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\begin{document}

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\section*{Möbius inversion}

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

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

[[!include combinatorics - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{mbius_inversion_for_posets}{M\"o{}bius inversion for posets}\dotfill \pageref*{mbius_inversion_for_posets} \linebreak
\noindent\hyperlink{IncidenceAlgebrasZetaFn}{Incidence algebras and the zeta function}\dotfill \pageref*{IncidenceAlgebrasZetaFn} \linebreak
\noindent\hyperlink{the_mbius_function_and_the_mbius_inversion_formula}{The M\"o{}bius function and the M\"o{}bius inversion formula}\dotfill \pageref*{the_mbius_function_and_the_mbius_inversion_formula} \linebreak
\noindent\hyperlink{properties_of_the_mbius_function}{Properties of the M\"o{}bius function}\dotfill \pageref*{properties_of_the_mbius_function} \linebreak
\noindent\hyperlink{duality}{Duality}\dotfill \pageref*{duality} \linebreak
\noindent\hyperlink{the_product_formula}{The product formula}\dotfill \pageref*{the_product_formula} \linebreak
\noindent\hyperlink{the_galois_coconnection_theorem}{The Galois coconnection theorem}\dotfill \pageref*{the_galois_coconnection_theorem} \linebreak
\noindent\hyperlink{examples_of_mbius_inversion}{Examples of M\"o{}bius inversion}\dotfill \pageref*{examples_of_mbius_inversion} \linebreak
\noindent\hyperlink{inclusionexclusion}{Inclusion-exclusion}\dotfill \pageref*{inclusionexclusion} \linebreak
\noindent\hyperlink{numbertheoretic_mbius_inversion}{Number-theoretic M\"o{}bius inversion}\dotfill \pageref*{numbertheoretic_mbius_inversion} \linebreak
\noindent\hyperlink{mbius_inversion_for_categories}{M\"o{}bius inversion for categories}\dotfill \pageref*{mbius_inversion_for_categories} \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}

The classical \emph{M\"o{}bius inversion formula} is a principle originating in [[number theory]], which says that if $f$ and $g$ are two (say, [[complex]]-valued) functions on the positive [[natural numbers]], then

\begin{displaymath}
f(n) = \sum_{d\mid n} g(d)
\end{displaymath}
if and only if

\begin{displaymath}
g(n) = \sum_{d\mid n} f(d)\mu(n/d)
\end{displaymath}
where the sums run over the divisors of $n$, and where $\mu$ is the \emph{M\"o{}bius function}:

\begin{displaymath}
\mu(n) = \begin{cases}(-1)^r & n\,\text{is the product of}\,r\,\text{distinct primes}\\0 & \text{otherwise}\end{cases}
\end{displaymath}
[[Gian-Carlo Rota]] \hyperlink{Rota64}{(1964)} described a vast generalization of the M\"o{}bius inversion formula, which was at the same time a generalization of the classical [[inclusion-exclusion]] principle in [[combinatorics]]. Rota's formulation of M\"o{}bius inversion applies to functions defined over [[partially ordered sets]], while further category-theoretic generalizations have also been proposed.

\hypertarget{mbius_inversion_for_posets}{}\subsection*{{M\"o{}bius inversion for posets}}\label{mbius_inversion_for_posets}

\hypertarget{IncidenceAlgebrasZetaFn}{}\subsubsection*{{Incidence algebras and the zeta function}}\label{IncidenceAlgebrasZetaFn}

The \textbf{incidence algebra} of a poset $P$ (relative to a [[commutative ring]] $R$) is an [[associative unital algebra]] defined as follows. Its elements are functions $f : P \times P\to R$ such that $x \nleq y$ implies $f(x,y) = 0$. Pointwise addition $f+g$ and scalar multiplication $r\cdot f$ are defined straightforwardly ($(f+g)(x,y) = f(x,y) + g(x,y)$, $(r\cdot f)(x,y) = r\cdot f(x,y)$), while the \emph{convolution product} $f*g$ is defined as follows:

\begin{displaymath}
(f*g)(x,y) = \sum_{x \le z \le y} f(x,z) \cdot g(z,y)
\end{displaymath}
Note that the unit of the convolution product is the [[Kronecker delta]]:

\begin{displaymath}
\delta(x,y) = \begin{cases}1 & x = y \\ 0 & \text{otherwise}\end{cases}
\end{displaymath}
We write $I(P)$ for the incidence algebra of $P$. A special and important element of $I(P)$ is given by

\begin{displaymath}
\zeta(x,y) = \begin{cases}1 & x \le y \\ 0 & \text{otherwise}\end{cases}
\end{displaymath}
referred to as the \textbf{zeta function} of $P$.

\hypertarget{the_mbius_function_and_the_mbius_inversion_formula}{}\subsubsection*{{The M\"o{}bius function and the M\"o{}bius inversion formula}}\label{the_mbius_function_and_the_mbius_inversion_formula}

A \textbf{M\"o{}bius function} of $P$ is defined as an inverse to the zeta function with respect to the convolution product. Such an inverse always exists if the poset is [[locally finite poset|locally finite]], that is, if every [[interval]] $[x,y] = \{z \mid x \le z\le y\}$ is finite.

\begin{prop}
\label{}\hypertarget{}{}
\hyperlink{Rota64}{(Rota 1964)}: If $P$ is a locally finite poset, then there exists a function $\mu$ such that $\zeta * \mu = \mu * \zeta = \delta$.

\end{prop}
\begin{proof}
$\mu(x,y)$ can be computed by induction on the number of elements in the interval $[x,y]$. In the base case we set $\mu(x,x) = 1$. Otherwise in the case that $x \ne y$, we assume by induction that $\mu(x,z)$ has already been defined for all $z$ in the half-open interval $[x,y)$, and set

\begin{displaymath}
\mu(x,y) = -\sum_{x \le z \lt y} \mu(x,z).
\end{displaymath}
From this definition, it is immediate that if

\begin{displaymath}
x_0 \lt x_1 \lt \cdots \lt x_n
\end{displaymath}
is a chain of length $n$ in $P$, where each $x_{i+1}$ [[covering relation|covers]] $x_i$, then for all $i \leq j$,

\begin{displaymath}
\mu(x_i,x_j) = \begin{cases}1 & i = j \\ -1 & j - i = 1\\ 0 & j - i \ge 2\end{cases}
\end{displaymath}
It follows that both of the sums $\sum_{x \le z \le y} \mu(x,z)$ and $\sum_{x \le z \le y} \mu(z,y)$ add to zero unless $x = y$, hence that $\zeta*\mu = \mu*\zeta = \delta$.

\end{proof}
We are now ready to state the M\"o{}bius inversion formula(s) for posets.

\begin{prop}
\label{}\hypertarget{}{}
\hyperlink{Rota64}{(Rota 1964)}: Let $f$ and $g$ be functions defined over a locally finite poset $P$. Then

\begin{displaymath}
f(y) = \sum_{x \le y} g(x)
\qquad\text{if and only if}\qquad
g(y) = \sum_{x \le y} f(x)\mu(x,y).
\end{displaymath}
Dually, we also have that

\begin{displaymath}
f(x) = \sum_{y \ge x} g(y)
\qquad\text{if and only if}\qquad
g(x) = \sum_{y \ge x} \mu(x,y)f(y).
\end{displaymath}
\end{prop}
\begin{proof}
An easy way of proving the M\"o{}bius inversion formulas \hyperlink{KungRotaYan}{(Kung et al.)} is to view $\zeta$ and $\mu$ as matrices acting on the column vectors $f$ and $g$ by multiplication. Then the first proposition simply says that $f = g * \zeta$ iff $f * \mu = g$, while the dual formulation says that $f = \zeta * g$ iff $\mu * f = g$.

\end{proof}
\hypertarget{properties_of_the_mbius_function}{}\subsection*{{Properties of the M\"o{}bius function}}\label{properties_of_the_mbius_function}

\hypertarget{duality}{}\subsubsection*{{Duality}}\label{duality}

The [[opposite poset]] $P^{op}$ has the same M\"o{}bius function as $P$ except with the arguments exchanged:

\begin{displaymath}
\mu_{P^{op}}(x,y) = \mu_P(y,x)
\end{displaymath}
\hypertarget{the_product_formula}{}\subsubsection*{{The product formula}}\label{the_product_formula}

The M\"o{}bius function of the [[cartesian product]] $P \times Q$ of two posets $P$ and $Q$ is the product of their M\"o{}bius functions:

\begin{displaymath}
\mu_{P\times Q}((x,y), (x',y')) = \mu_P(x,x') \cdot \mu_Q(y,y')
\end{displaymath}
for all $x,x' \in P$ and $y,y' \in Q$.

\hypertarget{the_galois_coconnection_theorem}{}\subsubsection*{{The Galois coconnection theorem}}\label{the_galois_coconnection_theorem}

Suppose $P$ and $Q$ are related by an [[adjunction]] (i.e., covariant [[Galois connection]]) $f \dashv g : Q \to P$. Then for all $x \in P$, $y \in Q$, the following equation holds:

\begin{displaymath}
\sum_{\substack{u \in P\\ f(u) = y}} \mu_P(x,u) = \sum_{\substack{v \in Q \\ g(v) = x}} \mu_Q(v,y)
\end{displaymath}
(This result essentially goes back to \hyperlink{Rota64}{Rota (1964)} but in a slightly different formulation; for the above formulation, see \hyperlink{Greene81}{Greene (1981)}, \hyperlink{AguiarFS}{Aguiar and Ferrer Santos (2000)}, \hyperlink{KungRotaYan}{Kung et al. (2009)}.)

\hypertarget{examples_of_mbius_inversion}{}\subsection*{{Examples of M\"o{}bius inversion}}\label{examples_of_mbius_inversion}

\hypertarget{inclusionexclusion}{}\subsubsection*{{Inclusion-exclusion}}\label{inclusionexclusion}

Let $\mathcal{P}X$ be the lattice of subsets of a finite set $X$. $\mathcal{P}X$ is isomorphic to the cartesian product of $|X|$ many copies of $\mathbf{2} = \{ 0 \lt 1 \}$, and so by the product rule for the M\"o{}bius function, we have

\begin{displaymath}
\mu(I,J) = (-1)^{|J|-|I|}
\end{displaymath}
for any pair of subsets $I,J \subseteq X$ such that $I \subseteq J$. The M\"o{}bius inversion formula then says that for any functions $f$ and $g$ defined on $\mathcal{P}X$, we have

\begin{displaymath}
f(J) = \sum_{I \subseteq J} g(I)
\qquad\text{if and only if}\qquad
g(J) = \sum_{I \subseteq J} (-1)^{|J|-|I|} f(I)
\end{displaymath}
or dually that

\begin{displaymath}
f(I) = \sum_{J \supseteq I} g(J)
\qquad\text{if and only if}\qquad
g(I) = \sum_{J \supseteq I} (-1)^{|J|-|I|} f(J)
\end{displaymath}
This is called the \emph{inclusion-exclusion} principle.

As a textbook example of the inclusion-exclusion principle in action, suppose we want to compute $g(I)$ = the number of permutations of $X$ fixing exactly the elements in $I$. Well, it is easy to compute the number of permutations of $X$ fixing \emph{at least} the elements in $I$,

\begin{displaymath}
f(I) = \sum_{J \supseteq I} g(J) = (|X|-|I|)!
\end{displaymath}
but then we can compute $g$ in terms of $f$ via M\"o{}bius inversion. As a special case, when $|X| = n$ and $I = \emptyset$, we recover the classical formula for the number of \emph{derangements} of $n$ elements:

\begin{displaymath}
!n = \sum_{k=0}^n (-1)^k \binom{n}{k} (n-k)! = \sum_{k=0}^n (-1)^k \frac{n!}{k!}
\end{displaymath}
where a derangement is defined as a permutation fixing no elements.

\textbf{Exercise:} Prove that $n! = \sum_{k=0}^n (-1)^k \binom{n}{k} (n-k)^n$.

\hypertarget{numbertheoretic_mbius_inversion}{}\subsubsection*{{Number-theoretic M\"o{}bius inversion}}\label{numbertheoretic_mbius_inversion}

The classical M\"o{}bius inversion formula is recovered by considering the set of positive natural numbers ordered by divisibility.

\hypertarget{mbius_inversion_for_categories}{}\subsection*{{M\"o{}bius inversion for categories}}\label{mbius_inversion_for_categories}

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

\begin{itemize}%
\item [[Euler characteristic]]
\item [[incidence algebra]]
\item [[inclusion-exclusion]]
\item [[zeta polynomial]]

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

The classical reference on M\"o{}bius inversion for posets is:

\begin{itemize}%
\item [[Gian-Carlo Rota]], \emph{On the foundations of combinatorial theory I: theory of M\"o{}bius functions} , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340--368.

\end{itemize}
and a more modern reference is:

\begin{itemize}%
\item Joseph P. S. Kung, [[Gian-Carlo Rota]], Catherine H. Yan. Combinatorics: The Rota Way. Cambridge, 2009.

\end{itemize}
M\"o{}bius inversion for categories is discussed in:

\begin{itemize}%
\item [[Tom Leinster]], \emph{Notions of M\"o{}bius inversion}, Bulletin of the Belgian Mathematical Society 19 (2012) 911-935, \href{http://arxiv.org/abs/1201.0413}{arXiv:1201.0413v3}


\item $n$Cafe: \href{http://golem.ph.utexas.edu/category/2011/05/mbius_inversion_for_categories.html}{M\"o{}bius inversion for categories}



\end{itemize}
Other references include:

\begin{itemize}%
\item Curtis Greene, \emph{The M\"o{}bius Function of a Partially Ordered Set}, NATO Advanced Study Institute, Series C (1981), 555-581.

\end{itemize}
\begin{itemize}%
\item [[Joachim Kock]], \emph{Incidence Hopf algebras}, \href{http://mat.uab.es/~kock/seminars/incidence-algebras.pdf}{pdf}


\item Imma G\'a{}lvez-Carrillo, [[Joachim Kock]], Andrew Tonks, \emph{Decomposition spaces, incidence algebras and M\"o{}bius inversion}, \href{http://arxiv.org/abs/1404.3202}{arxiv/1404.3202}


\item Metropolis, N.; Rota, Gian-Carlo, \emph{Witt vectors and the algebra of necklaces}, Advances in Mathematics \textbf{50} (2): 95--125 (1983) \href{http://dx.doi.org/10.1016%2F0001-8708%2883%2990035-X}{doi}


\item [[Marcelo Aguiar]] and Walter Ferrer Santos, \emph{Galois connections for incidence Hopf algebras of partially ordered sets}, Adv. Math. 151 (2000), 71-100.


\item wikipedia \href{http://en.wikipedia.org/wiki/M%C3%B6bius_function}{M\"o{}bius function}, \href{http://en.wikipedia.org/wiki/Moreau%27s_necklace-counting_function}{necklace polynomial}



\end{itemize}
category: combinatorics

[[!redirects Mobius inversion]] [[!redirects Möbius function]] [[!redirects Moebius function]] [[!redirects Moebius inversion]] [[!redirects necklace polynomial]]



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