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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-2
votes
0answers
10 views

How many ways shuffle n1 and n2 balls when we but them together?

I have $n_1$ white balls and $n_2$ black balls and i want to know how many ways i can make a distinct arrangement from them. for example , $n_1 = 2$, $n_2 = 1$ then there are three distinct ...
0
votes
1answer
11 views

Counting weakly connected graphs with outdegree of exactly one.

If we count all graphs of $N$ labelled vertices, where each vertex has an outdegree of exactly $1$ with no self-loops allowed, we'll find that there are exactly $(N-1)^N$ of them (for every of $N$ ...
1
vote
2answers
34 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...
0
votes
1answer
27 views

Numbers of factors of (n)(n+1)/2 is product of exponents?

I was trying to find the number of factors of $n(n+1)/2$, and I read this blog article, and it says that the number of factors of it is the product of its prime factor's exponents with one added to ...
-2
votes
0answers
42 views

Counting the maximum number of intersections.

Let $n$ be a positive integer. Points $A_1,A_2, \cdots, A_n$ lie on a circle. For $1 \le i <j \le n$, we construct $\overline{A_iA_j}$. Let $S$ denote the set of all such segments. Determine the ...
1
vote
0answers
29 views

How to Evaluate this Summation to Find a Closed Form

While taking the incomplete Bell Polynomil of $x^a$ i found out that: $$ B_{n,k}^{x^a}(x) = x^{ak-n} \sum_{m=0}^k \frac{(am)!(-1)^{k-m}}{m!(k-m)!(am-n)!} $$ Now, what i am wondering is, what is the ...
0
votes
0answers
33 views

Stirling transform of $(k-1)!$

While reading about combinatorial mathematics, I found this article about the Stirling transform which caught my attention. So, if I wanted to find the Stirling transform of, for instance, $(k-1)!$, ...
0
votes
0answers
34 views

Prove the function is nondecreasing

Lets take: $A_1,...,A_n$ family of finite, nonempty sets. Define: $$f(t)=\sum_{k=1}^n\left( \sum_{1\le i_1<...<i_k\le n}(-1)^{k-1}t^{|A_{i_1} \cup ... \cup A_{i_k}|} \right)$$ for $t \in [0,1]$. ...
1
vote
0answers
25 views

Number of elements in discrete $n$-dimensional simplex such that $x_1 \leq \ldots \leq x_n$

For positive integers $n,d$, how many elements are there in the set $S = \{(x_1,\ldots,x_n) \in \mathbb{Z}^n\ |\ 0 \leq x_1 \leq \ldots \leq x_n \wedge \sum_i x_i = d \}$? I'm hoping that the order ...
2
votes
1answer
39 views

How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?

I know that $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}=\frac 12((H_k^s)^2-H_k^{(2s)})$$ and $H_k^s=\sum_{n=1}^kn^{-s}$ But, how would I go about finding identities in terms of the harmonic number like ...
3
votes
4answers
82 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
0
votes
3answers
31 views

Combinations and Double Factorials

In a village, there are 10 boys and 10 girls. The village matchmaker arranges all the marriages. In how many ways can she pair off the 20 children, if homosexual marriages (male-male or female-female) ...
0
votes
1answer
19 views

How many n-permutations have no substrings of the type (j,j+1)?

How many n-permutations have no substrings of the type $(j,j+1)$? $$1\leq j\leq n-1 \text{ and } n\geq 2$$ For example, let n be 5: [3 2 1 5 4] is one of the permutations we have to count. [4 ...
2
votes
1answer
25 views

Combinatorics strategie for order

At the moment I have to deal a bit with Combinatorics but I have some problems with it. Let's say I have following situation: Spend 1500 Euro to 4 people so that everyone has a multiple of 100 ...
2
votes
1answer
34 views

Distribution of K balls in N Cells with limitations

In how many ways can i distribute $k$ balls in $n$ numbered cells with the following limitations: 1.Each cell has different number of balls in it 2.Given each cell has more balls than the cell ...
0
votes
1answer
21 views

Counting functions and stirling numbers

Let S= { f | f: A $\rightarrow$ B, |Image(f)|=k}. |A|=m, |B|=n. where k $ \le n, k \le m $ |S|=$ {n \choose k} $ S(m,k) k!. where S(m,k) are the striling numbers of the second kind. What I can't ...
3
votes
1answer
42 views

An inequality relating to moves to P-positions in Nim

I have been researching this variant of Nim. I have been unable to prove the following claim. What is annoying is that I feel I am missing something really obvious. Does anyone have any ideas on how ...
1
vote
2answers
31 views

Arrangements in which only two of the three empty chairs are next to each other

While studying, I got stuck on this problem: "Seven identical chairs in a row are to be seated by four students. How many arrangements are there such that the only two of the three empty chairs are ...
1
vote
1answer
26 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with ...
0
votes
3answers
50 views

Proof for number of completely odd and even subsets.

While studying, I read this: "A subset of integers $1,2,...,n$ has the property that the sum of its members is odd. The number of such subsets is $2^{n-1}$." I also read this: "A subset of integers ...
1
vote
1answer
42 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...
1
vote
0answers
14 views

Available lists of all latin squares up to order 5?

There are available online lists of the number of all latin squares up to order 11, e.g.: https://oeis.org/A002860. For a permutation-based test of a latin square design, one option is to fit the ...
1
vote
4answers
47 views

Proof for number of ways to select k non-consecutive elements from n consecutive terms. [duplicate]

While studying, I found a formula that found the number of ways to select k non-consecutive elements from n consecutive terms, not necessarily the first n consecutive terms, but any n consecutive ...
4
votes
0answers
26 views

Proving that the intersection of two closed sets is closed in a matroid

I am stuck on a little homework problem I have. Here, $M$ is a matroid with rank function $R$. I am given this definition: In a matroid $M$, a set $A$ is closed if $R(A \cup e) > R(A)$ for all $e ...
2
votes
1answer
25 views

Tuples in cartesian product without duplicates

I have $n$ sets $S_1,\ldots,S_n$ and I would like to count the number of tuples $(i_1,\ldots,i_n)\in S_1\times\cdots\times S_n$ such as $i_h\neq i_k$ $\forall h,k\in \{1,\ldots, n\}$. Is there a ...
2
votes
0answers
43 views

Unique ways to distribute k1, k2, .. colored balls into n boxes uniquely

Example: Uniquely distribute 2 Red Balls and 4 Blue Balls into 3 boxes: [B][BB][RRB] [B][BBB][RR] [B][R][RBBB] [B][RB][RBB] [BB][R][RBB] [BBB][R][RB] Answer: ...
6
votes
2answers
58 views

$\sum_{k=1}^n \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$

(Own) Let $n,m$ be positive integers such that $m>n$. Prove that $$\sum_{k=1}^n \sum_{a_1+a_2 + \cdots +a_k=n} \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$$ ...
1
vote
0answers
30 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
1
vote
1answer
48 views

Name of the numbers defined by $T(p,q) = T(p-1,q) + T(p,q-1)$?

I came across these numbers : $$ T(p,q)= \sum_{k=0}^{q-1} {p+k-1 \choose p-1} + \sum_{l=0}^{p-1} {q+l-1 \choose q-1} \quad p,q \in \mathbb{N} $$ While trying to solve this recurrence relation : $$ ...
4
votes
2answers
61 views

Simplifying a combinatorial expression

Find \begin{eqnarray} \sum_{i=1}^{k-1}i(2k-2-i)\binom{2k}{2i+1} \end{eqnarray} I know how to find $\sum_{i=1}^{k-1}a_i\binom{2k}{2i+1}$ if $a_i$ is linear in $i$, but got stuck when $a_i$ is ...
4
votes
1answer
92 views

A Combinatorial Sum!

Is there a closed form formula for the following sum \begin{equation} F(x;n,m)=\sum_{k=0}^{\min\{n,m\}} {n \choose k}{m \choose k}k!\ x^{k}=n! \, m!\sum_{k=0}^{\min\{n,m\}}\frac{1}{k!(n-k)!(m-k)!} ...
-1
votes
1answer
43 views

Calculating distance between two squares of a board

Given an $n\times n$ board, for example a chess board 8x8, with the squares ordered in a Little-Endian Rank-File Mapping. Is there a direct way to calculate the distance between two squares using ...
0
votes
0answers
34 views

Converting base 10 to base 52 using a bijective function [migrated]

I was recently asked in an interview the following question: "How would you design a URL shortener?" My response was to store the URL into a database which provides a unique key of maximum length 10 ...
2
votes
3answers
35 views

Probability theory combinatoric problem

A total of $n$ bar magnets are placed end to end in a line with random independent orientations. Adjacent ends with equal polarities repel each other, and adjacent ends with opposite polarities ...
0
votes
0answers
62 views

Number of graphs with 5 vertices

Let $v_i$ where $i=1,2,3,4,5$ be vertices of a graph. Each vertex makes only one directed edge to any other vertex. For instance $v_1 \to v_2 \to v_3 \to v_4 \to v_5 \to v_1$ and $v_1 \to v_3 \to v_4 ...
1
vote
1answer
40 views

Cartesian product with all elements

I have two sets A and B with $A = \{1,2,3\} \\ B = \{ A, B, C, D, E \}$ Now I need to get something similar to the Cartesian product. If my understanding is correct, the Cartesian product would ...
0
votes
0answers
32 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
0
votes
1answer
29 views

How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
0
votes
0answers
32 views
+50

Mixing up seating charts: Measuring “mixedness” over time

Background: My class has $10$ students and $3$ tables; naturally, the students are distributed with $3, 3,$ and $4$ seated at the individual tables. On the second day of class, students sat in the ...
1
vote
0answers
43 views

How to give a rigorous proof of a fact about convex polygon?

I claim that there exists universal constants $0<\delta_1(m), \delta_2(m)<1$ such that for any convex polygon $P$ in $\mathbb{R}^n$ with $m$ faces, \begin{equation} \frac{\mathcal{H}^{n-1}(\{x ...
1
vote
1answer
20 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
1
vote
2answers
80 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
0
votes
0answers
17 views

Minimal posets and chains [on hold]

Given a poset (X, P ) we can say that an element x ∈ X is minimal if it doesn’t cover any other element y ∈ X. Think about the relation between finding a maximal chain and the minimal elements. Isn't ...
0
votes
1answer
33 views

Chains of people names

Consider the set $X$ of possible names for people. Let $(x, y) \in P$ (in the partial order) if and only if $x$ ends in a consonant and $y$ ends in a vowel. What is the length of the longest ...
1
vote
1answer
42 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
-1
votes
1answer
42 views

Probability of winning consecutively [on hold]

India and USA play $7$ football matches. No match ends in a draw. Both the countries are of same strength. Find the probability that India wins at least $3$ consecutive matches.
3
votes
1answer
30 views

How many ways can I connect labeled trees into a tree.

Suppose I have the labeled trees $T_{1}, \ldots, T_{n}$ with $b_{1}, \ldots, b_{n}$ vertices respectively. I would like to know how many ways I can compose a tree from these trees by using all trees? ...
0
votes
4answers
43 views

All possible combinations

I have two sets (1,2,3) and (A,B,C,D,E). I want to calculate all possible combinations. This would be my approach: combinations with a single 1: ...
1
vote
1answer
41 views

Permutations for a set of rules

The question is from - http://www.iarcs.org.in/inoi/2015/zio2015/zio2015-question-paper.pdf - Q.2 I tried solving it but I have no clue how to go about doing it. The question says that a railway ...
5
votes
1answer
64 views

Function equation, find the function evaluated at the certain point.

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ The constant term, $a_0 = f(0) = 1$. Let: ...