Questions Tagged [Generating-Functions]
Ask Question Generating Functions Are Formed by Making a Series $\Sum_{N\Geq 0} A_N X^n$ out of a Sequence $A_N$. They Are Used to Count Objects in Enumerative...
![Questions Tagged [Generating-Functions]](https://www.alledaags.nl/uploads/questions-tagged-generating-functions_1200.jpg)
Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.
3,961
questions
1
vote
0
answers
16
views
Exponential generating function of the relation $B(n,k)=B(n-1,k-1)+(n-1)B(n-2,k-1)$ —just over $n$—
Let $B(n,k)$ the number of permutations of the set $[n]=\{1,\ldots,n\}$ that are decomposable in $k$ disjoint cycles of order $1$ or $2$. For example, $\mu=(1,3)(2,5)(4)(6,7)(8)$ is counted by $B(8,5)$...
8
votes
1
answer
142
views
A new series for $\frac{1}{\pi}$
Let $C_n$ denote the $n$-th Catalan number defined by
$${\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}}\quad \left(n\geqslant 0\right).}$$
Next, we define ...
1
vote
2
answers
64
views
Product of $n$ terms of sequence where the $n^{th}$ term is of the form $(x^{a^n}+1)$
While practicing from a book I found a product in the form $$(x^{a^1}+1)\cdot(x^{a^2}+1)\cdot(x^{a^3}+1)\cdot(x^{a^4}+1)$$ and was immediately curious if I could a formula to solve the product for $n$ ...
1
vote
1
answer
52
views
Why does my solution of the closed form formula for the Fibonacci sequence differ from the actual solution by a sign?
I have $a_{n}=a_{n-1}+a_{n-2}$ with $a_0=0$ and $a_1=1$.
I've found the generating function $G(x)=\frac{x}{1-x-x^2}$. Solving $1-x-x^2=0$ gives the solutions $\alpha_1=\frac{-1+\sqrt{5}}{2}$ and $\...
0
votes
1
answer
30
views
Exponential families in combinatorics and in probability: where is the connection?
I'm familiar with the concept of Exponential Family as it appears in probability theory (see e.g. the wikipedia page).
Lately, while reading "generatingfunctionology" by H. Wilf, I stumbled ...
0
votes
1
answer
21
views
Find probability generating function when P(Y=r)=kP(X=r)
The variable Y can take only the values 1,2,3,... and is such that
P(Y=r)=kP(X=r), where X~Po($\lambda$). Show that the probability
generating function of Y is given by: $$G_Y(t)=\frac{e^{\lambda
t}-1}...
1
vote
1
answer
26
views
Counting Partitions of $k$ into $j$ distinct parts, with sizes restricted by a sequence.
This question has come up in my research, but since I usually do not do combinatorics, I am struggling to find out any information regarding these sequences of numbers.
Let $L_n = L_1, L_2\dots$ be a ...
2
votes
1
answer
50
views
Flajolet & Sedgewick: symbolic inclusion-exclusion example error?
I'm reading Analytic Combinatorics by Flajolet and Sedgewick, and I have an issue with the following argument from page 208:
The authors claim to derive $P(z, u) = e^{(u-1)z}/(1-z)$, the "BGF of ...
3
votes
1
answer
68
views
How many numbers between 4000 and 9999 have sum of digits equal to ten (why is exponential generating functions not correct)?
How many numbers in between 4000 and 9999 have sum of digits equal to ten?
My attempt
My thinking was to use exponential generating functions because 4510 is not equal to 5401 so the order should ...
-1
votes
0
answers
24
views
Number of unlabeled and unordered rooted trees with n vertices and k leaves
I am interested in calculating the number of possible unlabeled and unordered rooted trees with $n$ nodes and $k$ leaves. Based on one of the answers to Number of unlabeled rooted trees with n ...
0
votes
0
answers
31
views
positive to non-negative integers - partition and generating function
Question: How many solutions in positive integers are there of the equation $3u + 5v + 7w + 9t = 40$?
The answer is the number of solutions of $3U + 5V + 7W + 9T = 16$
in non-negative integers, as ...
-3
votes
1
answer
47
views
How to calculate the number of subsets, elements of which add up to 25 in the set {1,2,3,4,5….,24,25} using generating functions
For example: one such subset would be:
{1,2,3,5,6,8} because 1+2+3+5+6+8=25
{25} would be another subset
I want to know how to count the total number of these in the total(2^25 total subsets)
I tried ...
2
votes
0
answers
22
views
Show that $Q(s) = \frac{1-P(s)}{1-s}$ for $|s|<1$
Let $X$ be a random variable with pmf $P(X=j)=p_j$
Set $P(X>j)=q_j= p_{j+1} + p_{j+2}+ \cdots $
Then the series for $Q(s)$ converges in $|s| < 1$
Show that $Q(s) = \frac{1-P(s)}{1-s}$ for $|s|&...
0
votes
0
answers
33
views
Two sequences with the same pairwise sums [duplicate]
Find all integers $n \geq 2$ for which there exist distinct sequences of integers $a_1, a_2, \dots a_n$ and $b_1,b_2,\ldots,b_n$ such that the set of sums $a_i + a_j$, $1\leq i < j \leq n$, is a ...
1
vote
1
answer
36
views
Prove that number of partitions of $n$ into parts $2,5,11$ modulo $12$ is the same as the number of partitions into distinct parts $2,4,5$ modulo 6
Let $A(n)$ denote the number of partitions of the positive integer $n$ into parts congruent to $2$, $5$, or $11$ modulo $12$. Let $B(n)$ denote the number of partitions of $n$ into distinct parts ...
Chloe Bennett explores the intersection of pop culture, streaming entertainment, digital trends, and contemporary lifestyle. Her weekly commentary reaches thousands of culture enthusiasts.
