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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4
votes
1answer
68 views

Knight movement on chess field

I had this task in programming competition: There are two knights, which are $(p_1,q_1)$ and $(p_2, q_2)$. $(p,q)$ knight is figure, with p(q)-length first step, and q(p)-length second step in ...
5
votes
3answers
437 views

How many possible words of this type can be formed?

We are making $10$ letter words using the letters $A,C,G,T$. How many possible words are there of the form $A...AC...CG...GT...T$ This is where all of the $A's$ go before the all of the $C's$ and ...
3
votes
0answers
63 views

Is it 3-D Catalan numbers?

I am studying Catalan numbers recently but I think that how about 3-D Catalan? So that I imagine following situation ; A man travel through the path-way parallel to $ x, y, z $ axis from O ...
2
votes
1answer
42 views

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV?

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV? Not sure how to do this one. Obviously there are $\frac{10!}{4!}$ anagrams of KOLMOGOROV but I'm not sure how to account ...
1
vote
0answers
38 views

A Combinational identity using permutations

For a distribution {$p_1,p_2, …,p_m$}, with $p_i>0$ and$\sum_1^m{p_i}=1$ , let $J$ be a subset of size $j$, and $m>j\geq1$. It holds that: $$\int_0^1\prod_{i \in J} (x^{-p_i}-1) dx = ...
2
votes
3answers
46 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
votes
1answer
50 views

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

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
votes
1answer
24 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
votes
1answer
51 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 ...
-4
votes
1answer
29 views

Number of different possible armies in Clash of Clans [closed]

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
vote
1answer
31 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
votes
1answer
26 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
votes
1answer
32 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
votes
0answers
53 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
vote
1answer
29 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
votes
0answers
38 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
votes
2answers
21 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
vote
0answers
23 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
vote
1answer
21 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
votes
1answer
33 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
votes
1answer
41 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 ...
4
votes
2answers
47 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
vote
1answer
30 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
votes
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
votes
2answers
31 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 ...
1
vote
2answers
32 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
vote
0answers
45 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
votes
0answers
46 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. ...
0
votes
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
votes
1answer
44 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 ...
0
votes
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: $${n\choose r}={n-1\choose r-1}+{n-1\choose r}$$ ...
0
votes
0answers
16 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 ...
0
votes
1answer
34 views

partition of integers proof

For each partition $\sigma = (\lambda_1,\ldots,\lambda_k)$, define the weight function $w^∗(σ) = k$. Let $\Phi^∗P_n (x)$ be the generating function for $P_n$ with respect to $w^*$. Prove that for all ...
0
votes
0answers
51 views

Solve the following recurrence relation in two variables

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
votes
1answer
29 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
vote
1answer
54 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 ...
0
votes
1answer
72 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 ≥ ··· ≥ ...
1
vote
0answers
28 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 ...
0
votes
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
85 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
votes
2answers
30 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 ...
4
votes
1answer
79 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 ...
2
votes
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
votes
0answers
87 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
vote
1answer
52 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? ...
2
votes
2answers
35 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
votes
3answers
54 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
votes
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
vote
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 ...