Identifier
Values
[1] => [1] => [1,0] => [1] => 0
[1,1] => [1,1] => [1,0,1,0] => [1,2] => 0
[2] => [2] => [1,1,0,0] => [2,1] => 0
[1,1,1] => [1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 0
[1,2] => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 0
[2,1] => [1,2] => [1,0,1,1,0,0] => [1,3,2] => 0
[3] => [3] => [1,1,1,0,0,0] => [3,2,1] => 0
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,3] => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
[2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
[3,1] => [1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 0
[4] => [4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[1,1,2,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 0
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,2,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
[1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
[2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
[3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 0
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0
[5] => [5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 0
[1,1,1,2,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 0
[1,1,2,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[1,1,2,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 1
[1,1,3,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 0
[1,2,1,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 0
[1,2,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 1
[1,2,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 0
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 1
[1,3,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 0
[1,4,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 1
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 0
[2,1,1,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
[2,1,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
[2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 0
[2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
[2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 0
[2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 0
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 0
[3,1,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 0
[3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 0
[3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 0
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 0
[4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 0
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 0
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 0
[6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,2,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 1
[1,1,1,2,1,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => 1
[1,1,1,3,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => 1
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => 1
[1,1,2,2,1] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => 1
[1,1,3,1,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,5,4,7] => 1
[1,1,4,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 1
[1,2,1,1,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => 1
[1,2,1,2,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => 1
[1,2,2,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => 1
[1,2,3,1] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => 1
[1,3,1,1,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => 1
[1,3,2,1] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => 1
[1,4,1,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => 1
[1,5,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 1
[2,1,1,1,1,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => 0
[2,1,1,2,1] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => 1
[2,1,2,1,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => 0
[2,1,3,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => 0
[2,2,1,1,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => 0
[2,2,2,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => 0
[2,3,1,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => 0
[2,4,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => 0
[3,1,1,1,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => 0
[3,1,2,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => 0
[3,2,1,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => 0
[3,3,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => 0
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Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
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..
Map
rotate back to front
Description
The back to front rotation of an integer composition.