Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St000366
(load all 46 compositions to match this statistic)
Mapping
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of double descents of a permutation. A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001172
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 76%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 4
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 2
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 4
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 3
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 3
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[7,6,5,8,3,2,1,4] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[8,7,6,5,1,2,3,4] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3
[8,7,5,6,1,2,3,4] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[8,7,6,4,2,1,3,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[8,7,6,1,2,3,4,5] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[8,6,5,4,3,1,2,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[8,6,5,4,2,1,3,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[8,6,4,3,2,1,5,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[8,6,4,2,1,3,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3
[8,3,2,1,4,5,6,7] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[7,6,4,5,3,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[6,5,4,7,3,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[7,6,4,3,5,2,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 3
[7,5,4,3,6,2,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 3
[6,5,4,3,2,1,7,8] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4
[5,4,3,2,1,6,7,8] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3
[4,3,2,1,5,6,7,8] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[3,2,1,8,7,6,5,4] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 4
[3,2,1,6,5,8,7,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[4,3,2,1,8,7,6,5] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 4
[5,4,3,2,1,8,7,6] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 4
[6,5,4,3,2,1,8,7] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 4
[4,3,2,1,6,5,8,7] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[5,4,3,2,7,6,1,8] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 3
[4,3,2,7,5,6,8,1] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[4,3,2,8,5,7,6,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[6,3,2,5,4,1,8,7] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,3,2,5,4,7,6,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[8,3,2,7,6,5,4,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 4
[7,4,3,6,5,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[7,6,3,5,4,2,1,8] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St000931
(load all 5 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 75%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1,0]
 => ? = 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
 => 0
[2,1] => [1,2] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 0
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,4,5,3,2] => [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 6
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[6,5,8,7,4,3,2,1] => [1,2,3,4,7,8,5,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4
[3,5,8,7,6,4,2,1] => [1,2,4,6,7,8,5,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4
[4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4
[2,3,5,6,8,7,4,1] => [1,4,7,8,6,5,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[3,2,6,5,4,8,7,1] => [1,7,8,4,5,6,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[1,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 5
[1,7,8,6,5,4,3,2] => [2,3,4,5,6,8,7,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4
[1,6,8,7,5,4,3,2] => [2,3,4,5,7,8,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4
[1,7,6,8,5,4,3,2] => [2,3,4,5,8,6,7,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[1,6,7,8,5,4,3,2] => [2,3,4,5,8,7,6,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,5,8,7,6,4,3,2] => [2,3,4,6,7,8,5,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4
[1,7,6,5,8,4,3,2] => [2,3,4,8,5,6,7,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[1,3,5,7,8,6,4,2] => [2,4,6,8,7,5,3,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,4,3,8,7,6,5,2] => [2,5,6,7,8,3,4,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[1,6,5,4,3,8,7,2] => [2,7,8,3,4,5,6,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,4,3,6,5,8,7,2] => [2,7,8,5,6,3,4,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
[2,1,8,7,6,5,4,3] => [3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4
[1,2,6,7,8,5,4,3] => [3,4,5,8,7,6,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[3,2,1,6,5,8,7,4] => [4,7,8,5,6,1,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,4,2,1,7,8,6,5] => [5,6,8,7,1,2,4,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[2,4,3,1,6,8,7,5] => [5,7,8,6,1,3,4,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[3,2,4,1,6,8,7,5] => [5,7,8,6,1,4,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[2,1,4,3,8,7,6,5] => [5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[2,1,4,3,6,8,7,5] => [5,7,8,6,3,4,1,2] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[5,4,3,2,1,8,7,6] => [6,7,8,1,2,3,4,5] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[3,2,5,4,1,8,7,6] => [6,7,8,1,4,5,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[3,2,1,4,5,8,7,6] => [6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,8,4,7,6,5,3,2] => [2,3,5,6,7,4,8,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[1,8,5,4,7,6,3,2] => [2,3,6,7,4,5,8,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[1,8,7,4,5,6,3,2] => [2,3,6,5,4,7,8,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,8,7,4,6,5,3,2] => [2,3,5,6,4,7,8,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[1,8,6,5,4,7,3,2] => [2,3,7,4,5,6,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,8,7,5,4,6,3,2] => [2,3,6,4,5,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,8,7,5,6,4,3,2] => [2,3,4,6,5,7,8,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[2,1,8,5,4,7,6,3] => [3,6,7,4,5,8,1,2] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
Description
The number of occurrences of the pattern UUU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000731
(load all 40 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
 => [2,1] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0
[2,1,3,4,7,5,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 0
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[2,1,6,3,4,5,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 0
[2,1,7,3,4,5,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 0
[2,1,7,6,5,4,3] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,3] => ? = 3
[3,1,2,4,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[3,2,4,5,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[4,1,2,3,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[4,1,6,2,3,5,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 0
[4,2,3,5,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[4,2,5,1,3,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 0
[4,3,2,5,6,7,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,3,1,4,5,7,6] => ? = 1
[4,3,5,2,6,7,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 0
[4,3,6,5,2,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => ? = 2
[5,1,2,3,4,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[5,1,2,3,6,4,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 0
[5,1,2,3,6,7,4] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[5,1,2,3,7,4,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 0
[5,1,2,6,3,4,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[5,1,6,2,3,4,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 0
[5,1,6,2,3,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 0
[5,2,3,4,6,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[5,2,6,7,1,3,4] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[5,3,2,4,6,7,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,3,1,4,5,7,6] => ? = 1
[5,3,2,6,1,4,7] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,1,5,4,6,7] => ? = 1
[5,3,6,7,1,2,4] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[5,4,3,2,6,1,7] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => ? = 2
[5,4,3,6,2,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => ? = 2
[5,4,6,3,2,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => ? = 2
[5,4,6,7,1,2,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[6,1,2,3,4,7,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[6,1,2,3,7,4,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 0
[6,1,2,7,3,4,5] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[6,2,3,4,5,7,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 0
[6,2,4,5,3,1,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 1
[6,2,5,4,3,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => ? = 2
[6,3,2,5,4,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => ? = 2
[6,3,4,2,5,1,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 0
[6,3,4,5,2,1,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 1
[6,3,4,7,1,2,5] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[6,3,5,4,2,1,7] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => ? = 2
[6,4,3,2,5,1,7] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => ? = 2
[6,4,3,2,7,1,5] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => ? = 2
[6,4,3,5,2,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => ? = 2
[6,4,5,7,1,2,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 0
[6,4,7,2,5,1,3] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 0
[6,4,7,3,5,1,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 0
[6,5,3,4,2,1,7] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => ? = 2
[6,5,7,3,4,1,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 0
[7,2,3,1,4,5,6] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 0
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000052
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 3
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 3
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 4
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 2
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 3
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 2
[6,7,8,5,2,3,4,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 1
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 4
[8,6,7,5,4,3,1,2] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 3
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 3
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 3
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0
[7,6,8,5,4,2,1,3] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 2
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 0
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[7,8,4,5,6,1,2,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[6,7,4,5,8,1,2,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0
[7,6,5,8,3,2,1,4] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 3
[7,5,6,8,2,3,1,4] => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[6,7,5,8,3,1,2,4] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[6,5,7,8,2,1,3,4] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[8,7,5,6,1,2,3,4] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[7,8,5,6,1,2,3,4] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[8,5,6,7,1,2,3,4] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[5,6,7,8,1,2,3,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 0
[8,7,6,1,2,3,4,5] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[8,6,7,1,2,3,4,5] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,8,1,2,3,4,5] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,5,4,2,1,3,6] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 3
[7,8,5,4,1,2,3,6] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[8,7,1,2,3,4,5,6] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[7,8,1,2,3,4,5,6] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000776
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000776: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => [2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 0 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 0 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1 = 0 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4 + 1
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4 + 1
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[6,7,8,3,4,5,2,1] => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[6,7,8,5,2,3,4,1] => [3,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[8,5,6,7,2,3,4,1] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[7,8,3,2,4,5,6,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4 + 1
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[3,4,5,2,6,7,8,1] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[5,2,3,4,6,7,8,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[4,3,2,5,6,7,8,1] => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[3,4,2,5,6,7,8,1] => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[4,2,3,5,6,7,8,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[3,2,4,5,6,7,8,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[2,3,4,5,6,7,8,1] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 5 + 1
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4 + 1
[8,6,7,5,4,3,1,2] => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[7,8,5,6,3,4,1,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[5,6,7,8,3,4,1,2] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[7,8,3,4,5,6,1,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[3,4,5,6,7,8,1,2] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 5 + 1
[7,6,8,5,4,2,1,3] => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 1
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4 + 1
[6,7,8,5,4,1,2,3] => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[8,5,6,7,4,1,2,3] => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 1
[8,6,7,4,5,1,2,3] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
[8,7,4,5,6,1,2,3] => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 1
[7,8,4,5,6,1,2,3] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St001167
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[6,7,8,3,4,5,2,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[6,7,8,5,2,3,4,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
[7,8,3,2,4,5,6,1] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[4,3,2,5,6,7,8,1] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[3,4,2,5,6,7,8,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[8,6,7,5,4,3,1,2] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 0
[3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[7,6,8,5,4,2,1,3] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
[7,8,4,5,6,1,2,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. The top of a module is the cokernel of the inclusion of the radical of the module into the module. For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[6,7,8,3,4,5,2,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[6,7,8,5,2,3,4,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
[7,8,3,2,4,5,6,1] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0
[3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[4,3,2,5,6,7,8,1] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[3,4,2,5,6,7,8,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[8,6,7,5,4,3,1,2] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 0
[3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[7,6,8,5,4,2,1,3] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
[7,8,4,5,6,1,2,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 0
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details. The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001066
(load all 6 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 1
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[6,7,8,3,4,5,2,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 + 1
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 1
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[6,7,8,5,2,3,4,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0 + 1
[7,8,3,2,4,5,6,1] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 1
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0 + 1
[3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0 + 1
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[4,3,2,5,6,7,8,1] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
[3,4,2,5,6,7,8,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 1
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4 + 1
[8,6,7,5,4,3,1,2] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 0 + 1
[3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0 + 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
[7,6,8,5,4,2,1,3] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[7,8,4,5,6,1,2,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 1
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[6,7,8,3,4,5,2,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 + 1
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[8,7,5,6,4,2,3,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 1
[8,6,5,7,3,2,4,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[6,7,8,5,2,3,4,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0 + 1
[7,8,3,2,4,5,6,1] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[6,5,7,4,3,2,8,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 1
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0 + 1
[3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0 + 1
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[4,3,2,5,6,7,8,1] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
[3,4,2,5,6,7,8,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 0 + 1
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4 + 1
[8,6,7,5,4,3,1,2] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[8,7,5,6,3,4,1,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 0 + 1
[3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0 + 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
[7,6,8,5,4,2,1,3] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[7,8,4,5,6,1,2,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000359The number of occurrences of the pattern 23-1. St000732The number of double deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000365The number of double ascents of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000039The number of crossings of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset.