Identifier
Values
[1] => [1,0] => [1] => {{1}} => 1
[2] => [1,0,1,0] => [1,2] => {{1},{2}} => 3
[1,1] => [1,1,0,0] => [2,1] => {{1,2}} => 2
[3] => [1,0,1,0,1,0] => [1,2,3] => {{1},{2},{3}} => 6
[2,1] => [1,0,1,1,0,0] => [1,3,2] => {{1},{2,3}} => 4
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => {{1,2,3}} => 3
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => {{1},{2},{3},{4}} => 10
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => {{1},{2},{3,4}} => 7
[2,2] => [1,1,1,0,0,0] => [3,2,1] => {{1,3},{2}} => 5
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => {{1},{2,3,4}} => 5
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => {{1,2,3,4}} => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => 15
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => {{1},{2},{3},{4,5}} => 11
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => {{1},{2,4},{3}} => 8
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => {{1},{2},{3,4,5}} => 8
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => {{1,3,4},{2}} => 6
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => {{1},{2,3,4,5}} => 6
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => 5
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => 21
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}} => 16
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => {{1},{2},{3,5},{4}} => 12
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}} => 12
[3,3] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => {{1,4},{2},{3}} => 9
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => {{1},{2,4,5},{3}} => 9
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => {{1},{2},{3,4,5,6}} => 9
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => {{1,4},{2,3}} => 7
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => {{1,3,4,5},{2}} => 7
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => {{1},{2,3,4,5,6}} => 7
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => 6
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => {{1},{2},{3},{4,6},{5}} => 17
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => {{1},{2,5},{3},{4}} => 13
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => {{1},{2},{3,5,6},{4}} => 13
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => {{1,4,5},{2},{3}} => 10
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => {{1},{2,5},{3,4}} => 10
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => {{1},{2,4,5,6},{3}} => 10
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => {{1,4,5},{2,3}} => 8
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => {{1,3,4,5,6},{2}} => 8
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => {{1},{2},{3,6},{4},{5}} => 18
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => {{1,5},{2},{3},{4}} => 14
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => {{1},{2,5,6},{3},{4}} => 14
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => {{1},{2},{3,6},{4,5}} => 14
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => {{1,5},{2},{3,4}} => 11
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => {{1,4,5,6},{2},{3}} => 11
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => {{1},{2,5,6},{3,4}} => 11
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => {{1,5},{2,3,4}} => 9
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => {{1,4,5,6},{2,3}} => 9
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => {{1},{2,6},{3},{4},{5}} => 19
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => {{1,5,6},{2},{3},{4}} => 15
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => {{1},{2,6},{3},{4,5}} => 15
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => {{1,5},{2,4},{3}} => 12
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => {{1,5,6},{2},{3,4}} => 12
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => {{1},{2,6},{3,4,5}} => 12
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => {{1,5,6},{2,3,4}} => 10
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => {{1,6},{2},{3},{4},{5}} => 20
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => {{1,6},{2},{3},{4,5}} => 16
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => {{1},{2,6},{3,5},{4}} => 16
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => {{1,5,6},{2,4},{3}} => 13
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => {{1,6},{2},{3,4,5}} => 13
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => {{1,6},{2,3,4,5}} => 11
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => {{1,6},{2},{3,5},{4}} => 17
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => {{1,6},{2,4,5},{3}} => 14
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => {{1,6},{2,5},{3},{4}} => 18
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}} => 15
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Description
Sum of the maximal elements of the blocks of a set partition.
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..
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].