Identifier
-
Mp00128:
Set partitions
—to composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000423: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1,0] => [1] => 0
{{1,2}} => [2] => [1,1,0,0] => [1,2] => 0
{{1},{2}} => [1,1] => [1,0,1,0] => [2,1] => 0
{{1,2,3}} => [3] => [1,1,1,0,0,0] => [1,2,3] => 1
{{1,2},{3}} => [2,1] => [1,1,0,0,1,0] => [3,1,2] => 0
{{1,3},{2}} => [2,1] => [1,1,0,0,1,0] => [3,1,2] => 0
{{1},{2,3}} => [1,2] => [1,0,1,1,0,0] => [2,3,1] => 0
{{1},{2},{3}} => [1,1,1] => [1,0,1,0,1,0] => [3,2,1] => 0
{{1,2,3,4}} => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 4
{{1,2,3},{4}} => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
{{1,2,4},{3}} => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
{{1,2},{3,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
{{1,2},{3},{4}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 0
{{1,3,4},{2}} => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
{{1,3},{2,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
{{1,3},{2},{4}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 0
{{1,4},{2,3}} => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
{{1},{2,3,4}} => [1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
{{1},{2,3},{4}} => [1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 0
{{1,4},{2},{3}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 0
{{1},{2,4},{3}} => [1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 0
{{1},{2},{3,4}} => [1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 0
{{1},{2},{3},{4}} => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 0
{{1,2,3,4,5}} => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 10
{{1,2,3,4},{5}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
{{1,2,3,5},{4}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
{{1,2,3},{4,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
{{1,2,3},{4},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
{{1,2,4,5},{3}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
{{1,2,4},{3,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
{{1,2,4},{3},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
{{1,2,5},{3,4}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
{{1,2},{3,4,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
{{1,2},{3,4},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1,2,5},{3},{4}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
{{1,2},{3,5},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1,2},{3},{4,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0
{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 0
{{1,3,4,5},{2}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
{{1,3,4},{2,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
{{1,3,4},{2},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
{{1,3,5},{2,4}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
{{1,3},{2,4,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
{{1,3},{2,4},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1,3,5},{2},{4}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
{{1,3},{2,5},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1,3},{2},{4,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0
{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 0
{{1,4,5},{2,3}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
{{1,4},{2,3,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
{{1,4},{2,3},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1,5},{2,3,4}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
{{1},{2,3,4,5}} => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
{{1},{2,3,4},{5}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
{{1,5},{2,3},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1},{2,3,5},{4}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
{{1},{2,3},{4,5}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 0
{{1},{2,3},{4},{5}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 0
{{1,4,5},{2},{3}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
{{1,4},{2,5},{3}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1,4},{2},{3,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0
{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 0
{{1,5},{2,4},{3}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 0
{{1},{2,4,5},{3}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
{{1},{2,4},{3,5}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 0
{{1},{2,4},{3},{5}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 0
{{1,5},{2},{3,4}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 0
{{1},{2,5},{3,4}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 0
{{1},{2},{3,4,5}} => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 1
{{1},{2},{3,4},{5}} => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 0
{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 0
{{1},{2,5},{3},{4}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 0
{{1},{2},{3,5},{4}} => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 0
{{1},{2},{3},{4,5}} => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 0
{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 0
{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 20
{{1,2,3,4,5},{6}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 10
{{1,2,3,4,6},{5}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 10
{{1,2,3,4},{5,6}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 4
{{1,2,3,4},{5},{6}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 4
{{1,2,3,5,6},{4}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 10
{{1,2,3,5},{4,6}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 4
{{1,2,3,5},{4},{6}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 4
{{1,2,3,6},{4,5}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 4
{{1,2,3},{4,5,6}} => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 2
{{1,2,3},{4,5},{6}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 1
{{1,2,3,6},{4},{5}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 4
{{1,2,3},{4,6},{5}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 1
{{1,2,3},{4},{5,6}} => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 1
{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 1
{{1,2,4,5,6},{3}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 10
{{1,2,4,5},{3,6}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 4
{{1,2,4,5},{3},{6}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 4
{{1,2,4,6},{3,5}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 4
{{1,2,4},{3,5,6}} => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 2
{{1,2,4},{3,5},{6}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 1
{{1,2,4,6},{3},{5}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 4
{{1,2,4},{3,6},{5}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 1
{{1,2,4},{3},{5,6}} => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 1
{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 1
{{1,2,5,6},{3,4}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 4
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Description
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
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
bounce path
Description
The bounce path determined by an integer composition.
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