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Results tagged with co.combinatorics
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user 7076
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
1
vote
How do I determine the number of "second degree" derangements?
The answer depends on the cycle structure of $D_1$. Let $n:=|S|$ and $c_i$ be the number of cycles of length $i$ in $D_1$ (with $\sum_i ic_i=n$). Since $D_1$ is a derangement, we have $c_1=0$, but wha …
- 29.5k
0
votes
Count of binary matrices that avoids a certain sub-matrix
It is worth to mention that if we forbid also $2\times 2$ submatrices of all zeros, then there will be no such matrices as soon as $m,n\geq 5$.
In other words, every binary $5\times 5$ matrix contains …
- 29.5k
0
votes
What is this restricted sum of multinomial coefficients?
Another way to approach the original problem is to recall the formula:
$$\cos(y)^k = \frac{1}{2^k} \sum_{j=0}^k \binom{k}{j}\cos((k-2j)y).$$
Plugging in $y=\frac{\pi}{2} - x$ would give an expansion f …
- 29.5k
3
votes
Sequences without repeated objects
See
L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, Sequences with adjacent elements unequal. Aequationes Mathematicae 6:2-3 (1971), 256-262.
http://dx.doi.org/10.1007/BF01819761
P.S. I have a PARI/GP …
- 29.5k
3
votes
Accepted
Sum over integer compositions of $m$ with $n$ parts of a fixed monomial in the parts
I'm not sure if there is anything simpler than $f_m(a_1,\dots,a_n)$ being the coefficient of $x^m$ in the product of polylogarithms:
$$\mathrm{Li}_{-a_1}(x)\cdot \mathrm{Li}_{-a_2}(x)\cdots \mathrm{Li …
- 29.5k
1
vote
Accepted
Intersection of members in a separating union-closed family of sets
Statement (3) is easier to prove directly by induction on $n=|U(\mathcal{F})|$.
The base case $n=1$ is trivial.
To make the induction step for $n>1$, let $x\in U(\mathcal{F})$ be an element that bel …
- 29.5k
6
votes
An Operation on Multisets
But I don't have a clear proof that the sequence always terminates in a loop. – Martin Erickson
Here is a proof that the sequence always terminates in a loop.
Let $A, B$ be consecutive arrays …
- 29.5k
5
votes
Accepted
Order of a combination when mapping them to whole numbers
Let $N(n;a_1,\dots,a_k)$ where $0\leq a_1 < a_2 < \dots < a_k < n$ be the order number of $(a_1,\dots,a_k)$ as a combination from ($n$ choose $k$).
Since there are exactly $\binom{n-1}{k-1}$ combinat …
- 29.5k
2
votes
A Graph-Theory Related Question
Let $p$ be a path consisting of $m$ shaded unit squares (where every two adjacent shaded squares share a side). Define a binary string $B_p=b_0b_1b_2\dots b_m$, where $b_0b_1=10$ and for $i>1$, $b_i=1 …
- 29.5k
5
votes
Long identity-free sequences of permutations
Construct a bipartite graph $G$ where one part is $[n]$ and the other is $[k]$ such that there is an edge between $i\in[n]$ and $j\in[k]$ iff $i\in A_j$. Then $A_1, \dots, A_k$ are identity free if $G …
- 29.5k
5
votes
A Bernstein-like Combinatorial Sum
First off, it is always worth to remove common factors (not depending on the index of summation) from the summands. The given sum is reduced to (I also assume $k>0$ to have summation start from $j=0$) …
- 29.5k
1
vote
Hitting set problem variant
Let $\mathcal{E} = \bigcup_{k=1}^m E_k.$
For each $j\in\mathcal{E}$, let $A_j = \{ k\in [1,m] : j\in E_k \}$. Then the anticipated subset $I\subset\mathcal{E}$ should satisfy the following requiremen …
- 29.5k
1
vote
Accepted
Deriving a closed form for rolling a sum $n$ with $k$ dice using stars and bars
Answer is given by the coefficient of $z^n$ in
$$(z+z^2+\dots+z^6)^k = \left(z\frac{1-z^6}{1-z}\right)^k = z^k (1-z^6)^k(1-z)^{-k}.$$
An explicit formula for this coefficient is:
$$\sum_{i=0}^{\min(k …
- 29.5k
2
votes
Resolution of multiple edges
First off, let me reformulate the problem. I call edges of $G$ black. Let $K_{k,n}$ be the complete graph on the same partite sets $V_1, V_2$, whose edges I will refer to as red. Let $H$ be the superp …
- 29.5k
2
votes
How to do the sum over integer compositions
Let $n$ be fixed.
The sum in question can rewritten as
$$S_k:=\frac{1}{(n-1)!}\sum_{L=1}^k\sum_{r_1+\dots+r_L=k} (n+3k-L)!\cdot \alpha^{k-L}\cdot f(n,k,L),$$
where $\alpha:=-\frac{a}{a+1}$ and
$$f(n,k …
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