SubexcedantSeq

The singular-inhabitant constructor for a subexcedant sequence, a word whose entry $i$ (from 1) lies in $0 \le a_i \le i - 1$ .

SubexcedantSeq(list)The singular-inhabitant constructor for a subexcedant sequence, a word whose entry

i

(from 1) lies in

0 \le a_i \le i - 1

Typed signatures

Type	Package	Applies
`((list<integer>) -> subexcedant_seq) & ((permutation) -> subexcedant_seq)`	`combinatorics`	always

Domain: Collections

Details

A SubexcedantSeqs element.
SubexcedantSeq(permutation) is the permutation's inversion sequence: entry $i$ counts the earlier entries larger than the $i$ -th. Its total is the inversion count, and Permutation(sequence) is the way back.
The two enumerate in different orders, so the conversion is a bijection and not a rank-preserving one: SymmetricGroup runs its last two positions fastest, SubexcedantSeqs its last entry.

Maps

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

PermutationSubexcedantSeq → PermutationinverseThe permutation a subexcedant sequence is the inversion sequence of.

SubexcedantSeqPermutation → SubexcedantSeqinverseEntry i counts the earlier entries larger than p(i): the permutation's inversion sequence.

Basic #

From a permutation: $2$ has two larger entries before it, $1$ one, $4$ and $3$ none

\operatorname{SubexcedantSeq}({\operatorname{Permutation}}([3, 1, 4, 2]))

Out

{\operatorname{SubexcedantSeq}}([0, 1, 0, 2])

And back

\operatorname{Permutation}({\operatorname{SubexcedantSeq}}([0, 1, 0, 2]))

Out

{\operatorname{Permutation}}([3, 1, 4, 2])

About

Examples	2
Checked against	Wolfram Language 2 differJulia 2 differRust 2 differ
Crosswalk	OEIS