For questions about the study of finite or countable discrete structures, specifically 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.

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71
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0answers
3k views

Making Friends around a Circular Table

I have $n$ people seated around a circular table, initially in random order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every ...
25
votes
0answers
803 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
17
votes
0answers
194 views

Moving half of the nuts

An even number of nuts is divided into three nonempty piles. In each step, we are allowed to take half the nuts from a pile with an even number of nuts, and put them on another pile. Can we always ...
17
votes
0answers
718 views

How to find a total order with constrained comparisons

There are 25 horses with different speeds. My goal is to rank all of them, by using only runs with 5 horses, and taking partial rankings. How many runs do I need, at minumum, to complete my task? As ...
17
votes
0answers
618 views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for $i,j\in\{...
14
votes
0answers
363 views

Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
14
votes
0answers
416 views

Symmetric polynomials

I've got a seemingly simple question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots \...
12
votes
0answers
112 views

Covering pairs with permutations

Consider an $n \times n$ matrix $M_n$ with the following properties: Each row is a permutation of $A_n \equiv \{1, 2, ..., n\}$. Every ordered pair $(i,j)$, $i,j \in A_n$, $i \neq j$, appears as a ...
12
votes
0answers
215 views

When can we quit a game of War?

Consider the game of War. (The rules are below.) It would be nice to be able to end the game early. Suppose, for example, one player has 50 of the 52 cards. It is very likely that he's going to win. ...
12
votes
0answers
281 views

Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers?

Does anyone know a nice (combinatorial?) proof and/or reference for the following identity? $$\left( \frac{\alpha}{1 - e^{-\alpha}} \right)^{n+1} \equiv \sum_{j=0}^n \frac{(n-j)!}{n!} |s(n+1, n+1-...
11
votes
0answers
637 views

Minimal time gossip problem

The gossip problem (telephone problem) is an information dissemination problem where each of $n$ nodes of a communication network has a unique piece of information that must be transmitted to all the ...
11
votes
0answers
181 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
10
votes
0answers
370 views

Distributing groups of objects into boxes

How can I enumerate the number of ways of distributing distinct groups of identical objects (but various cardinality) into $k$ boxes such that at most one box is empty $(1)$ and no combination of ...
10
votes
0answers
150 views

In how many ways can the integers from $1$ to $n$ be divided into two groups with the same sum?

In how many ways can the integers $1,2,\ldots,n$ be divided into two groups with the same sum? I have tried calculating some of these values for small $n$, but cannot seem to find a pattern. Any ...
10
votes
0answers
223 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
10
votes
0answers
202 views

On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with $...
10
votes
0answers
180 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} e_{h}(\...
10
votes
0answers
132 views

Generalization of Gray codes

A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ...
10
votes
0answers
229 views

The right way to motivate lattice theory in a combinatorics class

I am attending a course on combinatorics. I was asked to present Möbius functions on lattices for this course. I was trying to look for a simple non-trivial problem that illustrates the need for ...
10
votes
0answers
229 views

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
10
votes
0answers
193 views

Reconstruction Conjecture and Partial 2-trees

Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old. Searching relevant literature,...
9
votes
0answers
152 views

Who is the mathematician “Jacques” in this anecdote?

Who is the mathematician "Jacques" in this anecdote, which I read on p. 260 of The Mathematical Magpie by Clifton Fadiman, who quotes it from the 1942 memoir The Last Time I Saw Paris by Elliot Paul? ...
9
votes
0answers
249 views

Interpretation of $\frac{22}{7}-\pi$

Integral and series proofs that $\frac{22}{7}>\pi$ We can prove that $\frac{22}{7}$ exceeds $\pi$ by using Dalzell integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ or its ...
9
votes
0answers
395 views

How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that ...
9
votes
0answers
72 views

Number of paths of a certain type in a triangular array

Consider the $X_n$ set of finite integer sequences $(x_1, \ldots, x_{n})$ of length $n$ for which $x_1 = 0$ and $|x_k - x_{k+1}| = 1$ for each $k \in \{1, \cdots, n-1 \}$. Obviously $|X_n| = 2^{n-1}$. ...
9
votes
0answers
248 views

Number of sets of vertices whose union of neighbours contains exactly $k$ vertices

Suppose a bipartite graph $g$ consisting of $2n(n-1),n\in\Bbb N,n>1$ vertices, is divided equally into two colors: red and blue, and is constructed as follows: For example, $g$ for $n=3$: If ...
9
votes
0answers
149 views

How many different paths to reach $(p,q,r)$ from $(0,0,0)$ without intersection

Problem: In $\mathbb{Z}^3$ starting from $(0,0,0)$ we try to reach $(p,q,r)$ with a sequence of moves. In each step we make a move from a point to another point under following conditions: You can ...
9
votes
0answers
279 views

History of a combinatoric problem: exchanging numbers by throwing stones

Another user recently asked a question on the Puzzling stack: Two spies throwing stones into a river. Suitably generalised, it becomes: Two spies (Alice and Bob) need to exchange a message. Each ...
9
votes
0answers
157 views

To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?

In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees. The idea to construct such a bijection stems from looking at some ...
9
votes
0answers
245 views

Combinatorial Proofs of Real Analysis Identity

In this question,a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
9
votes
0answers
289 views

Extracting an (almost) independent large subset from a pairwise independent set of Bernoulli variables

Let $n>1$, and let $X_1,X_2, \ldots ,X_n$ be non-constant random variables with values in $\lbrace 0,1 \rbrace$. Let us say that a subset of variables $X_{i_1},X_{i_2}, \ldots,X_{i_d}$ is complete ...
9
votes
0answers
546 views

How strong is the statement that Thompson F is amenable?

Justin Moore's proof turned out to have an error I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
9
votes
0answers
176 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = \...
8
votes
0answers
95 views

How many ways can I arrange the numbers $1$ to $N$ with this divisibility condition?

For the numbers $1, \ldots, N$, how many ways can I arrange them such that either: The number at $i$ is evenly divisible by $i$, or $i$ is evenly divisible by the number at $i$. Example: for N = 2$...
8
votes
0answers
93 views

The smallest number that if multiplied by 2 forms a permutation of itself

I am looking for the smallest number larger than $0$ which when multiplied by $2$, forms a permutation of itself. I quickly remembered that the number $142,857$ does that, as well as with all numbers ...
8
votes
0answers
107 views

If $n={k^2 \choose k}$, then what is $k$?

Given that: $$n={k^2 \choose k}$$ what is $k$ as a function of $n$? So far, I have found the following approximation: $$ n \approx (k^2)^k = (k^k)^2 $$ $$ \sqrt{n} \approx k^k $$ If we take this ...
8
votes
0answers
87 views

A combinatorial proof by tesselation of the plane.

Some days ago the following problem was posed in the site: given a set of $N$ points in the plane such that for each pair of points $p,q$ we have $\lVert p-q\rVert >1$, prove there is a subset of ...
8
votes
0answers
294 views

Rotations of a tetrahedron

Let $P$ be a tetrahedron inside an sphere such that all of its vertices are on the surface of the sphere. Suppose that three quarters of sphere's surface is colored black. Show that there is a ...
8
votes
0answers
366 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any ...
7
votes
0answers
109 views

A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$ H_n=\sum_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}. $$ Recently, I have found ...
7
votes
0answers
87 views

What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix $...
7
votes
0answers
128 views

Minimal number of animals in a matching card game

I saw a card game designed for small children. Each card has a picture of 6 animals on it, and there are 31 cards. When any two cards are compared to each other, they share exactly one animal. The ...
7
votes
0answers
113 views

Counting the size of the largest sets of independent strings

This question derives from a PPCG coding challenge I posed previously. For a given positive integer $n$, consider all binary strings of length $2n-1$. For a given string $S$, let $L$ be an array of ...
7
votes
0answers
84 views

Showing that poset of set of supports of a vector space is semimodular

Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by $\text{supp}(...
7
votes
0answers
192 views

How to maximize the number of operations in process

In my research project I have encountered the following problem, concerning a tuple of words in the formal language $L=\{0,1\}^*$, with $\epsilon$ denoting the empty word. If we are given an ordered ...
7
votes
0answers
166 views

Check whether a polynomial ideal is prime in the power series ring

I would like to know whether the ideal $I = \langle y^{2}(y^{2}-x^{2}) + w^{7}, y^{2}(y^{4}-x^{4}) + z^{7}\rangle$ is prime in $\mathbb{C}[[x,y,z,w]]$, the ring of formal power series in the ...
7
votes
0answers
89 views

Has this subset-sum game been studied?

Consider the following game: two players, Yolanda (who always goes first) and Zachary, take turns selecting (not yet chosen) numbers between $1$ and $9$. The first player who can make three of their ...
7
votes
0answers
108 views

Finding a separating family of subsets of $[n]$ of size $n+1$.

I have this friend who always tells me problems I can't solve. Here is the latest one. We are given a family $\mathcal F$ of at least $2^{n-1}+1 $ subsets $[n]$. We must prove that we can $\color{...
7
votes
0answers
53 views

Need a feedback on my solution (logic) of the combinatorial problem which involves ordinary deck of cards

So, here is the problem: An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. Compute the probability that each pile has exactly 1 ace. My solution: $$\frac { {4 \...
7
votes
0answers
113 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...