Identifier
Values
[1,0] => [1,0] => [2,1] => [2,1] => 2
[1,0,1,0] => [1,1,0,0] => [2,3,1] => [3,2,1] => 3
[1,1,0,0] => [1,0,1,0] => [3,1,2] => [3,1,2] => 2
[1,0,1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => 3
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 2
[1,1,1,0,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => [4,1,3,2] => 3
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,5,2,3,1] => 3
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,2,1,4] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,4,1] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 2
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,2,1,4,3] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,5,3,1] => 3
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => 2
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,5,4,2,1] => 4
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 2
[1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,1,2,4,3] => 3
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,1,3,2,4] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,5,3,2] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,1,3,4,2] => 3
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,5,2,3,4,1] => 4
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,6,2,3,1,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,6,2,3,5,1] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,6,2,1,4,5] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,6,2,1,5,4] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => 3
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,6,2,4,1,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,2,5,3,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,2,1,3,5,4] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,4,5,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,2,1,4,3,5] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,1,6,4,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,2,1,4,5,3] => 3
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,6,3,4,1] => 3
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,6,3,1,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,6,3,5,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,2,3,1,5,4] => 3
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,6,4,2,1,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,1,2,3,5,4] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,6,4,2,5,1] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,1,2,4,3,5] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,2,6,4,3] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,1,2,4,5,3] => 3
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,6,4,1] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 2
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [4,2,6,5,3,1] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,1,3,2,4,5] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,1,3,2,5,4] => 3
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,6,4,5,2,1] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,6,3,2,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,1,6,3,4,2] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,6,3,5,2] => 3
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 3
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,1,3,4,2,5] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,1,3,6,4,2] => 3
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [4,1,6,5,3,2] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,1,3,4,5,2] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => 2
[1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [5,4,1,2,8,3,6,7] => [4,8,2,3,5,1,6,7] => 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [8,7,1,5,2,3,4,6] => [5,7,2,8,4,3,6,1] => 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [8,2,1,3,4,5,6,7] => 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [2,3,8,1,4,5,6,7] => [8,2,3,1,4,5,6,7] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 2
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [4,1,2,5,6,8,3,7] => [8,1,2,4,5,6,3,7] => 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => [9,2,1,3,4,5,6,7,8] => 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,9,1,4,5,6,7,8] => [9,2,3,1,4,5,6,7,8] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 2
[] => [] => [1] => [1] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => [10,2,1,3,4,5,6,7,8,9] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [11,1,2,3,4,5,6,7,8,9,10] => 2
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Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Map
promotion
Description
The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.