Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
St000373: Permutations ⟶ ℤ
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
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Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and 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.
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