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.

learn more… | top users | synonyms (4)

1
vote
0answers
5 views

Mathematics of Magic Squares

I have seen many popular accounts of simple magic squares but I would like to find a proper mathematical background to understanding magic squares. What background knowledge do I need. I am a retired ...
1
vote
1answer
18 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...
0
votes
5answers
40 views

Picking (and replacing) among five balls in an urn

An urn contains 5 balls numbered from 1 to 5. A ball is chosen at random and its number is noted the ball is then returned to the urn. this is done a total of 5 times. What is the probability that ...
3
votes
0answers
17 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
1
vote
3answers
57 views

Probability that two numbers differ by one bit

Assuming that t is the bit length of the numbers and that we can pick 2 random numbers (the same number cannot be chosen twice), which is the probability that the two numbers will differ by exactly ...
1
vote
1answer
41 views

Given $N$, is there a formula for $card( \{(m,n)\, s.t.\, m\cdot n \leq N \} )$?

The formula is also equivalent to : $$ \sum_{m=1}^N \left \lfloor \frac{N}{m} \right \rfloor $$ An interpretation would be to count the discrete rectangles with total area inferior to N. But aside ...
1
vote
2answers
24 views

How many arrangements of the integers 1,2, .., n such that from (ALAN TUCKER Applied Combinatorics)

I was solving a question from alan tucker's applied combinatorial book and got stuck at this question: How many arrangements of the integers 1,2, .., n are there such that each integer differ by ...
-2
votes
0answers
36 views

Summation of prime multiples less than n [on hold]

How can I sum the following $$ \sum (2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i})) $$ with $$2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i}) \le n$$ where $p_i \ge 11$ are list of fixed ...
0
votes
0answers
15 views

Why $|N(P_{i, j})| \cong [0, 1]^n$ as stated in page 21 of HTT?

maybe this is an idiot question, however I could not solve this after thinking for a while. I added the tag about higher categories simply because of the nature of the question, however this is just a ...
-2
votes
0answers
19 views

mapping of integer to unit circle through function $f(k)=k\theta \pmod{2\pi}$ [on hold]

Let $N$ be a positive integer and $\theta$ an angle in $(0,2\pi)$. Consider the map $f\colon\{0,1,\ldots,N\}\to\text{unit circle}$, defined by $f(k)=k\theta \pmod{2\pi}$. Show that the image of $f$ ...
-2
votes
1answer
20 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
1
vote
0answers
31 views

How to calculate combinations by drawing out the spaces?

I'm learning about probability on khanacademy. They teach a certain method (they draw out the spaces) to calculate combinations. Two Examples: 1. Take the question "What is the probability to get ...
0
votes
1answer
24 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...
2
votes
1answer
26 views

Given $n>0$, let $S$ be a set whose elements are positive integers $\leq 2n$ such that:

S is a set with the property that for all a,b∈S with $a<b$, a doesn't divide b. What is the maximum number of integers that $S$ can contain ? I thought it was the number of prime numbers smaller ...
1
vote
0answers
24 views

Count the number of strings containing $ac$ or $ca$ for a fixed length over ternary alphabet $A = \{a,b,c\}$ using rational series

This question is a continuation the one asked here, and which already received good answers. Here I am asking for a solution using rational series of formal languages as suggested by the user J. E. ...
2
votes
1answer
48 views

NP combination puzzle (Klotski)

I've written a C++ program to solve sliding puzzles games such as UnblockMe and Car Parking. I'm quite happy about it, since it solves various schemes in less than a second. Recently I fed the game ...
-2
votes
0answers
32 views

combinatorial nullstellensatz [on hold]

I was wondering if there is any trick for selecting the polynomial in Combinatorial Nullstellensatz method by Alon. This could be a powerful tool provided we choose right polynomial.
2
votes
3answers
88 views

High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at ...
4
votes
1answer
39 views

Arrangement counting problem

This is my son's exercise: How many ways that 6 rabbits can be put in 10 cages. I count in 2 different ways: The first rabbit can be in any of 10 cages. Same for the second and so on. So in total, ...
6
votes
2answers
89 views

Describe and count the set of sequences containing $20$ or $02$

Let $X = \{ 0,1,2 \}$ be a ternary alphabet and denote by $X^*$ the set of finite sequences (i.e. strings) with three symbols. For $w \in X^*$ with $n$ the length of $w$ and $w = w_1 w_2 \cdots w_n$ ...
0
votes
0answers
17 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
3
votes
2answers
84 views

a vector inequality and combinatorics related question

This question is a similar restatement of this question which has been recently closed. Let $$A=\{\ (x,y,z)\in\mathbb{N}^3\ |\ 0\leq x,y,z\leq7\}$$ and $$B\subset A \text{ with } ...
0
votes
1answer
19 views

Isomorphic relation between Catalan representations

There is an unanswered question at MathOverflow: Intersecting Family of Triangulations This article at Wikipedia explains the concept of non-intersecting partitions of a polygon: Catalan number So ...
0
votes
2answers
33 views

A question on probability of choosing coins

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3 \over 4$ and tails with probability $1 \over 4$. Three coins are chosen from ...
2
votes
2answers
82 views

Algebraic proof that $\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$

I'm looking for an algebraic proof of this identity for $n, k \in \mathbb{N}$: $$\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$$ So far, I've turned the left hand side of the equality into ...
1
vote
2answers
31 views

Find the total number of matchings in a complete graph with even vertices

I am trying to solve questions from a Walk through combinatorics.., I came across this proof which I was unable prove: Determine the number of perfect matchings for a graph with 2n vertices. I don't ...
3
votes
1answer
25 views

Median of waiting time for $k$-th ace from bridge cards

I can't figure out how to get median of a waiting time from the exercise 36 from W. Feller's book An Introduction to Probability Theory and Its Applications Vol.1 (bold in the quote): ...
3
votes
1answer
41 views

Choose 8 distinct integers from $\{1, 2,\dots,16,17\}$. Show that the eight contain at least three pairs with a common difference for _any_ choice.

This problem is from the 1999 Canada National Olympiad. I am stuck trying to prove the first part using the pigeonhole principle. Is there a refinement that will allow it to be used more sharply, or ...
3
votes
2answers
40 views

number of triangles determined by a rectangular grid

Suppose we are given an $m\times n$ rectangular grid of lattice points, such as $S=\{(k,l): 0\le k\le n-1,\; 0\le l\le m-1, \;k,l\in\mathbb{Z}\}$, and we want to determine the number of ...
0
votes
1answer
29 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
2
votes
1answer
33 views

Number of distinct permutations given a character set.

How many distinct three-letter sequences with at least one $T$ can be formed by using three of the six letters of $TARGET$? One such sequence is $T-R-T$. [MathCounts 2005 National Countdown] The ...
2
votes
2answers
87 views

Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
0
votes
0answers
23 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
1
vote
1answer
71 views

Interesting property of Pascal's Triangle

I was looking at the Pascal's Triangle and saw that for all central numbers in even length row $a \gt 17$, the number $\dbinom{a}{b-2}$ is greater than $\dbinom{a-1}{b}$. This is where $b$ is equal to ...
0
votes
1answer
30 views

Interesting Combinatorics question relating the coefficients of variables in Pascal's Triangle

I tried this problem for a while by canceling the factorials on either side but for whatever reason, wasn't able to solve it. Could someone please help me? Is there a proof that ...
0
votes
0answers
27 views

All-pairs top-k min-cost flow paths

I am using a directed multigraph to model network flow. For example: Associated with each edge is: a cost of sending flow down that edge (red) a maximum capacity which the amount of flow sent ...
4
votes
1answer
42 views

Identities involving binomial coefficients, floors, and ceilings

I found the following four apparent identities: $$ \begin{align} \sum_{k=0}^n 2^{-\lfloor\frac{n+k}{2}\rfloor} {\lfloor\frac{n+k}{2}\rfloor\choose k} &= \frac{4}{3}-\frac{1}{3}(-2)^{-n},\\ ...
0
votes
1answer
47 views

consider a graph of a gameboard

Consider a graph of a game board. Rounds in the game result in a token moved from a game board location to a game board location, possibly returning to the same one. Let the game board location at the ...
4
votes
2answers
40 views

Number of divisors of the form $(4n+1)$

Find the number of divisors of $$2^2\cdot3^3\cdot5^3\cdot7^5$$ which are of the form $(4n+1)$ I know how to find the total number of divisors. But, to find the number of divisors of the form ...
2
votes
1answer
42 views

How many height arrangements are there for people?

Let's suppose $n$ people of different height stand in line, and the observer (who is smaller than the people in line) looks at them from the side. The observer sees a person unless there is a taller ...
0
votes
0answers
24 views

A question on choosing numbers to form geometric sequence

So the question states: In how many ways can you choose three numbers from $1,2,...,100$ to form a geometric sequence $k,{km \over n}, {km^2 \over n^2}$ such that $n \gt 1$,$m \gt n$,$n^2|k$ and ...
1
vote
0answers
49 views

Optimize for happiness and equality

I'm trying to solve an optimization problem: There are $N$ students who can choose to enroll into $C$ courses, each of them has a set of 3 preferences $P = \{c_1, c_2, c_3\}$ about the courses they ...
1
vote
1answer
28 views

Argue that $\binom{n}{n_1,n_2,…,n_r} = \binom{n-1}{n_1-1,n_2,…,n_r} + \binom{n-1}{n_1,n_2-1,…,n_r}+…+\binom{n-1}{n_1,n_2,…,n_r-1} $

Argue that $\binom{n}{n_1,n_2,...,n_r} = \binom{n-1}{n_1-1,n_2,...,n_r} + \binom{n-1}{n_1,n_2-1,...,n_r}+...+\binom{n-1}{n_1,n_2,...,n_r-1} $ Each term on the right hand side is the number of ways ...
2
votes
1answer
60 views

Probability of getting the same vector result

This is part of a mathematical puzzle I was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
0
votes
0answers
60 views

I have a 8 DIP Switches on a device. How many combinations and permutations are there? [on hold]

I do not know how to figure this out. It is a head scratcher for me. $\begin{array}{cc} \text{DIP Switch} & \text{Possible Settings} \\ 1 & 4 \\ 2 & 4 \\ 3 & 9 \\ 4 & 4 \\ 5 ...
0
votes
1answer
19 views

Number of transformations of rank $m$

Let $\mathcal{T}_n$ be the set of all transformations on $\mathbb{N}_n = \{1, 2, \ldots, n\}$. For all $\phi \in \mathcal{T}_n$ let $\text{rank}(\phi) := |\phi(\mathbb{N}_n)|$. What is the cardinality ...
2
votes
2answers
37 views

How to find the number of all the possible ordered trees with n edges and k leaves?

We know that a tree with n edges have n+1 nodes.So if $|B_{n+1}|$ is the number of all possible ordered trees with n+1 nodes then its true that $C_{n+1} = |B_{n+1}|$ where $C$ is the Catalan ...
1
vote
3answers
52 views

Ordered Sum of Odd Numbers

EDIT: The vectors can be any length. That is $k$ is not fixed. For a given natural number $n$, let $S_1(n)$ be the number of vectors $(a_1, a_2, \ldots, a_k)$ such that $$a_1 + a_2 + \cdots + a_k = ...
4
votes
0answers
44 views

Terminology in graph theory

Let $G$ be a finite graph with the following property: For any vertex $a$ and edge $\{b, c\}$ of $G$, there is an edge connecting them: there is one of $\{a,b\}$ or $\{a, c\}$ in $G$. Is there ...
1
vote
1answer
19 views

Number of decompositions in sum of $s$ elements

Let $E=\{ 3^k+3^l; 0\leq k\leq l\}$. For all $n\in \mathbb{Z}$ and $s \geq 1$ denote $r_s(E,n)$ the cardinality of $$ \{(n_1, \ldots ,n_s) \in E^s, n_1+\ldots +n_s=n \}.$$ I'm looking for an upper ...