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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0answers
11 views

Diagonalizing a matrix arising in a simple combinatorial situation

Maybe I'll return to this question a few hours from now and possibly even post an answer then. This concerns a matrix that I described in this answer. Start with a $\dbinom n2\times n$ matrix $B$ ...
-1
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0answers
29 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall exclusively discuss finite commutative unital ...
3
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1answer
55 views

Combinatoric Birthday Paradox

There is likely a closed form solution for this problem but it's had me puzzled for days. This is about a variant on the classic birthday paradox. To recap, the birthday paradox is where given only 23 ...
0
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2answers
25 views

Why is this counting function finite? (It is used Probability)

Why is this counting function finite? I don't understand this interpretation of the author. Can you explain more about this? Please.
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3answers
42 views

Evaluate: binomial theorem

Show: $$(x+1)^m=\sum_{k=1}^{m}\binom{m}{k}x^k$$ Can somebody help me in showing the above stated problem?
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0answers
43 views

Is it possible to solve sudoku without backtracking?

I occasionally solve sudoku puzzles on smartphone in spare time. My approach is based on the belief that at each stage in solving a sudoku puzzle there is at least one cell where there in only one ...
0
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2answers
50 views

How many permutations of the word TOMORROW can be made if the O's can't be together?

I'm trying to answer this question. This is my attempt of solution: First we distiguish the O's and R's, then we have the word: $TO_1MO_2R_1R_2O_3W$. We have $8!-7!\cdot3!-6!\cdot 3!$ different ...
2
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0answers
26 views

If $G$ is shellable, then $G \backslash \{x_i\}$ is shellable?

A simplicial complex $\triangle$ is shellable on the vertex set $\{x_1,\ldots,x_n\}$, if the facets of $\triangle$ can be ordered, say $F_ 1 , . . . , F _s$ , such that for all $1 ...
2
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1answer
29 views

Reference for a combinatorial theorem

Is there a reference for this theorem https://en.wikipedia.org/wiki/Schur%27s_theorem#Combinatorics? I am unable to locate a reference. Google search does not spot this particular theorem well.
2
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5answers
966 views

Hot dog combinatorics

A hot dog stand has 12 different toppings available. How many different kinds of hot dogs can be made, assuming the order of the toppings does not make a difference. I believe the correct answer is ...
1
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1answer
28 views

A question on arithmetic progressions

Is it true that for every $n \in \mathbb N$ , $\exists N \in \mathbb N$ such that for any subset $A \subseteq \{1,2,...,N\}$ , either $A$ or $\{1,2,..,N\} \setminus A$ contains an arithmetic ...
3
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3answers
35 views

Combinatorics Question - Permutations and Supersets

I had a question that seems pretty straightforward, but I can't seem to wrap my mind around it. Let's say I have a bunch of elements in a set. {A, B, C, D, E}. How many permutations are there of ...
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1answer
22 views

How many option are there to divide n people into any number of groups of any size?

I found only something like this: $$f(0) = 1$$ $$ f(n) = \sum_{i=0}^{n-1} \binom{n-1}{i} * f(i) $$ Now I wonder if there are other (faster) ways to calculate this.
2
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2answers
47 views

Count all possible combinations

I want to check how many combinations of $2$ numbers I can generate from $20$ different numbers when the same number can be picked twice. I calculated it like this and answer is $20 \cdot 20 =400$. Is ...
1
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0answers
59 views

Find the number of “p-safe numbers”

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is ...
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0answers
50 views

Permutation of numbers from multiple sets [May contain duplicate numbers among other sets], resulting in Non-Duplicate Set

We have 3 Data Sets. From each set we will be selecting few numbers. 3 from Set 1, 2 from Set 2, 3 from Set 3. Totally, we will get 8 Numbers from 3 Sets. The resulting sets shouldn't contain any ...
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0answers
30 views

Number of triangles possible in android lock patterns?

I recently starting using the patternlock on my android phone and i play around with it a lot, just drawing lines until im locked out for 30 secs. I thought i'd make it into a pointless game of ...
0
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3answers
36 views

Why is it true that $(a_1, a_2, \dots, a_r) = (a_1, a_r)(a_1, a_{r-1})\dots(a_1, a_3)(a_1, a_2)$?

In the theory of permutation, a $r$-cycle $(a_1,a_2,...,a_r)$ is defined in the following way: Start from $a_i$, a permutation function $f$ sends $a_i$ to $a_{i+1}$. When $i=r, a_i \text{ will be ...
0
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0answers
28 views

Let $T$ be the set of all positive integer divisors of $2004^{100}$. Size of largest subset $S$ of $T$ such that no element in $S$ divides another?

I am getting an answer slightly over $100^2$. Is this right (working below), or is there a better way of selecting elements of S? The following question appeared on the 2004 Canada National Olympiad: ...
3
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0answers
64 views

Help with binomial identity

In my work, I was led to the following identity. I would be grateful if someone could provide an easy proof. $$ \sum_{j = 0}^d (-1)^{d-j} \cdot \binom{d}{k} \cdot \binom{n-j-1}{n-d-1} \cdot ...
0
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1answer
43 views

Probability problem 1

I just wanted to double check to see if I'm doing this problem correctly. 3 kids (Alice, Bob, and Carol) have to divide 15 different toys among themselves in a way that each kid gets 5 toys. How many ...
6
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5answers
133 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
0
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0answers
34 views

How many distinct non-negative integer solutions to $x+y+z=S$ are there, without variable naming?

How many distinct non-negative integer solutions to $x+y+z=S$ are there, without variable naming? Any two solutions $(x_0,y_0,z_0)$ and $(x_1,y_1,z_1)$ are considered equivalent if $x_0,y_0,z_0$ can ...
1
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1answer
46 views

Block of integers: Divisibility

Let a < b be natural numbers. Prove that every block of b consecutive natural numbers contains two distinct elements whose product is divisible by ab. (I've proved this) Suppose now a < b < ...
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2answers
33 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
1
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2answers
66 views

How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q_m(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
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0answers
17 views

combination problem in coding [on hold]

consider we have K input symbol of length 1 bit and also from this K input symbols we can produce symbols which are XOR of 2 input symbols, chosen uniformly from these K input symbols which gives us ...
0
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0answers
40 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
0
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1answer
20 views

Closed solution to double recursion

I have a problem, where a subproblem is counting how many ways there are to interleave two words from disjoint alphabets while keeping the relative order of the letters within each word. For example, ...
1
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1answer
19 views

Amount of match combinations of creating a 5 v 5 team from a pool

This question is inspired by the popular games: Dota 2, Heroes of the Storm and League of Legends; where players have to create two teams of 5 from a pool of "Heroes" in each match. How many ...
0
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0answers
16 views

Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? [duplicate]

Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we ...
0
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1answer
28 views

Simplify the formula for the number of distributions leaving none of the $n$ cells empty

I'd like to help with the following problem: $$ \binom{x}{r-1} + \binom{x}{r} = \binom{x+1}{r} \tag{8.6}\label{8.6} $$ 7. Let $A(r, n)$ be the number of distributions leaving none of the n cells ...
0
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1answer
29 views

Shortest Path Length as mathematical function/expression

I have a graph (unweighted and undirected) of n vertices. My objective is to express the following constraints as inequalities. The degree of any node should be at least 3. The shortest path length ...
1
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1answer
27 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
0
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1answer
28 views

combination of 5 digit numbers

looking at 5-digit number when digits can be $1,2,...,9$ and with repetition, $|\Omega|=9^5$ the event of $5$ distinct digits is $9\times 8\times 7\times 6\times 5$? and the event 2 digits the same ...
0
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0answers
34 views

A residue question in integers

Given $N\in\Bbb N$, is it possible to find $9$ positive integers $A_j,N_i$ with $j\in\{1,2,3\}$, $i\in\{1,2,3,4,5,6\}$ such that following holds? $(1)$ $N\log N < A_j < cN\log N$ at every $j$ ...
0
votes
1answer
32 views

Prove that $\sum_{t \vert n} d^3(t) = (\sum_{t \vert n}d(t))^2$ for all $n \in \mathbb{N}$ [duplicate]

here $d(n)$ counts the number of positive divisors of $n$. I've tried 2 things: Using Bell series. But then again it just showed me that the bell series of the square of a function is not the ...
10
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1answer
86 views

Asymptotic Behavior of a Sum with Binomial Coefficients

The Problem: Find the asymptotic behavior (with respect to $n$) of the following sum $$\sum\limits_{j = 3}^n \binom{n}{j} \frac{(j - 1)!}{2\cdot n^j}. $$ Where the Problem Comes From: If we ...
1
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3answers
54 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
2
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0answers
24 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 ...
2
votes
2answers
79 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. ...
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5answers
56 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
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0answers
20 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
66 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 ...
2
votes
1answer
45 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
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2answers
28 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 ...
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0answers
43 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
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0answers
24 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
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0answers
20 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 $f$ ...
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1answer
22 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 ...