Questions tagged [combinatorics]

Question 1

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 ...

Question 2

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 ...

Question 3

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 '...

Question 4

Suppose we have some permutation of the integers from $1$ to $n$. We start reading the permutation from left to right, and only select a number if it is greater than all the numbers to the left of it. ...

Question 5

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 ...

Question 6

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 ...

Question 7

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 "...

Question 8

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 ...

Question 9

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 $U$ ...

Question 10

For positive integers $n$, $m$, define $$[n]:=\{0, 1, \cdots n-1\}, [m]^{[n]}:=\{\text{Functions }f:[n]\to [m]\}$$ For functions $f\in [p]^{[q]}, g\in [q]^{[r]}$: $f\circ g\in [p]^{[r]}$ is the ...

Question 11

I have been trying to find a decent form for the generating function $\sum_{n=0}^\infty{n+k \brace n}x^n$ but it has remained largely untenable. There doesn't appear to be any useful recurrence here ...

Question 12

Let $0 \in \Bbb{N}$. Prime number theorists will be familiar with the formula: $$\rho(x) = \pi(x) - \pi(\sqrt{x}) + 1 = \sum_{d \ \mid\ \sqrt{x}\#}\mu(d)\lfloor\frac{x}{d}\rfloor$$ It is usually taken ...

Question 13

I have done research and found many articles and examples on counting, but didn't find something that handle counting under the constraint given in the following question. Could you please provide any ...

Question 14

In the study of some lattice paths I came across the following identities. I would be interested in a simple algebraic proof. $\sum_{j=0}^{n}(2j)\binom{n-j}{k}\binom{n+j}{k}=(k+1)\binom{n}{k+1}\binom{...

Question 15

this question is intended as a standard place to gather combinatorial interpretations, properties/relations and literature references to be linked to from other questions, in a similar vein to The sum ...

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Questions tagged [combinatorics]

For which $n$ and $k$ does there exist a "cursed" centrifuge arrangement?

In how many ways a mouse can get in the $j$-th cell in the $i$-th row?

How many points are needed to cover a whole lattice?

choosing numbers from a permutation

The number of strings with bounded partial sums

A problem about permutation group to find the minimum cardinality of a set

Ramsey theorem where we only ask for "polychromatic" set?

Possible game states of card game Naishi

Ultrafilter and relation to finite colouring [closed]

Bias amplification using constraint overlap [closed]

Ordinary generating functions for the sequence $a_n={n+k \brace n}$ where $k$ is fixed [closed]

Since $\pi(x) - \pi(\sqrt{x}) + 1 = \sum_{d \mid \sqrt{x}\#}\mu(d)\lfloor\frac{x}{d}\rfloor$ is discontinuous, is it everywhere-discontinuous?

Counting Number of Combinations under a constraint

Proof of some binomial identities [closed]

the sum $\sum_{j=k}^n{n\brack j}{j\brace k}x^j$

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