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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3answers
31 views

Different ways of giving away 35 coins to 5 people?

The first part of the problem asks how many ways there are to give away 35 identical coins to 5 people, and I've concluded that it's ${35 \choose 5}$ because you're selecting how many ways you can ...
-4
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1answer
39 views

Solving $x+2y+3z=100$ in nonnegative integers. [on hold]

Solving for number of solution in set of non-negative integer of $$x+2y+3z=100$$ by generating function but finding problem in writing partial fraction of ...
0
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1answer
22 views

What's the least number of combinations you need to determine who the most efficient members are?

Not sure if this question fits here, but it's something I was thinking about last night. Maybe someone can throw some light on it. Let's say I have a group of people doing some shared task. Let's ...
2
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0answers
23 views

Counting Spanning Trees Needed to cover Edges

This is in the same spirit as this stackexchange post, but I am seeking a more general answer. The goal is, given a graph $G$, give a method of counting the minimum number of spanning trees needed ...
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1answer
29 views

Number of different possible armies in Clash of Clans [on hold]

Suppose we are given a set of sixteen different units. How many different armies of $200$ units exist ? In other words, how many $16$-uplets $(c_1, \cdots, c_{16})$ exist such that for each $i$, ...
1
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1answer
23 views

How many different ways can you choose a group of 4 people?

You have a total of 9 people to choose from. Of these 9 people you are supposed to create a group of 4. How many different ways can the new group look? This is my reasoning: To the new group, the ...
3
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1answer
21 views

Generating Constrained Random Distributions

I am trying to help another StackExchange user. We are attempting to fill a 6x6 matrix with 12 A's, 12 B's, and 12 C's subject to the constraint that each row contains 2 A, 2B and 2 C and each column ...
2
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1answer
22 views

How to answer this graph theory question?

Okay so let me define some terms before I ask my problem: Let $K_n$ denote the complete graph on $n$ vertices with $n\geq 2$ and let $C_3$ be a cycle of length $3$ (a triangle). Suppose $x,y,z$ ...
0
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0answers
40 views

Mega-straight flush with a huger hand

Three days ago I asked about the probability of drawing a straight flush when being dealt $26$ out of the $52$ cards of the deck, which Michael wisely solved. Now I'd like to make things more ...
1
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1answer
27 views

Combinatorial optimization problem

I'm having trouble writing a general algorithm for what at first glance appears to be a simple problem. If I have a volume $V_{required}$ that can be made from two smaller, different volumes how can ...
0
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0answers
28 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
0
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1answer
15 views

Partition set of $n$ elements until each partition contains $1$ element. Must terminate after exactly $n-1$ iterations?

Suppose I have a set of $n$ elements and I want to partition the set (split into two) until each partition contains a single element. How do I see that the terminating case must occur after exactly ...
1
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0answers
22 views

Is there a name for the relationship between matching combinations?

Is there a term that describes the relationship between $\binom 3 1 = \binom 3 2$ or $\binom 5 2 = \binom 5 3$? Symmetric comes to mind, but I was wondering if a specific term is used to describe ...
1
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1answer
15 views

A conjecture on binomial factors

Can any one help me prove the following conjecture: \begin{equation} \sum_{p=1}^{\min(n,m+1)}C_{m+1}^p C_{n-1}^{p-1}=\sum_{p=1}^{\min(m+1,n+1)}C_n^{p-1}C_m^{p-1}=C_{m+n}^n \end{equation} Here ...
2
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1answer
30 views

Counting the functions with f(i) ≤ f(i+1) for all i=1,..,n-1

How can I determine how many functions are weakly monotone increasing from $[n]\equiv \{1,..,n\}$ to itself: $$ f:[n] \to [n] \text{ so that } f(i) \leq f(i+1) \; \forall i\in[n-1]$$ Thank you for ...
4
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1answer
32 views

Variant Generating Function related to Euler Numbers

The generating function $$\frac{2e^x}{e^{2x}+1}=\sum_{n\ge 0}E_k\frac{x^k}{k!}$$ counts the number of alternating permutations of a set with an even number of elements. My question is this, if we ...
3
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2answers
29 views

Triangular Array's Recursive Formula Breakdown

I have the following polynomials: $$1$$ $$z-1$$ $$z^2-2z+3$$ $$z^3-3z^2+9z-15$$ $$z^4-4z^3+18z^2-60z+93$$ $$z^5-5z^4+30z^3-150z^2+465z-725$$ $$...$$ They are generated both recursively and explicitly. ...
1
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1answer
25 views

How many different towers, with regards to colour, can be built?

You are going to build a tower with coloured blocks. There are ten available blocks, of which three are white, two are red, two are yellow, one is green, one is blue and one is black. The tower you ...
0
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1answer
13 views

Which is the more likely outcome when dealing cards.

Suppose you are given 6 cards. Which is more likely, you get $3$ different value cards with value having $2$ suits. (e.g. two aces two kings and two jacks). Or $2$ different value cards with $3$ ...
2
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2answers
30 views

Number of ways to place $K$ objects in $N^3$ cube

On how many ways I can place $K$ objects in $N \times N \times N$ cube, assuming that in every coordinate $x$, $y$, $z$ (i.e. in every "row") may be at most one object? For example, if $N = 2$ and $K ...
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2answers
31 views

How many 3 letters-long codes can be made by 5 different letters?

You have five letters: C, H, E, S, T How many different codes, consisting of three letters, can be made from the above letters? I'd say ${5}\choose{3}$ is the correct answer, since the order of the ...
1
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0answers
34 views

Partitioning real numbers with sum $1$ to sets

If the sum of a finite number of positive real numbers is $1$ and each of them is less than $x$, then those real numbers can be partitioned into $50$ sets (some of which may be empty) such that the ...
2
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0answers
44 views

Different coloured bottles of two different sizes

Question The table below shows the distribution by colour (green, blue and red) and size (small and large) of a collection of $20$ bottles. All other features of the bottles are exactly the same. ...
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0answers
26 views

Sum of products of K numbers taken from N numbers in closed form

Let's say i have 5 numbers, $A,B,C,D,E$. I want to know the sum of all the possible products of some or all of these numbers each taken at most once. Instead of a lot of multiplications and additions ...
2
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1answer
35 views

What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples: The set of all finite ...
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3answers
50 views

How to approach combinatorics proofs like these.

Okay so I have been trying to solve problems for my course and keep running into persistent issues with proofs. For example. Prove the following: $${n\choose r}={n-1\choose r-1}+{n-1\choose r}$$ ...
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0answers
13 views

Enumerating set combinations in an order that maximises the number of previously unseen subsets

Consider a set $S=\{a,b,c,d,e,f,g,h,i,j,k\}$, $\left|S\right|=11$. There are ${11 \choose 5} = 462$ combinations of $S$'s members of size $5$. There are $462! \approx 1.419 × 10^{1032}$ possible ...
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0answers
15 views

partition of integers proof

For each partition σ = (λ1, . . . , λk), define the weight function $w^∗(σ)$ = k. Let $Φ^∗P_n (x) $be the generating series for $P_n$ with respect to $w^*$. Prove that for all n ∈ N, $Φ_{P_n} ...
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0answers
32 views

Solve the following recurrence relation in two variables [on hold]

How to solve this recurrence $$S(m,n)=S(m,n-1)+S(m-1,n-1)+S(m-1,n)$$ with base conditions $$S(1,1)=3,\; S(0,n)=S(m,0)=1.$$ This recurrence came up when I tried to solve this problem: Find the ...
0
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1answer
28 views

q-binomial Identity

Unfortunately I am not able to solve the following problem: I tried finding a bijection similar to the prove of this binomial identity: $$\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$$ ...
1
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1answer
53 views

The numbers of functions : There are not exist $f(i) < f(i+1) <f(i+2)$

I solved this problem some days ago. Find the numbers of functions $f$ that satisfy these three conditions: (1) $f$ is a bijection (2) $f : \{1, 2, 3,4\} \to \{1, 2, 3,4\}$ (3)We do not ...
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0answers
55 views

Generating series using partitions

A partition of $n$ is a monotone decreasing sequence of positive integers which sum up to $n$; i.e. $(\lambda_1,...,\lambda_k)$ where $\lambda_1 +···+\lambda_k = n$ and $\lambda_1 ≥ \lambda_2 ≥ ··· ≥ ...
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0answers
26 views

Cardinality of a set of permutations of integers mod $p$.

Let $p$ be a prime number. I wonder if there is a closed formula for the number of permutations $\sigma$ of $ \{0,1,\ldots ,p-1\}$ such that $$\sum_x x \cdot \sigma(x) \equiv 0 \mod p$$ Obs: The ...
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0answers
11 views

How big can a $k$-sum free set be?

Let $S \subset [1, \dots, n]$. Say that $S$ is $k$-sum free if, for any $\{z_i\} \in \mathbb{Z}$ such that this equation holds: $z_1 s_1 + \dots + z_{|S|}s_{|S|} = 0$ we have $|z_1| + \dots + ...
3
votes
1answer
80 views

Power series as fractions

This is what I did: \begin{equation*} (x^3-x^6)x^6[x+x^2+x^3+..], \\ \frac{(x^3-x^6)x^6}{1-x}. \end{equation*} What mistake did I make? And, How to solve this: $1+3x^2+9x^4+27x^6+...+3^{157}x^{314}$ ...
0
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2answers
29 views

How to show $x,y,z \in A$ - Functions, Combinatorics

If $A \subseteq \{1,2,3,4,5,6\}$, how to show that for every $A$ there are $x,y,z \in \{1,2,3,4,5,6\}$, where $x,y,z$ can also be the same or at least not different from each other, and the following ...
0
votes
1answer
23 views

How many ways there are to arrange a boolean $2\times5$ matrix such that there won't be two zeros one above the other

How many ways there are to arrange a boolean $2\times5$ matrix such that there won't be two zeros one above the other. For example, this is not allowed ...
0
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0answers
33 views

Combinatorial Identity, combinatorial proof

Let $n$, $m$, $r$ and $s$ be in $\mathbb{N}$. Give $x$ and $y$ in terms of $n$, $m$, $r$ and $s$ and give an combinatorial proof of: \begin{align} {y \choose x} = \sum\limits_{k=-m}^n {r \choose ...
4
votes
1answer
78 views

How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?

Given $n\in \mathbb{N}^+$, what is the possible number of combinations $s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$ Additionally, is there an efficient way to find ...
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2answers
29 views

One-to-one and binary strings [on hold]

Assume $T$ be the set of binary strings of length $30$ with $10$ $1$’s and $20$ $0$’s. Let $X$ be the set of the first $30$ positive integers $\{1,2,3,…,30\}$. Let $Y$ be the set of all subsets of $X$ ...
2
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1answer
31 views

Questions on integer-valued polynomials

An integer-valued polynomial or numerical polynomial is a polynomial $f \in \mathbb Q[x]$ with the property that $f(\mathbb Z)\subseteq \mathbb Z$. The set of numerical polynomials forms a subring ...
6
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0answers
80 views

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$?

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$ so that each column and each row has exactly $n $ $1$'s and $n$ $-1$'s ? I tried for cases $n=1 , 2$ but the solutions were ...
1
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1answer
49 views

transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
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1answer
26 views

How many pairs of digits can you make using 7 digits? (0-6) [on hold]

P.S (0,1) is the same as (1,0) and so on.
2
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2answers
31 views

Drawing colored balls

I have a sack with $15$ red balls, $15$ blue balls, $15$ green balls and $15$ yellow balls (balls of the same color are indistingishable). In how many ways can I take $30$ balls from the sack? $\\ ...
4
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3answers
52 views

Proving that $r{n \choose r}=n{n-1\choose r-1}$

For proving that: $r{n \choose r}=n{n-1\choose r-1}$ I attempted it with: $r{n\choose r}=\frac{rn!}{r!(n-r)!}=\frac{n!}{(r-1)!(n-r)!}$ $n{n-1\choose ...
0
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3answers
38 views

Questions about Two Identities in Derangements

I found in Wiki following identities. I think these are very nice. But I don't know how prove these identities. $!n = \left[\dfrac{n!}{e}\right] = \left\lfloor \dfrac{n!}{e} + ...
1
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1answer
21 views

Quick question for proof on unimodal sequence formula in Enumerative Combinatorics

I am looking at page 238 of Stanley's Enumerative Combinatorics where he says that $\#V_n = \#D_n - \#V_n^1$ because every element in $V_n^1$ appears twice as a value of $\Gamma_1$. Can someone ...
3
votes
2answers
77 views

Formal power series coefficient problem

Find the coefficient of: $[x^{33}](x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}$ I have figured out that I need to use this identity: $(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n $ But I ...
2
votes
2answers
40 views

Prove this binomial identity using induction

prove this identity: $(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n $ using induction. Verification for k=1 is trivial. assuming k= i, proving the identity when k=i+1 is something i ...