MajorIndex

The major index of $p$: the sum of the descent positions $\sum_{p_i > p_{i+1}} i$.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000290via BinaryWord✓↗St000769via IntegerComposition✓↗St000004via Permutation✓↗St000330via StandardTableau✓↗

MajorIndex(p)the major index of a one-line permutation $p$

CombinatoricaIndex[p]SageMathenumeratio_stat("Permutation", p, "MajorIndex")

Typed signatures

Type	Package	Applies
(binary_word | composition | dyck_path | list | permutation | standard_tableau) -> integer | number	combinatorics	always

Domain: Permutation statistics

Details

• Introduced by MacMahon, who proved it equidistributed with Inversions over $S_n$
• Both are Mahonian, so their common generating function is the q-factorial $[n]_q!$
• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation, and also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• The distinction from Descents is the entire content of the statistic.
• A separate definition exists for DyckPath (The sum of the descent positions of the step word — where an up step is followed by a down step.) but is not the one declared: one head, one owner.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else Sum(Filter(
                                     Range(1, Length(_x) - 1),
                                     Function(_x[i] > _x[i + 1], i),
                                   ))

on DyckPath, over _x

0 if Length(_x) < 2 else Sum(Filter(
                                     Range(1, Length(_x) - 1),
                                     Function(_x[i] == 1 && _x[i + 1] == 0, i),
                                   ))

on IntegerComposition, over _x

0 if Length(_x) < 2 else Sum(Filter(
                                     Range(1, Length(_x) - 1),
                                     Function(_x[i] > _x[i + 1], i),
                                   ))

on BinaryWord, over _x

0 if Length(_x) < 2 else Sum(Filter(
                                     Range(1, Length(_x) - 1),
                                     Function(_x[i] > _x[i + 1], i),
                                   ))

Examples

Basic #

one descent, at position 1

live ↗

In$\mathrm{MajorIndex}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$1$

descents at positions 1 and 2

live ↗

In$\mathrm{MajorIndex}(\bigl\lbrack3, 2, 1\bigr\rbrack)$Out$3$

no descents

live ↗

In$\mathrm{MajorIndex}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$0$

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{MajorIndex}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$1$

the one-line word $2413$

live ↗

In$\mathrm{MajorIndex}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$2$

the standard tableau with rows $13 \mid 24$, whose descents are $1$ and $3$

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1, 3], [2, 4]]), \text{MajorIndex})$Out$4$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{MajorIndex})$Out$0$

Known value · FindStat St000004

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{MajorIndex})$Out$0$

Known value · FindStat St000004

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{MajorIndex})$Out$1$

Known value · FindStat St000004

About

Examples	10
Checked against	Combinatorica 6 agreeSageMath 9 agree
Crosswalk	FindStat ×4CombinatoricaSageMath

Implementation

referenceenginenotatio

Checked against the implementation over every permutation of 1..6.

nativeenginetypescriptpackages/symbols/combinatorics/combinatorics/collections/src/stats.ts:majorIndex

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica's Index is the major index: the sum of the descent positions, 1-based, as ours.

mappedsagepackages/oracle/src/mappings.ts

See also: Inversions, Descents, SymmetricGroup