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Results tagged with co.combinatorics
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user 9025
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
1
vote
Maintaining a search-optimal tree
See this survey paper for several types of binary tree balancing. You will see things quite similar to your suggestions. However, I'll also note that what you call "search optimal" is not much use a …
- 37k
11
votes
Combinatorial Interpretation
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 …
- 37k
4
votes
Accepted
Hitting set problem variant
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 …
- 37k
3
votes
An asymptotic question
The OP doesn't need this case so I'll just give a sketch. The values
$$p_k = (1-x)^{n-1} x^k\binom{n+k}{n}$$
sum to 1 over $k\ge 0$, so they are the probabilities of a nonnegative integer random vari …
- 37k
4
votes
How many binary $N \times N$ matrices exist with a given row and column sum
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 …
- 37k
5
votes
Number of tuples satisfying the following condition
First I will assume that you don't count $0!$ and $1!$ as different.
If $s$ is a fixed number, and $M\to\infty$, the asymptotic number of solutions is
$$\binom{M}{\lfloor \log_2 s\rfloor}.$$
Proof: …
- 37k
4
votes
Asymptotic number of certain functions without fixed points
Here is a generalization of the derangements case.
Take a Latin rectangle with $n$ columns and $k\le n^{1-\epsilon}$ rows, such that, for each $i$, the value $i$ does not appear in the $i$-th column. …
- 37k
1
vote
Restriction sum of multinomial coefficients
Here is how to estimate
$$Q(n,t) = \sum \binom{n}{k_1,\ldots,k_t}$$
with the sum over $k_1,\ldots,k_t\ge 1$ such that $k_1+\cdots+k_t=n$.
Let $X_1,\ldots,X_t$ be independent random variables with trun …
- 37k
-1
votes
Homomesy in perfect matchings
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 …
- 37k
5
votes
Accepted
Number of transitive relations on a set
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 …
- 37k
1
vote
What is this restricted sum of multinomial coefficients?
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 …
- 37k
2
votes
Given A set $U$ and a set $\mathcal O$ of subsets of $U$, how many subsets of $\mathcal O$ h...
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 …
- 37k
6
votes
Accepted
Growth constant limit for sum of products of two binomial coefficients
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
$$ \rho = \frac{729(2+3\alpha)^2}{64(1+\alpha)^2}
\left(\fra …
- 37k
4
votes
Counting equivalence relations with marked classes
If $B_{n,t}$ is the number of partitions of a set of size $n$, with $t$ parts marked (hopefully as desired, though I find the description unclear), then
$$ \sum_{n=0}^\infty \sum_{t=0}^\infty \frac{B_ …
- 37k
5
votes
Accepted
A multinomial-type sum over compositions of an integer
$$ 1 + \frac{(n+1)^{n-1}-1}{n!}.$$
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) …
- 37k