LehmerCodes

The Lehmer code of every permutation of $\{1, …, n\}$ — inversion tables $(a_0, …, a_{n-1})$ with $0 \le a_i \le n-1-i$ .

LehmerCodes(n)the Lehmer code of every permutation of

\{1, …, n\}

— inversion tables

(a_0, …, a_{n-1})

with

0 \le a_i \le n-1-i

Typed signatures

Type	Package	Applies
`(integer<0..>) -> indexed_collection<permutation_inversion>`	`combinatorics`	always

Domain: Collections

Details

Count is $n!$ (A000142); order-isomorphic to SymmetricGroup via the Lehmer code bijection.
Each element is the code itself, not the permutation it decodes to.

Definition

Epsil, as the implementation runs it.

count(_n)

_n!

unrank(_n, _r)

Map(
     Function((_r - _r % (_n - j)!) / (_n - j)! % (_n - j + 1), j),
     Range(1, _n, 1),
   )

rank(_n, _x)

Fold(Function(lr * (_n - lj + 1) + _x[lj], lr, lj), 0, Range(1, _n, 1))

valid(_n, _x)

Fold(
      Function(ok_lj && 0 <= _x[lj] && _x[lj] <= _n - lj, ok_lj, lj),
      True,
      Range(1, _n, 1),
    ) if Length(_x) == _n else False

Into and out of PermutationInversion, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

ToLehmerCodePermutation → PermutationInversionEntry i counts the later entries smaller than p(i).

1 more kept as data, checked against the oracles

Basic #

$Count(LehmerCodes(4)) = 24$

\mathrm{Count}(\mathrm{LehmerCodes}(4))

Out

24

The first element in enumeration order

\mathrm{LehmerCodes}(4)[1]

Out

{\operatorname{PermutationInversion}}([0, 0, 0, 0])

About

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