This tag is for basic questions about the study of finite or countable discrete structures ā€” 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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Tao's proof of Szemeredi-Trotter theorem on incidences

In Tao's proof of Szemeredi-Trotter theorem on Point-Line incidences , he uses a lemma which decomposes the real plane into cells. During the proof of this lemma, the final step of getting rid of the ...
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91 views

Graph composed of matchings and K_4

Let $Gā€² = (V, E_1 \cup E_2 \cup E_3)$ be a graph, where $E_1$ and $E_2$ are (nonempty) matchings and $E_3$ is the set of edges of a nonempty collection of pairwise vertex disjoint copies of $K_4$. ...
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157 views

What is known about this jigsaw combinatorics problem?

The problem: How many valid reconfigurations, as defined below, are there for a given puzzle? My question is, what, if anything, is known about this problem already? For example, does this ...
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46 views

The largest regular m-gon that fits inside a regular n-gon

This question just popped into my head while doing some "for fun" math. More precisely: Let $m,n\in\Bbb{Z};m,n>2$. Let $P$ be a regular $n$-gon (let's say $P$ is the convex hull of the $n$ $n$th ...
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133 views

Combinatorial Identity

I have to validate the following identity which is defined: $$ \sum_{k=1}^n (-1)^{k-1}*q^{\frac{k(k-1)}{2}} *\frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1 $$ where $0<q<1$. I ...
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120 views

Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such ...
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135 views

The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
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75 views

Selecting k numbers out of N sorted numbers to a minimize a condition/Formula

A sorted list of $\mathbf N$ numbers is given. $X_1$ $\le$ $X_2$ $\le$ $X_3$ $\le$ .... $\le$ $X_N$ Select $\mathbf K$ Numbers - $Y_1$ , $Y_2$ , $Y_3$ , ..... , $Y_K$ - Such that the following ...
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94 views

Simplifying a sum?

Define polynomials $P_{j,s}^{(r)}$ via the generating series $$\left(\frac{d^s}{dz^s}f(z)\right)^r=\sum_{j=0}^{\infty} P_{j,s}^{(r)}z^j,$$ where $r\geq 1$. Here, $f(z)=z+a_2z^2+a_3z^3+\cdots.$ I was ...
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84 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
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94 views

Extracting Taylor coefficients of a quotient

I was wondering if anybody has come across functions of the form $$\Phi_n(z):=\frac{f(z)^{n+1}}{zf'(z)-f(z)}\quad (n\geq 1).$$ Here, $$f(z)=\sum_{k=0}^{\infty} a_kz^k$$ is holomorphic on the open unit ...
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235 views

Which unordered partition of $n$ gives rise to the largest number of ordered partitions?

A quick look at the wikipedia article on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim ...
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69 views

What's the name for this unimodal sequence?

Let $a_0, a_1, \ldots, a_n$ be an increasing sequence of positive numbers, and consider the sequence $s_1,\ldots,s_n$, where $$ s_k \;=\; \frac{a_0+\cdots + a_k}{k}. $$ So $$ s_1 \;=\; a_0+a_1,\quad ...
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87 views

A combinatorial problem.

Let be $(X, \mathbb{A}, \mu)$ a measure space, a partition of $ X $ is a disjoint family $\xi=\{P_1,\ldots,P_k \}$ of measurable sets such tath $\bigcup P_i=X\pmod0).$ If $\xi=\{P_1,\ldots,P_k ...
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66 views

In Pursuit of a Broader Understanding of Complicated Binomial Coefficient Sums

$$\sum_{k=0}^{n}\binom{n}{k}\frac{k!}{(n+k+1)!}$$ The above identity was posted once before by me, however, all results were obtained numerically exploring the identity rather than understanding ...
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79 views

Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
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78 views

Sequences of Integers such that no one divides the product of two others.

I'm working on of a problem of Bollobas' Modern Graph Theory, but I can't seem to get the last part of the problem: Let $1<a_1<a_2< \cdots a_k \leq x$ be natural numbers. Suppose no $a_i$ ...
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147 views

Using Bernoulli distribution approximate the $q$-th moment

Let $x$ be vector in $R^n$. Let $\pi(ā‹…)$ be a permutation on the set $\{1,\ldots,n\}$ with a uniform distribution. Let $|m|\leq n, m \in Z$. Using Bernoulli (or maybe some other) distribution ...
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131 views

Function composition and Bell polynomials

Suppose that we have the Taylor expansions : $$f(x)=\sum_{n=1}^{\infty}\frac{a_{n}}{n!}x^{n}$$ $$g(x)=\sum_{n=1}^{\infty}\frac{b_{n}}{n!}x^{n}$$ Then we have the standard result : ...
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162 views

Efficiently count possible nim-like moves

Consider $n$ piles of coins, with pile $i$ having $a_i$ coins. A valid move is to remove zero or more coins from each of the piles, with the constraint that atleast one pile should remain unchanged, ...
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89 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1] For ...
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336 views

Number of labelled graphs with $n$ nodes, $k$ edges and $t$ triangles

How many labelled undirected graphs are there with precisely $n$ vertices, $k$ edges, and $t$ triangle subgraphs? (By triangle I mean a graph with three vertices and three edges.) (Clarification: I ...
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115 views

A general Combinatorics problem (Coefficients of the q factorial)

I was solving a combinatorics problem when I encountered difficulties. The problem was: $x_1 \in \{0,1\}$ $x_2 \in \{0,1,2\}$ . . $x_{n-1}\in\{0,1,2..,n-1\}$ We have to find the number of ways ...
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72 views

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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204 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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87 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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207 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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192 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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321 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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248 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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189 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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60 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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223 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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228 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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691 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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95 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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116 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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78 views

A curious identity of weighted sums over multi-set permutations.

Suppose we have $n$ balls which are the same except colors, denote $S$ to be the set of all different permutations of the balls.(i.e. the swap of two balls with the same color will be the same ...
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Building a 3D matrix of positive integers

I'm trying to build a 3D matrix made up of positive integers that has very specific properties. The matrix dimensions are $N \times N \times (N+1)$ where $N$ is a positive integer. The matrix has two ...
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51 views

Probability of $m$ out of $n$ rolls of a die being among the numbers $1,2,\ldots,m-1$, for some $m$.

Suppose I have a $k$ sided die with the numbers $1,2,\ldots,k$ on each side, and that I roll it $n$ times ($n<k$). What is the probability that there exists an $m\leq n$, so that $m$ of the $n$ ...
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71 views

Prime factors of binomials

Is it true that for each $n\geq 2$ there are two primes $p, q$ such that (at least) one of them divides $\binom{n}{k}$ for each $1\leq k\leq n-1$? Examples: For $n=6: \binom{6}{1}=6; ...
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38 views

Trying to prove a congruence for Stirling numbers of the second kind

I am struggling with a demonstration for this: When $n$ and $m$ are 2 natural integers such that $n-m$ is odd, then the following congruence holds for Stirling number of the second kind ${n \brace ...
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35 views

Number of Secret Santa directed graph with a largest cycle of given size

Secret Santa is a Western Christmas tradition in which members of a group are randomly assigned another member of a group for whom they are to buy a gift. While we were doing the random assignment ...
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43 views

How many ways to arrange 12 identical apples and five distinct oranges in a row so no two oranges are side by side?

My first intuition to solve this problem was to use the separator technique with the apples acting as separators. $$_1_1_1_1_1_1_1_1_1_1_1_1_$$ Since there are now 13 blank spaces for the oranges to ...
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66 views

In how many different ways can one color the vertices of a regular pentagon into four colors?

I am trying to find the number of ways to color a pentagon with 4 colors up to symmetries. I know that I should be using Burnside's Theorem, and so far I know that the group $D_5$ should act on the ...
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38 views

How many ways to place 3 non-attacking bishops given the following conditions

How many ways are they to place 3 non-attacking bishops on an $n \times n$ board such that $2$ of these bishops are placed within the $(n-1) \times (n-1)$ board and the other 1 is placed outside of ...
3
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58 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = ...
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55 views

The number of partitions by distinct positive numbers

Let $N>0$ be a natural number and let $P(N)$ denote the number of ways to write $N$ as a finite sum of $a_i$ such that the $a_i$ are strictly decreasing positive natural numbers. There is a paper ...