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Your data matches 99 different statistics following compositions of up to 3 maps.
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Matching statistic: St000153
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 3
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St001498
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 86%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 86%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {2,3}
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {2,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3,3,4}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3,3,4}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3,3,4}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3,3,4}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3,3,4}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3,3,4}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,3,3,3,3,3,4,4,4,5}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001355
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 80%●distinct values known / distinct values provided: 71%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 80%●distinct values known / distinct values provided: 71%
Values
[1] => [1,0]
=> []
=> => ? = 1
[1,2] => [1,0,1,0]
=> [1]
=> 10 => 1
[2,1] => [1,1,0,0]
=> []
=> => ? = 2
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 110 => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 100 => 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> => ? ∊ {2,3}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> => ? ∊ {2,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,2,2,2,3,4}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,2,2,2,3,4}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,2,2,2,3,4}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,2,2,2,3,4}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,2,2,2,3,4}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,2,2,2,3,4}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 3
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,5}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 1010011010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 1001011010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 1010100110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1001100110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 1010010110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 1001010110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1000110110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1000110110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6}
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.
Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Matching statistic: St001632
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 77%●distinct values known / distinct values provided: 57%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 77%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1
[1,2] => [1,2] => [2,1] => ([],2)
=> ? = 2
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,2,3}
[1,3,2] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,2,3}
[2,1,3] => [2,1,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,2,3}
[2,3,1] => [2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,4}
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,5,4] => [2,1,5,4,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,4,3,5] => [2,1,5,4,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,5,3,4] => [2,1,5,4,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,5,4,3] => [2,1,5,4,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,1,4,5] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,3,1,5,4] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,3,4,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,3,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,3,5,4,1] => [2,5,4,3,1] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,4,1,5,3] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,4,3,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,4,3,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,4,5,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,5,1,4,3] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,5,3,1,4] => [2,5,4,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,5,3,4,1] => [2,5,4,3,1] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,5,4,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,1,2,5,4] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,1,4,2,5] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,1,4,5,2] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,1,5,2,4] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,1,5,4,2] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,2,1,4,5] => [3,2,1,5,4] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,4,1,5] => [3,2,5,1,4] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [3,2,5,1,4] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0
[4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000620
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 1
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,2,1,5,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,4,1,5,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,4,2,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,4,2,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,4,5,2,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[3,5,1,4,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[3,5,2,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,5,2,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,1,2,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,1,3,2,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,1,3,5,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,2,1,5,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[4,2,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,2,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,3,5,2,1] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[4,5,1,3,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[4,5,2,1,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,5,2,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[4,5,3,1,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[5,1,2,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[5,1,3,2,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000567
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 67%●distinct values known / distinct values provided: 57%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 67%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1]
=> []
=> ? = 1
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {1,2}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {1,2}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,1,1,2,2,3}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St001099
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 67%●distinct values known / distinct values provided: 57%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 67%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1]
=> []
=> ? = 1
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {1,2}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {1,2}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,1,1,2,2,3}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
Description
The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
Matching statistic: St001017
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 86%
Values
[1] => [1,0]
=> []
=> []
=> ? = 1
[1,2] => [1,0,1,0]
=> [1]
=> [1,0]
=> 1
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 2
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {2,3}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {2,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,2,2,3,3,4}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,2,2,3,3,4}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,2,2,3,3,4}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,2,2,3,3,4}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,2,2,3,3,4}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,2,2,3,3,4}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,5}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000137
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 65%●distinct values known / distinct values provided: 57%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 65%●distinct values known / distinct values provided: 57%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 1
[1,2] => [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,2}
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,2}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,2,2,3}
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,2,2,3}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,2,2,3}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,2,2,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000693
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000693: Standard tableaux ⟶ ℤResult quality: 43% ●values known / values provided: 61%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000693: Standard tableaux ⟶ ℤResult quality: 43% ●values known / values provided: 61%●distinct values known / distinct values provided: 43%
Values
[1] => [1]
=> []
=> []
=> ? = 1
[1,2] => [1,1]
=> [1]
=> [[1]]
=> ? ∊ {1,2}
[2,1] => [2]
=> []
=> []
=> ? ∊ {1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [[1]]
=> ? ∊ {0,1,2,2,3}
[2,1,3] => [2,1]
=> [1]
=> [[1]]
=> ? ∊ {0,1,2,2,3}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,1,2,2,3}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,1,2,2,3}
[3,2,1] => [2,1]
=> [1]
=> [[1]]
=> ? ∊ {0,1,2,2,3}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,2,3] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[2,4,3,1] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[3,1,2,4] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[3,4,1,2] => [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[4,1,3,2] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,1,3] => [3,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,4}
[4,3,2,1] => [2,2]
=> [2]
=> [[1,2]]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,3,4] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,5,3,4] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,3,4,1,5] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,5,4,1] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,1,3,5] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,5,1,3] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,1,4,3] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,3,1,4] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,4,3,1] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,1,4,2,5] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,1,5,4,2] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,5,1,4] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,4,2,1,5] => [4,1]
=> [1]
=> [[1]]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,5,2,1] => [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
Description
The modular (standard) major index of a standard tableau.
The modular major index is the usual major index [[St000330]] modulo $n$, where $n$ is the number of boxes in the standard tableau.
The following 89 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000929The constant term of the character polynomial of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001432The order dimension of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001651The Frankl number of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000260The radius of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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