Identifier
Values
[1,0] => [1,1,0,0] => [2,3,1] => [1,2,3] => 1
[1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => [1,4,2,3] => 1
[1,1,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 2
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,4,5,2,3] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,2,5,3,4] => 2
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,3,4] => 2
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 3
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 3
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,5,6,3,2,4] => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,6,4,5,2,3] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,6,3,2,4,5] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,4,5,6,2,3] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,2,6,4,3,5] => 3
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,2,5,6,3,4] => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,6,4,2,3,5] => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,6,3,4,2,5] => 3
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,5,6,2,3,4] => 3
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,2,3,6,4,5] => 3
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,2,6,3,4,5] => 3
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,6,2,3,4,5] => 3
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [1,2,6,7,4,3,5] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [1,2,7,5,6,3,4] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [1,2,7,4,3,5,6] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [1,2,5,6,7,3,4] => 4
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [1,2,3,7,5,4,6] => 4
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [1,2,3,6,7,4,5] => 4
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [1,2,7,5,3,4,6] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [1,2,7,4,5,3,6] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [1,2,6,7,3,4,5] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [1,2,3,4,7,5,6] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [1,2,3,7,4,5,6] => 4
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [1,2,7,3,4,5,6] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 5
[] => [1,0] => [2,1] => [1,2] => 0
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Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.