Identifier
Values
[1,0] => [2,1] => 0
[1,0,1,0] => [3,1,2] => 0
[1,1,0,0] => [2,3,1] => 0
[1,0,1,0,1,0] => [4,1,2,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => 0
[1,1,0,0,1,0] => [2,4,1,3] => 0
[1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 1
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => 2
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [8,1,2,7,6,3,4,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [6,1,8,2,3,4,5,7] => 0
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [8,1,4,2,6,3,5,7] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,3,1,2,4,5,6,7] => 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [8,3,1,2,4,7,5,6] => 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [8,4,1,2,3,5,6,7] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [8,3,1,5,6,2,4,7] => 3
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [2,6,7,1,3,4,8,5] => 0
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [6,3,8,1,2,7,4,5] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => 0
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [7,8,4,1,6,2,3,5] => 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [8,3,4,1,6,7,2,5] => 4
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [2,7,6,5,1,3,8,4] => 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [8,3,6,5,1,7,2,4] => 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [8,7,4,5,6,1,2,3] => 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => 0
>>> Load all 120 entries. <<<
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [9,3,1,2,4,5,6,7,8] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [9,7,8,1,2,3,4,5,6] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 0
[] => [1] => 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [10,9,8,1,2,3,4,5,6,7] => 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 0
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Description
The number of mid points of decreasing 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 $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also St000119The number of occurrences of the pattern 321 in a permutation..
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.