Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000214: Permutations ⟶ ℤ
Values
[1] => [1,0,1,0] => [2,1] => [2,1] => 1
[2] => [1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 1
[1,1] => [1,0,1,1,0,0] => [2,3,1] => [2,3,1] => 0
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,2,4,3] => 1
[2,1] => [1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,1,4,3] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,1,4,2] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => 0
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,1,3,5,4] => 2
[3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,4,3,2] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,4,3,1] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => 1
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => [2,1,3,4,6,5] => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,2,5,4] => 2
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,3,1,5,4] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => 3
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,5,1] => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,4,1,5,2] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [3,4,2,5,1] => 0
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => [3,2,4,5,6,1] => 1
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 1
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => [1,3,2,4,6,5] => 2
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => [2,3,1,4,6,5] => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,2,1,5,4] => 3
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,3,5,4,1] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [4,2,1,5,3] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [4,3,1,5,2] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => [2,4,3,5,6,1] => 1
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,4,5,2,1] => 1
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [3,4,2,5,6,1] => 0
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => 1
[1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,8,1] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 1
[7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,8,7] => 2
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,3,6,5] => 2
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => [3,2,1,4,6,5] => 3
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => [2,3,4,1,6,5] => 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [1,2,5,3,6,4] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [2,1,5,4,3] => 3
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,1,5,4,2] => 1
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,2,5,4,1] => 2
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => [2,3,5,4,6,1] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,1,5,3,2] => 1
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [4,2,5,3,1] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,3,5,2,1] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [4,3,2,5,6,1] => 2
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [3,4,5,1,6,2] => 0
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [3,4,5,2,6,1] => 0
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,7,1] => [3,4,2,5,6,7,1] => 0
[2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,8,1] => 1
[1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,9,1] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 1
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,1,5,6] => [2,3,4,1,5,7,6] => 1
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 2
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => [2,1,4,3,6,5] => 3
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,1,4,2,6,5] => 1
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => [3,2,4,1,6,5] => 2
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 1
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => [2,1,5,3,6,4] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,5,4,3,2] => 3
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [2,5,4,3,1] => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,2,1] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,2,5,4,6,1] => 2
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => 2
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [4,2,5,3,6,1] => 0
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => [4,3,5,1,6,2] => 1
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => 1
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [4,3,2,5,6,7,1] => [4,3,2,5,6,7,1] => 2
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [3,4,5,6,2,1] => 1
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,7,1] => [3,4,5,2,6,7,1] => 0
[2,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,8,1] => 0
[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,9,1] => 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,10,1] => 0
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => 1
[6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,5,1,6] => [2,3,4,5,1,7,6] => 1
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => 0
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => [2,1,3,6,5,4] => 3
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => [1,4,3,2,6,5] => 3
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => [2,4,3,1,6,5] => 2
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => [3,4,2,1,6,5] => 2
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,2,4,6,5,1] => 2
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => [1,5,3,2,6,4] => 1
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => [2,5,3,1,6,4] => 0
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => [1,5,4,2,6,3] => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [2,5,4,3,6,1] => 2
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => [3,5,4,1,6,2] => 1
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [3,5,4,2,6,1] => 1
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => [4,5,2,1,6,3] => 1
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Description
The number of adjacencies of a permutation.
An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as small descents.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as small descents.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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