Finite Combinatorics: Notation

(i) Background. This setting extends Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices: its conventions, notation and routine facts are in force.

(ii) Counting. For a finite set $A$, $\#A$ is its number of elements.

(iii) Maps. For sets $A$ and $B$, $B^{A}$ is the set of maps from $A$ to $B$, and $\operatorname{Inj}(A,B)$, $\operatorname{Surj}(A,B)$ and $\operatorname{Bij}(A,B)$ are its subsets of injections, surjections and bijections. For a map $f:A\to B$ and $y\in B$, $f^{-1}\{y\}$ denotes the preimage $f^{-1}[\{y\}]$. For $n\in\mathbb{N}_{0}$, $S_{n}$ is the set of permutations of $[n]$ and $D_{n}$ is the set of derangements of $[n]$.

(iv) Subsets. For a set $A$ and $k\in\mathbb{N}_{0}$, $\mathcal{P}(A)$ is the power set of $A$, and $\mathcal{P}_{k}(A)$ is the set of its $k$-element subsets.

(v) Numbers. For $n,k\in\mathbb{N}_{0}$, $n!$ is the factorial of $n$, $\binom{n}{k}$ the binomial coefficient, $(n)_{k}$ the falling factorial, and $n^{k}$ the power of $n$ for the multiplication of $\mathbb{N}_{0}$; for $k\le n$, $n-k$ is their difference. For $m\in\mathbb{N}$ and $(k_{1},\dots,k_{m})\in\mathbb{N}_{0}^{m}$, $\binom{k_{1}+\dots+k_{m}}{k_{1},\dots,k_{m}}$ is the multinomial coefficient. For a finite set $A$ and a map $f:A\to\mathbb{N}_{0}$, $\sum_{x\in A}f(x)$ and $\prod_{x\in A}f(x)$ are the sum and the product over $A$ formed with the addition and the multiplication of $\mathbb{N}_{0}$, and likewise over intervals.

(vi) Signs. Natural numbers with zero and integers are read in $\mathbb{R}$ through the identification in force by (i).