Questions tagged [generating-functions]
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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.
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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 $B(8,5)$... combinatorics generating-functions combinatorial-proofs- 129
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 ... sequences-and-series number-theory generating-functions pi- 235
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 $n$ ... sequences-and-series algebra-precalculus summation generating-functions products- 23
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 $\... sequences-and-series discrete-mathematics generating-functions fibonacci-numbers- 694
Exponential families in combinatorics and in probability: where is the connection?
I'm familiar with the concept of Exponential Family as it appears in probability theory (see e.g. the wikipedia page). Lately, while reading "generatingfunctionology" by H. Wilf, I stumbled ... generating-functions exponential-family- 3
Find probability generating function when P(Y=r)=kP(X=r)
The variable Y can take only the values 1,2,3,... and is such that P(Y=r)=kP(X=r), where X~Po($\lambda$). Show that the probability generating function of Y is given by: $$G_Y(t)=\frac{e^{\lambda t}-1}... probability generating-functions poisson-distribution- 31
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 ... combinatorics generating-functions integer-partitions- 1,164
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 ... combinatorics generating-functions inclusion-exclusion analytic-combinatorics- 217
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 ... combinatorics discrete-mathematics generating-functions- 189
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 ... combinatorics graph-theory generating-functions trees- 1
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 ... combinatorics generating-functions integer-partitions- 251
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 ... discrete-mathematics set-theory generating-functions- 1
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|&... probability statistics generating-functions- 1,408
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 ... combinatorics permutations examples-counterexamples generating-functions- 2,557
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 ... generating-functions integer-partitions- 1,106
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