Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤ
Values
[1] => [1,0,1,0] => [2,1] => 2
[2] => [1,1,0,0,1,0] => [3,1,2] => 3
[1,1] => [1,0,1,1,0,0] => [2,3,1] => 3
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 4
[2,1] => [1,0,1,0,1,0] => [3,2,1] => 6
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 4
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 5
[3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 8
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 6
[2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 8
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 5
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 6
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 10
[3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 12
[3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 12
[2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 12
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 10
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 6
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => 7
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => 12
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 15
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 15
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 10
[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 24
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 15
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 10
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 15
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => 12
[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] => 7
[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] => 8
[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,3,4,5,6] => 14
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => 18
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => 18
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 20
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 30
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 20
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 20
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 20
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 30
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => 18
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 20
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 18
[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] => 14
[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] => 8
[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] => 16
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [7,3,1,2,4,5,6] => 21
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [7,2,3,1,4,5,6] => 21
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => 24
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => 36
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => 24
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 15
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 40
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 30
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 40
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => 24
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 30
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 30
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 40
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 36
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,6,7,1] => 21
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 15
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 24
[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] => 21
[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] => 16
[7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [8,3,1,2,4,5,6,7] => 24
[7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [8,2,3,1,4,5,6,7] => 24
[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [7,4,1,2,3,5,6] => 28
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [7,3,2,1,4,5,6] => 42
[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] => 28
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 30
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => 48
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => 36
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => 48
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 30
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => 30
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 60
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 60
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 60
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 48
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [5,2,3,4,6,7,1] => 28
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 20
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 60
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => 36
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => 30
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 48
[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] => 42
[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [4,2,3,5,6,7,8,1] => 24
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 30
[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] => 28
[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] => 24
[7,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [8,4,1,2,3,5,6,7] => 32
[7,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [8,3,2,1,4,5,6,7] => 48
[7,1,1,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [8,2,3,4,1,5,6,7] => 32
[6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [7,5,1,2,3,4,6] => 35
[6,3,1] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [7,4,2,1,3,5,6] => 56
[6,2,2] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [7,3,4,1,2,5,6] => 42
[6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [7,3,2,4,1,5,6] => 56
[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] => 35
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => 21
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => 60
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => 72
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Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
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].
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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