FoataInversionsToMajor

The inverse of Foata's major-to-inversions bijection: peel off the last letter, rotate the blocks of the rest back.

FindStatMp00316via BinaryWord↗Mp00315via IntegerComposition↗Mp00175via Permutation↗

FoataInversionsToMajor(Permutation)The inverse of Foata's major-to-inversions bijection: peel off the last letter, rotate the blocks of the rest back.

Typed signatures

Type	Package	Applies
((binary_word) -> binary_word) & ((composition) -> composition) & ((permutation) -> permutation)	combinatorics	always

Domain: Combinatorial maps

Details

• Takes a Permutation and returns a Permutation — a typed map, so a wrong carrier is a type error rather than a wrong answer.
• FindStat's Mp00175, Sage's foata_bijection_inverse. It sends the number of inversions to the major index.
• Also takes a IntegerComposition, returning a IntegerComposition: The inverse of Foata's bijection on compositions: peel off the last part, rotate the blocks of the rest back.
• FindStat's Mp00315.
• Also takes a BinaryWord, returning a BinaryWord: The inverse of Foata's bijection on words: peel off the last letter, rotate the blocks of the rest back.
• FindStat's Mp00316.

Examples

Basic #

the one-line word $231$

live ↗

In$\mathrm{FoataInversionsToMajor}({\operatorname{Permutation}}([2, 3, 1]))$Out${\operatorname{Permutation}}([2, 3, 1])$

the composition $2 + 1 + 3$

live ↗

In$\mathrm{FoataInversionsToMajor}({\operatorname{IntegerComposition}}([2, 1, 3]))$Out${\operatorname{IntegerComposition}}([2, 1, 3])$

the binary word $01100$

live ↗

In$\mathrm{FoataInversionsToMajor}({\operatorname{BinaryWord}}([0, 1, 1, 0, 0]))$Out${\operatorname{BinaryWord}}([1, 0, 1, 0, 0])$

About

Examples	3
Crosswalk	FindStat ×3