Questions tagged [combinatorics]
For questions about the study of finite or countable discrete structures, especially how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.
61,217 questions
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Number of sequences summing to $n=2026$ with $\gcd(a_i, a_{i+1}) = 1$ and $\gcd(a_1, a_k) = 2$ [closed]
Problem
Let $n = 2026$.
A sequence of positive integers $(a_1, a_2, \dots, a_k)$ is called valid if:
$a_1 + a_2 + \cdots + a_k = n$,
$\gcd(a_i, a_{i+1}) = 1$ for all $1 \le i < k$,
$\gcd(a_1, a_k)...
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12
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3
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579
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A problem about random permutations
The following question was asked by another user and then deleted. I liked my answer to the post, so I have made another post with the question and the answer. I am paraphrasing the question here, ...
- 465
0
votes
1
answer
59
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Estimate in Bollobás "Modern Graph Theory" [closed]
In page 230 of "Modern Graph Theory" by Béla Bollobás, the following is defined:
Let $d=d(n,p)$ be the greatest natural number for which
$$\mathbb{E}(X_d) = \binom{n}{d}p^{\binom{d}{2}} \...
- 19
1
vote
1
answer
63
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Number of Dyck paths avoiding DUDUDD and not ending with DUDUD
I wish to know how many Dyck paths of semilength $k$ both avoid DUDUDD and do not end with DUDUD. I would certainly welcome a proof, but am more interested in understanding the reasoning behind the ...
- 319
0
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0
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23
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Graph enumeration based on unlabelled pairwise distance distribution of vertices [closed]
Given a 3D piecewise segment with finite number of vertices (around 100-1000), and an unlabelled pairwise distance distribution. Is there a way to systematically enumerate (or mathematically describe) ...
- 135
4
votes
1
answer
93
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A contest problem about Tic-Tac-Toe
Problem: Mahtab and Shuvo want to play the $3 × 3$ ordered “Tic-Tac-Toe”, where the cells are numbered. Mahtab won the toss and decided to start the game with a cross. In this case, how many
total ...
- 887
3
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1
answer
41
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Caro Wei bound with second neighbors in triangle free graphs
I've been struggling a bit with the following problem
Prove that in any triangle free graph one has
$$\alpha(G)\geq\sum_{v\in V}\frac{d(v)}{1+d(v)+d_2(v)}$$
where $d_2(v)$ is the number of vertices ...
- 5,646
3
votes
0
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67
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For which $n$ and $k$ does there exist a "cursed" centrifuge arrangement?
For background, say that a centrifuge has $n$ slots arranged in a circle and $k$ tubes are placed within it. This is equivalent to choosing $k$ distinct $n$-th roots of unity. The centrifuge is ...
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1
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1
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In how many ways a mouse can get in the $j$-th cell in the $i$-th row?
Suppose we have the following structure: there is $1$ cell in the first row, $2$ cells in the second row, ..., $k$ cells in the $k$-th row, ... (first picture):
A mouse stays in the cell in the first ...
- 655
7
votes
1
answer
203
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How many points are needed to cover a whole lattice?
Let $p, q\in n^2 = \{0, 1, \dots, n-1\}^2$ be points on the plane. Say "$p$ covers $q$" if the line segment from $p$ to $q$ intersect $n^2$ in no points other than $p$ or $q$ (they are in '...
- 7,398
5
votes
1
answer
137
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The number of strings with bounded partial sums
Let $n$ and $l$ be positive integers. How to find the number of strings $(x_1, x_2, ..., x_l)$ such that:
$($1$)$ $x_j \in \{-1, 0, 1\}$
$($2$)$ $\forall d \in \{1, 2, ..., l-1\}$ the following ...
- 655
9
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3
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394
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A problem about permutation group to find the minimum cardinality of a set
Let $A = \left\{ 1, 2, 3, \ldots, n \right\}$. Let $B$ be the set of all bijections from $A$ to $A$ (i.e., the symmetric group $S_n$). Let $C$ be a non-empty subset of $B$ such that for every ...
- 109
5
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1
answer
143
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Ramsey theorem where we only ask for "polychromatic" set?
Has this kind of problem been studied in (finite) combinatorics: given $m$ what is the least $n$ s.t. any $3$-coloring of $K_n$ contains a $K_m$ that uses at most $2$ colors? Can this "...
- 300
1
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1
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75
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Possible game states of card game Naishi
While playing Naishi with a friend of mine, we where wondering what the maximum points combination would be in said game. (After googling: people say it's 96) Still, I was interested in finding the ...
- 11
0
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0
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27
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Ultrafilter and relation to finite colouring [closed]
Let $S$ be a family of subsets of $\mathbb{N}$. Prove that the following are equivalent:
(i) Whenever $N$ is finitely coloured, some member of $S$ is monochromatic.
(ii) There is an ultrafilter $U$ ...
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