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
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 $x,y,z$ ...
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
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: $$\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$$ ...
0
votes
0answers
37 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 ...
4
votes
1answer
40 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 ...
0
votes
4answers
114 views

Number of certain (0,1)-matrices, Stanley's Enumerative Combinatorics

Stanley's Enumerative Combinatorics (http://www-math.mit.edu/~rstan/ec/ec1.pdf) contains next fact: 1.1.3 Example. Let f(n) be the number of n × n matrices M of $0$’s and $1$’s such that every row and ...
1
vote
7answers
183 views

How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?

Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to ...
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 ...
2
votes
1answer
32 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 ...
1
vote
1answer
20 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 ...
0
votes
1answer
332 views

Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...
1
vote
0answers
44 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 ...
3
votes
2answers
80 views

Finding all k-size subgraphs

I have no experience with advanced combinatorics, but I have to solve a problem that I think I will need advanced combinatorial techniques, correct me if I am wrong. Let $G$ be a large directioned ...
1
vote
1answer
27 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 ...
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 ...
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 $3$ ...
1
vote
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 ...
5
votes
5answers
636 views

Given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

I tried to solve it using induction, but that got me no were, in the middle of the equation stat appearing ks that I don't see how to get out of the equation. I think the easiest way to prove it, it's ...
2
votes
1answer
292 views

Count ways to reach last layer

Consider directed graph which has $N + 2$ layers numbered from left to right by integers from $0$ up to $N + 1$. The leftmost ($0$) and the rightmost ($N + 1$) layers both contain only one vertex ...
0
votes
2answers
31 views

Number of ways to assign $8$ subjects to $4$ people s.t. one gets an odd number of subjects

I am asked to find the number of ways to assign $8$ subjects to $4$ people, such that the third person always gets an odd number of subjects. What I did was consider the problem as putting ...
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 ...
4
votes
1answer
267 views

a combinatorial exercise

The problem asks us to calculate: $$ \sum_{i = 0}^{n}(-1)^i \binom{n}{i} \binom{n}{n-i}$$ The way I tried solving is: The given sum is the coefficient of $x^n$ in $ (1+x)^n(1-x)^n $, which is $ (1 ...
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
15 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 ...
1
vote
1answer
35 views

How many $n$ x $n$ matrices with this given property?

I would like to know how many $n$ x $n$ matrices are there containing elements that are either $1$ or $-1$, such that the product of the elements in each row and column is $-1$?
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 ...
15
votes
1answer
2k views

Why are there only a few known Ramsey numbers?

Can someone explain in a simple way, why there are so few known exact Ramsey Numbers? I guess it's because there are no efficient algorithms for this task, but are there so many combinations to test? ...
1
vote
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 ...
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
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? ...
3
votes
2answers
88 views

Prove ten objects can be divided into two groups that balances each other when placed on the two pans of balance. [closed]

There are 10 objects with total weight 20, each of the weight being a positive integer. Given that none of the weights exceed 10, prove ten objects can be divided into two groups that balances each ...
1
vote
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 ...
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 + ...
0
votes
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 ...
2
votes
1answer
43 views

Proving $R(3,4)\le 9$

I am trying to prove $R(3,4)\le 9$. This is my approach: For any $K_9$ we have (WLOG) at least 4 red edges by the pigeonhole principle. Consider all of the edges between these 4 red edges, if ...
3
votes
1answer
84 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: $1+3x^2+9x^4+27x^6+...+3^{157}x^{314}$ ...
1
vote
2answers
187 views

Probability of drawing a pair from a poker hand, unordered with replacement?

I am wondering what is the probability with which you can draw a pair in a 5-card hand from a standard 52-card deck, if order does not matter in the context of cards in the hand, and if the cards can ...
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 ...
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 ...
3
votes
1answer
55 views

Game with stones

Alice and Bob are playing a game. There is a pile of 2014 stones. Alice and Bob alternate taking stones from the pile, with Alice going fi rst. The number of stones Alice takes must be a power of 2 ...
-2
votes
2answers
29 views

One-to-one and binary strings [closed]

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 $X$ ...
6
votes
0answers
86 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 ...
7
votes
1answer
222 views

Combinatorial Interpretation of Graph Theoretical Relation Involving Chebyshev Polynomials

Given a graph $G$ and its adjacency matrix $A$. The $(i,j)$-th element of $A^r$ gives the number of ways to get from vertex $i$ to $j$ in $r$ steps (including backtracking). Now, the number of ...
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 ...
2
votes
2answers
33 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? $\\ ...
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 ...
3
votes
2answers
85 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 ...