Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤ
Values
{{1,2}} => [2,1] => [2,1] => ([(0,1)],2) => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 3
{{1},{2,3}} => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3) => 1
{{1,2,4},{3}} => [2,4,3,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
{{1,3},{2,4}} => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => 1
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
{{1},{2,3,4}} => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 1
{{1,4},{2},{3}} => [4,2,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
{{1},{2,4},{3}} => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
{{1},{2},{3,4}} => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,4},{2,3,5}} => [4,3,5,1,2] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2
{{1,5},{2,3,4}} => [5,3,4,2,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
{{1},{2,3,4,5}} => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 5
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 3
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 10
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 4
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [2,6,1,4,5,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3
{{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6
{{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 5
{{1,2},{3,4,6},{5}} => [2,1,4,6,5,3] => [6,1,3,5,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,5},{3,6},{4}} => [2,5,6,4,1,3] => [5,2,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 5
{{1,2,5},{3},{4,6}} => [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 2
{{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => [6,5,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11
{{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => [4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => [5,1,3,6,2,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3
{{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => [6,3,5,1,4,2] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
{{1,2},{3,6},{4,5}} => [2,1,6,5,4,3] => [6,5,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
{{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 4
{{1,2},{3,6},{4},{5}} => [2,1,6,4,5,3] => [4,6,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5
{{1,2},{3},{4,6},{5}} => [2,1,3,6,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => [5,1,3,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4
{{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => [6,1,4,3,2,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => [6,1,4,3,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => [6,1,2,5,4,3] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 7
{{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1,3},{2,6},{4,5}} => [3,6,1,5,4,2] => [6,1,5,3,4,2] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
{{1,5},{2,3,4,6}} => [5,3,4,6,1,2] => [2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1},{2,3,4,6},{5}} => [1,3,4,6,5,2] => [6,2,3,5,1,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6
{{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
{{1,5},{2,3,6},{4}} => [5,3,6,4,1,2] => [2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 3
{{1,5},{2,3},{4,6}} => [5,3,2,6,1,4] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 2
{{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9
{{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => [4,2,3,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3
{{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1},{2,3,6},{4,5}} => [1,3,6,5,4,2] => [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
{{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
{{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1},{2,3,6},{4},{5}} => [1,3,6,4,5,2] => [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
{{1},{2,3},{4,6},{5}} => [1,3,2,6,5,4] => [6,2,5,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6
{{1},{2,3},{4},{5,6}} => [1,3,2,4,6,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,4,5},{2},{3,6}} => [4,2,6,5,1,3] => [5,1,6,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 4
{{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
{{1,4,6},{2},{3,5}} => [4,2,5,6,3,1] => [3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 2
{{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => [6,2,1,4,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => [6,2,1,3,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1,4},{2},{3,6},{5}} => [4,2,6,1,5,3] => [6,2,1,4,5,3] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,5},{2,4,6},{3}} => [5,4,3,6,1,2] => [5,3,2,1,6,4] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,5},{2,4},{3,6}} => [5,4,6,2,1,3] => [5,4,2,1,6,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => [3,2,4,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
{{1},{2,4,5},{3,6}} => [1,4,6,5,2,3] => [2,5,6,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1,6},{2,4},{3,5}} => [6,4,5,2,3,1] => [4,2,6,5,3,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
{{1},{2,4,6},{3,5}} => [1,4,5,6,3,2] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 3
{{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => [4,3,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1},{2,4,6},{3},{5}} => [1,4,3,6,5,2] => [2,6,3,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4
{{1},{2,4},{3,6},{5}} => [1,4,6,2,5,3] => [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
{{1},{2,4},{3},{5,6}} => [1,4,3,2,6,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
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search for individual values
searching the database for the individual values of this statistic
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
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