Factorial number system

A mixed-radix positional notation using factorials as place values: $n={\sum }_{i\ge 1}{a}_{i}i!$ with $0\le a_i\le i$. For lexicographic order on permutations, division by $(k-1)!$ selects the next symbol when $k$ symbols remain; the remainder gives the rank within that block.

Factoradic indexing uses the factorial number system to rank or unrank permutations of an ordered $n$-element set. For zero-based rank $r<n!$, write $r={\sum }_{k=1}^{n-1}{d}_{k}k!$ with $0\le d_k\le k$. Read the digits from highest factorial down, selecting the indexed item from the remaining ordered elements and removing it.

For rank $4$ among permutations of $(a,b,c)$, $4=2\cdot 2!+0\cdot 1!$. Select index $2$, giving $c$, then index $0$, giving $a$, then the remaining $b$: the result is $cab$. This produces a bijection between ranks $0,\dots ,n!-1$ and lexicographically ordered permutations.