| $f$-class | Any $f$ | Injective f | Surjective $f$ |
| Distinct $f$ | $n$-sequence in $X$
$x^n$ | $n$-permutation of $X$
$x^{\underline n}$ | composition of $N$ with $x$ subsets
$x!\{{n\atop x}\}$ |
$S_n$ orbits
$f\circ S_n$ | $n$-multisubset of $X$
$\binom{x+n-1}n$ | $n$-subset of $X$
$\binom xn$ | composition of $n$ with $x$ terms
$\binom{n-1}{n-x}$ |
$S_x$ orbits
$S_x∘f$ | partition of $N$ into ≤$x$ subsets
$\displaystyle\sum_{k=0}^x\left\{{n\atop k}\right\}$ | partition of $N$ into ≤$x$ elements
$[n\leq x]$ | partition of $N$ into $x$ subsets
$\left\{{n\atop x}\right\}$ |
$S_n×S_x$ orbits
$S_x∘f∘S_n$ | partition of $n$ into ≤$x$ parts
$p_x(n+x)$ | partition of $n$ into ≤$x$ parts 1
$[n\leq x]$ | partition of $n$ into $x$ parts
$p_x(n)$ |
hbghlyj
Posted at 2021-10-9 15:56:34
