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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.

108
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3k views

Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
50
votes
0answers
1k 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 ...
43
votes
0answers
2k views

Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? ...
37
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0answers
1k 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 ...
32
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0answers
455 views

Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
26
votes
0answers
1k 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\{...
23
votes
0answers
717 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 \...
21
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0answers
965 views

Blocking directed paths on a DAG with a linear number of vertex defects.

Let $G=(V,E)$ be a directed acyclic graph. Define the set of all directed paths in $G$ by $\Gamma$. Given a subset $W\subseteq V$, let $\Gamma_W\subseteq \Gamma$ be the set of all paths $\gamma\in\...
19
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0answers
289 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$, ...
19
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0answers
578 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. ...
17
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0answers
302 views

A combinatorial identity involving generalized harmonic numbers

[This problem has now an answer here.] 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}\...
16
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0answers
1k 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 ...
15
votes
0answers
616 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 ...
14
votes
0answers
364 views

Random sum in coupon collection

I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
14
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0answers
166 views

graph with many authomorphisms

Let $G(V,E)$ be an undirected graph, potentially with loops and multi-edges. Assume the following two properties hold: $\forall a \in V(G), \exists f \in aut(G), f(a) \ne a$ $\forall f\in aut(G), \...
14
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0answers
454 views

Who is the mathematician “Jacques” in this anecdote from Fadiman's “The Mathematical Magpie”?

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? ...
13
votes
0answers
361 views

Strategy for board game 2

In this question the following was asked: Alice and Bob are playing the following game: They have a $4 \times 4$ empty grid and take turns coloring one square each, starting with Alice, both using ...
13
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0answers
381 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 ...
13
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0answers
193 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. ...
13
votes
0answers
249 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,...
12
votes
0answers
311 views

When does a set of sums uniquely determine a set of values?

This is a follow up to a question about determining the set of values from a set of sums. We consider the following setup: Consider a vector $A$ (which you will not see) of $n$ positive integers. ...
12
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0answers
2k views

Does the “prime ant” ever backtrack?

A few mathematical questions have come up from the question "The prime ant 🐜" on the Programming Puzzles & Code Golf Stack Exchange. Here is how the prime ant is defined: Initially, we have ...
12
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0answers
403 views

How many distinct Unruly boards are there?

Unruly is a puzzle game played on a board of $2n × 2n$ squares. Each square must be colored either black or white, with the following constraints: Each column must have $n$ white squares and $n$ ...
12
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0answers
410 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}(\...
12
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0answers
305 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 ...
11
votes
0answers
204 views

All interval sequences mod integers

In music, an all-interval twelve-tone sequence is a sequence that contains a row of 12 distinct notes such that it contains one instance of each interval within the octave, 1 through 11. The more ...
11
votes
0answers
358 views

Every 4-regular simple graph contains a 3-regular subgraph

The following result was conjectured by Berge & Sauer, and proved by Tashkinov [T]. Theorem A. Every 4-regular simple graph contains a 3-regular subgraph. A simple graph is the one with no ...
11
votes
0answers
132 views

Given $a_1 \leq a_2 \leq \dots \leq a_n$, how many possible orders are there for $\{a_i + a_j\}$?

Suppose we have $n$ variables $a_1 \leq a_2 \leq \dots \leq a_n$. If the values of the variables are not known, in how many ways could $\{a_i + a_j\}_{\{i,j\} \in {[n] \choose 2}}$ be ordered using $&...
11
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0answers
449 views

Twin-prime sieve

My question concerns the following sieve (call it S), which was an exercise in applying some elementary aspects of Brun's sieve while reading Halberstam's text. Using the Chinese Remainder theorem ...
11
votes
0answers
348 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 ...
11
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0answers
346 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 ...
11
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0answers
208 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) = \...
10
votes
0answers
274 views

Maximum number of regions of a sphere partitioned by $\binom{n}{3}$ planes from $n$ points

We can place $n\in\mathbb{N}$ points on the surface of a sphere in a configuration so as to maximize the answer. A plane is defined by $3$ points. We create all $\binom{n}{3}$ planes from the $n$ ...
10
votes
0answers
176 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{...
10
votes
0answers
311 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 I ...
10
votes
0answers
345 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
205 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 ...
10
votes
0answers
370 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 ...
10
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0answers
243 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
736 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
271 views

Number of simultaneously solvable linear equations with nonnegative variables

Let $N$ variables $x_i \ge 0$ but not all of them be zero. One may fix $\sum_{i=1}^N x_i = 1$. There are $P$ equations which need to be solved, with coefficients $a^k_i$ indexed with superscripts $k =...
9
votes
0answers
134 views

Generating function for number of $r$-disjoint subsets each of size $k$

Fix $n, k$. Then $$ C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!} $$ is the number of ways to form $r$ disjoint subsets each of ...
9
votes
0answers
156 views

Prove that $10 \times 10$ grid filled with positive integers contains two elements sharing a side such that their difference is at least $6$.

I have a $10 \times 10$ grid filled with different positive integers and want to prove that some two numbers sharing a side in a grid differ by at least $6$. My solution is: Let $m$ be the smallest ...
9
votes
0answers
131 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 ...
9
votes
0answers
175 views

Given $100$ coplanar points, no $3$ collinear, then at most $70$ percent triangles formed using these points are acute-angled

(IMO-$1970$) Given $100$ coplanar points, no $3$ collinear, prove that at most $70$ percent of the triangles formed using these points are acute-angled. I know that one solution proceeds by showing ...
9
votes
0answers
101 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
236 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 ...
9
votes
0answers
195 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
915 views

Is there a Steiner system S(6,9,45)?

Background: the Belgian Lottery switched its main game this year to a draw of 6 balls out of a pool of 45, plus a bonus number which doesn't matter for the sake of this question. Assuming someone ...
8
votes
0answers
58 views

Can a subset be chosen that intersects with other subsets a given number of times?

Let $S$ be a set,$|S| = n$, with $n$ sufficiently large and divisible by $8$. Suppose that $A_1, \dots, A_{n/2} \subseteq S$, $|A_i| = \dfrac{n}{2}$ for all $i$. Is it always possible to choose a ...