n)$ is the number of partial orders, then $\log_2 P(n) = n^2/4 + o(n^2)$, an old result of Kleitman. Look in MathSciNet for many different sharpenings. Now if $T(n)$ is the number of transitive …

Let $I$ be a minimal set that intersects each $E_j$, where minimal means that no point can be removed from $I$ without it no longer intersecting each $E_j$. Take any $i\in I$. We know $i$ lies in some …