This tag is for basic 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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Expectation of Reciprocals Involving Permutations

Let $a_i$ be $n$ distinct real numbers. What is the expectation: $$\mathbb E_\sigma \left[ \sum_{i=1}^{n} \frac {1} {a_{\sigma(i)} - \sum_{j=1}^{i-1}a_{\sigma(j)}} \right] $$ where the expectation ...
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201 views

There are $n$ horses. At a time only $k$ horse can run in the single race. How many minimum races are required to find the top $m$ fastest horses?

There are $n$ horses. At a time only $k$ horses can run in the single race. How many minimum races are required to find the top $m$ fastest horses? Please explain your answer. PS: There is no timer.
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82 views

Optimal packing of number sets with limited overlap

When considering something completely different yesterday I came across the following problem: Let $X = \{ 1,2,3,...,n\}$. What is the maximal number of subsets of $X$ of order $m$ one can choose ...
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200 views

Generating function for product of binomial coefficients

In general, if $a(n)$ is an integer sequence with generating function $A(t)$ and $b(n)$ is an integer sequence with generating function $B(t)$, it is not easy to find the generating function $C(t)$ ...
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65 views

What impedances can be generated from combination of impedances?

There was a problem which asked to connect some resistors in some order(parallel and series) to achieve an equivalent resistor with a specified ohms. I solved that problem but I think we can argue ...
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182 views

What is the exponential generating function of the inverse matrix of an integer triangle?

Let ${Z}$ denote an integer triangle (like Pascal's), ${Z}_{n,k}$ for $0\leq k \leq n$ and let $f$ be an exponential generating function for polynomials $p_{n}(x)$ with $[x^k]p_{n}(x)= Z_{n,k}$. ...
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312 views

A product puzzle

This is from a math contest. I have solved it, but I'm posting it on here because I think that it would be a good challange problem for precalculus courses. Also, it's kind of fun. Write the ...
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228 views

Combinatorics and graph theory - counting connected graphs

We denote by $C(n,n+k)$ the number of connected graphs on $n$ vertices with $n+k$ edges. I have 2 problems I wish to prove, but after much effort have gotten nowhere with. I would greatly value some ...
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1k views

Optimal Resolvable Steiner Quintuple System covering with circles and ellipses

Here is a resolvable Steiner quintuple system. Every tuple from 1-25 appears in exactly one of the sets. {{1,2,3,4,5},{6,7,8,9,10},{11,12,13,14,15},{16,17,18,19,20},{21,22,23,24,25}, ...
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181 views

Simplifying an expression involving binomial coefficients

Consider $$ \prod_{j=0}^{i-1}\binom{n-2j}{2}\left[1+\sum_{k=0}^{N-1}{\left( \prod_{l=0}^{k} \dfrac{N-l}{M-l}\right)\left(1 + \sum_{m=k+1}^{N-1} \prod_{p=k+1}^m \dfrac{N-p}{M-p}\right)+1}\right] $$ ...
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58 views

Necessary and sufficient condition for $f(q^n)$ to be in $\mathbb{Z}[q,q^{-1}]$ when $f\in\mathbb{Q}(q)[x]$?

In this question, user begins shows that, for each $k\in \mathbb{N}$, there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, ...
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69 views

About $t$-analogue of the Euler polynomials.

A certain way to define the $t$-analogue of the Euler polynomials $C_n(x)$ is by $$ C_n(x,t)=\sum_{\pi\in S_n}x^{\text{des}(\pi)+1}t^{\text{maj}(\pi)} $$ where $des(\pi)$ is the descents in $\pi$, ...
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212 views

How fast does $\binom{n}{k}$ grow when $k \le n/2$?

How fast does $\binom{n}{k}$, $n$ fixed, grow when $k \le n/2$? Especially, I'm interested in the growth of the "inverse" of binomial coefficient $B_n(x) := \min \{k:\binom{n}{k} \ge x\}$. EDIT: ...
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215 views

Transversal of a family of sets

What I want to prove is: Let $U=(A_1\ldots,A_n)$ be a family of sets and let $P\subseteq A_1\cup \cdots \cup A_n$. Then $U$ has a transversal which includes the set $P$ if and only if (i) $U$ has ...
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665 views

Project Euler Problem 338

I'm stuck on Project Euler problem 338. This is a cross post from StackOverflow where I initially posted, however, it was suggested that I post it here too since the problem mostly relies on math. The ...
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94 views

Identify this combinatorial construction

I am no combinateur, but I stumbled across the following construction when studying an operad arising from information theory (actually it's a special algebra of an A$_\infty$-operad). It looked ...
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113 views

Are almost all rooted trees asymmetric?

It's well known that almost all graphs are asymmetric (have trivial automorphism group) and that almost all free trees are symmetric. By which argument do I see whether almost all rooted trees are ...
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46 views

Combinatorial interpretation of an equality

In a recent project, I came up with the following equality which turned out to be extremely useful for counting conjugacy classes in certain division algebras (I won't go into the details here, it's ...
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75 views

What's so special about binomial coefficients that someone decided to organize them in a triangle?

I know that binomial coefficients are related to figurate numbers (which were studied by Greeks a loooong time ago, because of its connections to geometry). I also understand how the Pascal's triangle ...
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34 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
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84 views

Clash of Clans Permutations

I'm not an expert mathematician (I'm 16) and I'm Italian, so please try to understand my question and forgive my poor language. Thank you. When I play Clash of Clans, I ask me "How many buildings ...
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41 views

Prove that the number of nonisomorphic ways to partition $E(k_n)$ into complete bipartite graphs at least $2^{n-4}$

Apparently the solution is exercise 1.4.5 in Babal and Frankl's "linear algebra methods in combinatorics (1992)" but it seems they never actually got around to publishing this. Or at least, I am ...
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52 views

How to show this expression is always a perfect square?

The number of tilings of an $m \times n$ board with $2 \times 1$ dominoes (each placed either horizontally or vertically on two squares of the board) has been shown to be $$\sqrt{\prod_{j=1}^m ...
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74 views

hat matching problem (Ross, p.41)

I'm studying Ross's probability book, and kind of got stuck on the matching problem. Suppose that each of $N$ men at a party throws his hat into the center of the room. The hats are first mixed up, ...
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33 views

Does there exist Latin square critical sets for which deleting any entry results in arbitrarily many completions?

For those familiar with Latin squares terminology, I'll get straight to the point: Q: For all $N \geq 2$, does there exists a critical set $C$ (for a Latin square of any finite order) such that ...
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146 views

Derangement bijection

This is a generalization of this question. An $(n,k)$ partial permutation is an injection from $[k]$ to $[n]$. It can be thought of as word of length $k$ in symbols in $[n]$ without duplications. ...
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52 views

Conjectured Value of Ramsey Number $R(3,10)$

It is known that the value of the Ramsey number $R(3,10)$ is either 40, 41, or 42. Have any experts in the field offered a conjecture as to which it might be?
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75 views

Combinatorical interpretation of $\binom{15}{5} = \binom{14}{6}$

I was reading up on Sigmaster's conjecture on repeated binomial coefficiencts and I read that $$\binom{15}{5} = \binom{14}{6}$$ Sure, it's possible to prove it non-combinatorically: ...
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44 views

an elementary problem on wreath product groups with combinatorial flavor

Embarrassingly, I got stuck in solving the following elementary exercise. Let $G=H\wr \Gamma$ be a wreath product groups, $H,\Gamma$ are countable discrete groups, when $\xi\in\oplus_{\Gamma}H$, then ...
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48 views

What is the number of labeled caterpillars?

A caterpillar is a tree in which all the vertices are within a distance 1 of a central path. (See the Wikipedia article: Caterpillar tree, for an example and some equivalent characterizations). The ...
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53 views

number of Lattice paths from origin to diagonal after removing vertices

i am stuck with the following problem: consider the quarter plane $\mathbb{N}_0^2$ with vertices $(i,j)\in\mathbb{N}_0^d$ and edges from each vertex $(i,j)$ to $(i+1,j)$ and $(i,j+1)$, i.e. one ...
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59 views

A function on sets which is constant for all permutations

Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions: ...
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86 views

Random $0-1$ matrices

I'm working my way through the Oxford notes in Probabilistic Combinatorics and came across this question in one of the question sheets; I'd like to stress that this is not my homework: I'm simply ...
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109 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 ...
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90 views

Sum of product partitions of divisors

Let $M(n)$ be the the set of the multiplicative partitions of $n$, and let $D(n)$ be the set of the sum of the multiplicative partitions of the divisors of $n$. eg $M(30)=\{\{30\},\{2,15\},\{3, ...
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117 views

Proving $\sum_{k=1}^{n}\binom{n-1}{k-1}{\binom{n+k}{k}}^{-1}=\frac 12$ combinatorially

Question : How can we prove the following equations combinatorially? $$\begin{eqnarray}\sum_{k=1}^{n}\frac{\binom{n-1}{k-1}}{\binom{n+k}{k}}&=&\frac ...
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77 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
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34 views

A question on combinations of a set of numbers

I have the set of the first $n$ primes $\{2,3,5,\ldots,p_n\}$. There are $n^n$ ways of selecting $n$ numbers from this set. Each combination has a number ($C_k$) associated with it and it is the ...
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21 views

4-net in binary hypercube

Consider a binary hypercube $\mathbb{F}_2^n$. What is the largest size of a subset $S$ such that $d(x,y)\geq 4$ for all $x,y\in S$ ($x\neq y$), where $d(x,y)$ is the Hamming distance between $x$ and ...
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0answers
25 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
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41 views

Division of space by balls in R^n

I would like to know the generalized proof of this result: http://mathworld.wolfram.com/SpaceDivisionbySpheres.html, for $n$ dimensions. What is the maximum number of regions divided by $q$ ...
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52 views

Proving the inclusion exclusion principle from the definition of the cardinality

I want to prove the inclusion exclusion principle: $|A\cup B| = |A| + |B| - |A\cap B|$ where $A$ and $B$ are finite sets. I proved the addition rule by contructing a bijection to a subset of ...
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59 views

About derangement problems

Two derangement problems are really confusing me a lot. Please help me. First Problem There are n people in the room and they are sitting on a round table. All of them went out the room and ...
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78 views

Combinatorial Proofs of Two Identities

I need a combinatorial proof for these two identites: (a) $\sum_{k=0}^{n} \binom{n}{k}^2 x^k = \sum_{k=0}^{n} \binom{n}{k} \binom{2n-k}{n} (x-1)^k$ (b) $\binom{n}{k} ^2 = \sum_{l=0}^{n-k} (-1)^l ...
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0answers
38 views

C0mbinatorics: Minimum number of node transitions

For n nodes, labeled 1,2...,n, a total of x nodes can be on at a time. Each node must "visit" each of the other nodes, and a node can only visit another node when both are on. Each time nodes are ...
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48 views

How prove this there exsit set $B$ such $card(B)>\dfrac{n}{3}$,

Question: Let $a_{i}\in N^{+}$, and the set $A=\{a_{1},a_{2},\cdots,a_{n}\}$, show that: There exists a set $B\subset A$, such that $card(B)>\dfrac{n}{3}$, and that for any $x,y\in B$, ...
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50 views

News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
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60 views

How to calculate the number of automorphisms of this set.

Let $N = \{1,2,\dots,n\}$ and define the set of functions $$X = \{ f:N \longrightarrow N : f(i) \leq i\}$$ Similarly let $S_n$ (the usual symmetric group on $n$ elements) act on $N$ by permuting ...
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0answers
72 views

Number of 'unique' one bit binary functions with N-bit inputs

Consider the set of binary functions that takes an N-bit input -> 1 bit output. There are 2^(2^N) elements in this set. Now potentially reduce this set by restricting to only considering functions ...
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113 views

Circular permutations - $n$ sitting at a round table without repeating neighbors

I hope this isn't a duplicate - the problem is to find the number of ways of sitting $n$ people (who initially were sitting in the order $1, 2, \dots,n$, with $1$ and $n$ being neighbors) at a round ...