Identifier
-
Mp00128:
Set partitions
—to composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000430: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1,0] => [1] => 0
{{1,2}} => [2] => [1,1,0,0] => [2,1] => 0
{{1},{2}} => [1,1] => [1,0,1,0] => [1,2] => 0
{{1,2,3}} => [3] => [1,1,1,0,0,0] => [3,2,1] => 0
{{1,2},{3}} => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 0
{{1,3},{2}} => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 0
{{1},{2,3}} => [1,2] => [1,0,1,1,0,0] => [1,3,2] => 0
{{1},{2},{3}} => [1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 1
{{1,2,3,4}} => [4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
{{1,2,3},{4}} => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
{{1,2,4},{3}} => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
{{1,2},{3,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
{{1,2},{3},{4}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 2
{{1,3,4},{2}} => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
{{1,3},{2,4}} => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
{{1,3},{2},{4}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 2
{{1,4},{2,3}} => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
{{1},{2,3,4}} => [1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 0
{{1},{2,3},{4}} => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 2
{{1,4},{2},{3}} => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 2
{{1},{2,4},{3}} => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 2
{{1},{2},{3,4}} => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 2
{{1},{2},{3},{4}} => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 4
{{1,2,3,4,5}} => [5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
{{1,2,3,5},{4}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
{{1,2,3},{4,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
{{1,2,3},{4},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
{{1,2,4,5},{3}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
{{1,2,4},{3,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
{{1,2,4},{3},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
{{1,2,5},{3,4}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
{{1,2},{3,4,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
{{1,2},{3,4},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1,2,5},{3},{4}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
{{1,2},{3,5},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1,2},{3},{4,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 7
{{1,3,4,5},{2}} => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
{{1,3,4},{2,5}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
{{1,3,4},{2},{5}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
{{1,3,5},{2,4}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
{{1,3},{2,4,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
{{1,3},{2,4},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1,3,5},{2},{4}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
{{1,3},{2,5},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1,3},{2},{4,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 7
{{1,4,5},{2,3}} => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
{{1,4},{2,3,5}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
{{1,4},{2,3},{5}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1,5},{2,3,4}} => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
{{1},{2,3,4,5}} => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0
{{1},{2,3,4},{5}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 3
{{1,5},{2,3},{4}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1},{2,3,5},{4}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 3
{{1},{2,3},{4,5}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 4
{{1},{2,3},{4},{5}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 7
{{1,4,5},{2},{3}} => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
{{1,4},{2,5},{3}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1,4},{2},{3,5}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 7
{{1,5},{2,4},{3}} => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 4
{{1},{2,4,5},{3}} => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 3
{{1},{2,4},{3,5}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 4
{{1},{2,4},{3},{5}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 7
{{1,5},{2},{3,4}} => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
{{1},{2,5},{3,4}} => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 4
{{1},{2},{3,4,5}} => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 3
{{1},{2},{3,4},{5}} => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 7
{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 7
{{1},{2,5},{3},{4}} => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 7
{{1},{2},{3,5},{4}} => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 7
{{1},{2},{3},{4,5}} => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 7
{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 10
{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 0
{{1,2,3,4,5},{6}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 0
{{1,2,3,4,6},{5}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 0
{{1,2,3,4},{5,6}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
{{1,2,3,4},{5},{6}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 4
{{1,2,3,5,6},{4}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 0
{{1,2,3,5},{4,6}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
{{1,2,3,5},{4},{6}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 4
{{1,2,3,6},{4,5}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
{{1,2,3},{4,5,6}} => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 0
{{1,2,3},{4,5},{6}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 6
{{1,2,3,6},{4},{5}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 4
{{1,2,3},{4,6},{5}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 6
{{1,2,3},{4},{5,6}} => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 6
{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 10
{{1,2,4,5,6},{3}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 0
{{1,2,4,5},{3,6}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
{{1,2,4,5},{3},{6}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 4
{{1,2,4,6},{3,5}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
{{1,2,4},{3,5,6}} => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 0
{{1,2,4},{3,5},{6}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 6
{{1,2,4,6},{3},{5}} => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 4
{{1,2,4},{3,6},{5}} => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 6
{{1,2,4},{3},{5,6}} => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 6
{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 10
{{1,2,5,6},{3,4}} => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
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Description
The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of 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.
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