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\begin{document}

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\section*{generating function}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{combinatorics}{}\paragraph*{{Combinatorics}}\label{combinatorics}

[[!include combinatorics - contents]]

\begin{quote}%
This entry is about the generating functions in the sense of algebraic combinatorics. For another notion see [[generating function in classical mechanics]].


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{generating function} is an element of $R[\![z]\!]$, the [[rig]] of formal [[power series]] over the rig $R$ (which is often taken to be the [[natural numbers]] or the [[rational number]]s), used for purposes of [[combinatorics]]. A general element takes the form

\begin{displaymath}
f(z) = \sum_{n=0}^{\infty}f_n z^n.
\end{displaymath}
If $R$ is taken to be the [[real numbers]] or the [[complex numbers]] (or any subrig), then we can ask whether the power series has a radius of convergence $r \ge 0$ and if there's an [[analytic continuation]]; if so, then we also say that the continuation is the generating function.

When $R = \mathbb{N},$ we can think of the coefficients on $z^n$ as counting the number of ways to put a particular structure on the [[finite set]] $n$. (You get [[structure types]] if you take this literally.)

Multiplying generating functions in the same variable gives

\begin{displaymath}
\left(\sum_{n=0}^{\infty}f_n z^n\right)\left(\sum_{n=0}^{\infty}g_n z^n\right) = \sum_{n=0}^{\infty}\sum_{k=0}^n f_k g_{n-k} z^n,
\end{displaymath}
which effectively says to split up the set into two parts, put the $f$ structure on the first part and the $g$ structure on the second part.

\emph{Ordinary} generating functions (OGFs) describe structures on [[totally ordered sets]], while \emph{exponential} generating functions (EGFs) apply to unordered [[sets]] (whose elements may be distinguished by having different \emph{labels}). For example, the generating function for being an unordered finite set is

\begin{displaymath}
e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots ,
\end{displaymath}
while the generating function for being a finite ordered set is

\begin{displaymath}
\frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots .
\end{displaymath}
If we assume that the zeroth term is $1$, then multiplying generating functions in different variables gives

\begin{displaymath}
\left(\sum_{n=0}^{\infty}f_n x^n\right)\left(\sum_{n=0}^{\infty}g_n z^n\right) = 1 + (f_1 x + g_1 z) + (f_2 x^2 + f_1 g_1 x z + g_2 z^2) + \cdots
\end{displaymath}
If we take the product of countably many generating functions $f^i(x_i)$ and then set $x_i = p_i^{-s},$ where $p_i$ is the $i$th prime, then we get

\begin{displaymath}
1^{-s} + f^1_1 2^{-s} + f^2_1 3^{-s} + f^1_2 4^{-s} + f^3_1 5^{-s} + f^1_1 f^2_1 6^{-s} + \cdots + \prod_{i} f^i_{e_i} (p_i^{e_i})^{-s} + \cdots,
\end{displaymath}
which is called the \emph{Dirichlet} generating function for the family $f^i.$

The product of two Dirichlet generating functions gives

\begin{displaymath}
\left(\sum_{n=1}^{\infty}f_n n^{-s}\right)\left(\sum_{n=1}^{\infty}g_n n^{-s}\right) = \sum_{n=1}^{\infty} \sum_{d|n} f_d g_{n/d} n^{-s},
\end{displaymath}
which effectively says to \emph{factor} the term into two dimensions, apply $f$ to the first and $g$ to the second.

Sometimes we take the \emph{exponents} on $z$ to be in a rig other than the natural numbers. For example, we might set $z^a \cdot z^b = z^{\max(a,b)}$ and $(z^a)^b = z^{a+b};$ such a system lets us talk about the [[rig of costs|cost]] of operations done in parallel (max) or sequentially (+). Similarly, we could take the exponents to be [[binary string]]s when considering instantaneous codes, or [[finite field]] elements when considering structures on a finite collection of objects.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The OGF $T(z)$ for binary [[trees]] (rooted in the plane, counted by number of internal nodes) satisfies $T(z) = 1 + z T(z)^2$, which can be solved as $T(z) = \frac{1-\sqrt{1-4z}}{2z}$. The coefficient of $z^n$ in $T(z)$ is the $n$th \emph{Catalan number} $\binom{2n}{n}/(n+1)$.

Fixed-point free [[involutions]] on a set have the EGF $e^{z^2/2}$, while arbitrary involutions have EGF $e^{z+z^2/2}$, and [[partition\#of\_sets|partitions]] have EGF $e^{e^z-1}$.

In [[quantum field theory]] generating functions often appear as [[partition functions]] and [[vacuum amplitudes]] as functions of [[source fields]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Bogoliubov's formula]] for [[quantum observables]] in [[perturbative quantum field theory]] generated from an [[S-matrix]] functional


\item [[species]]


\item [[Witten conjecture]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Early articles exploring the foundations of generating functions include:

\begin{itemize}%
\item [[Peter Doubilet]], [[Gian-Carlo Rota]], and [[Richard Stanley]], ``On the foundations of combinatorial theory (VI): The idea of generating function'', in \emph{Sixth Berkeley Symposium on Mathematical Statistics and Probability}, Vol. II: \emph{Probability Theory}, University of California, 1972, pp. 267--318. (\href{https://projecteuclid.org/euclid.bsmsp/1200514223}{euclid})


\item [[André Joyal]], \emph{Une th\'e{}orie combinatoire des s\'e{}ries formelles} , Advances in Mathematics 42:1-82 (1981) , \href{http://www.ams.org/mathscinet-getitem?mr=633783}{MR84d:05025}



\end{itemize}
Textbook accounts include:

\begin{itemize}%
\item [[Herbert Wilf]], \emph{generatingfunctionology} (Third Edition), A K Peters, 2006. (\href{https://www.math.upenn.edu/~wilf/DownldGF.html}{book webpage}) (\href{https://www.math.upenn.edu/~wilf/DownldGF.html}{author pdf})


\item [[Philippe Flajolet]] and [[Robert Sedgewick]], \emph{Analytic Combinatorics}, CUP, 2009. (\href{http://algo.inria.fr/flajolet/Publications/book.pdf}{author pdf})



\end{itemize}
[[!redirects generating functions]] [[!redirects ordinary generating function]] [[!redirects exponential generating function]]

[[!redirects generating functional]] [[!redirects generating functionals]]



\end{document}