# Tagged Questions

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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### Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} e_{h}(\...
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### Coefficients of (generating) function

If I have the generating function \begin{equation*} A(x)= \frac{1}{(1-x^{10})\cdot(1-x^5)\cdot(1-x) }\,, \end{equation*} what is a clean way to find the coefficients of $x^{n}$. This coefficient ...
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### Number of non-negative solutions to an equation - check my work

I tried to solve the following question and would love someone to check my work. Let $\displaystyle x_1,x_2,x_3,x_4$ $100$ digits long numbers where every digit is $1$ or $2$. Find the number ...
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### Extinction probability of binomial branching process tends to poisson one.

The folowing is stated and proved in the random graphs book by Luczak, Janson, Rucinski and this is on page 108 in the Giant component section. I can't understand why the conclusion follows from the ...
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### Writing a generating function

I came across the following question and it's a bit different from what I'm used to... Write a generating function for each of the following: 1) You are making an Easter basket with at most two ...
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### Why the moment-generating function, rather than the characteristic function?

I'm wondering why the moment-generating function is worth discussing (say, in basic probability courses, or in textbooks, rather than research), when the characteristic function appears to completely ...
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### A recursion formula related to *Catalan numbers*

When I was working on a problem related to Catalan Number, I deduced the following recursion formula: a_{l,r}=a_{l-1,r}+a_{l-1,r-1}+a_{l-1,r-2}+\ldots+a_{l-1,l-1},\\ where \quad r \ge ...
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### How to show that two probability generating functions are equal?

From Grimmett's Probability and Random Processes: Let $G_a(s) := \sum_0^\infty a_is^i$ where $a = \{a_i : i \geq 0\}$ is a real sequence. Uniqueness. If $G_a(s) = G_b(s)$ for $|s| < R'$ ...