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Your data matches 548 different statistics following compositions of up to 3 maps.
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Matching statistic: St001314
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
St001314: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
Matching statistic: St001021
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> []
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,0,0]
=> []
=> []
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 0
[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]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,3,5,5,5,6}
[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]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[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]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,3,5,5,5,6}
[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]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 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]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {1,3,5,5,5,6}
[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]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 0
[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]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {1,3,5,5,5,6}
[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]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 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]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,3,5,5,5,6}
[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]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 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]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[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]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[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]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[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]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? ∊ {1,3,5,5,5,6}
Description
Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000516
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> []
=> [1] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,0,0]
=> []
=> []
=> [1] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,0,0,0]
=> []
=> []
=> [1] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1] => ? = 0
[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]
=> [8,1,4,5,2,7,3,6] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [7,3,4,1,6,2,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [4,1,2,5,8,7,3,6] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [3,1,4,7,6,2,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 0
[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]
=> [7,1,4,8,2,3,5,6] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [6,3,7,1,2,4,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [4,1,2,8,7,3,5,6] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [3,1,7,6,2,4,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 0
[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]
=> [8,1,2,3,7,4,5,6] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[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]
=> [7,1,2,6,3,4,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[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]
=> [6,1,4,5,2,7,3] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 1
[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]
=> [4,1,2,5,6,7,3] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[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]
=> [7,1,4,6,2,3,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0
[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]
=> [4,1,2,7,6,3,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 4
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 3
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [1] => ? ∊ {0,0,1,2,2,2,3,3,3,3,4,5,5,5,5,6}
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000594
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000594: Set partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 71%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000594: Set partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> []
=> {}
=> ? = 0
[1,0,1,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,0}
[1,1,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,0}
[1,1,1,0,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,0}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,0}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> {{1,3,6,10},{2,5,9},{4,8},{7}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,0,0,1,1,2,3,3,5,5,5,5,6}
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block.
Matching statistic: St000225
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 66%●distinct values known / distinct values provided: 57%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 66%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 0
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0}
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0}
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,2,2}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,3,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 3
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000454
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 66%●distinct values known / distinct values provided: 43%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 66%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2}
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001877
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 43% ●values known / values provided: 64%●distinct values known / distinct values provided: 43%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 43% ●values known / values provided: 64%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([(0,1)],2)
=> ? = 0
[1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001398
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 62%●distinct values known / distinct values provided: 57%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 62%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 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]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {2,2}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {2,2}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
Number of subsets of size 3 of elements in a poset that form a "v".
For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Matching statistic: St001811
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 62%●distinct values known / distinct values provided: 43%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 62%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0}
[1,1,0,0]
=> []
=> []
=> [] => ? ∊ {0,0}
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? ∊ {2,2}
[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]
=> [2,5,1,3,7,4,6] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [4,1,2,6,3,5] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [2,3,1,4,7,5,6] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [2,1,3,6,4,5] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 2
[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]
=> [2,5,1,6,7,3,4] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [4,1,5,6,2,3] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [2,3,1,6,7,4,5] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [2,1,5,6,3,4] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[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]
=> [2,3,4,6,7,1,5] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[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]
=> [2,3,5,6,1,4] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[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]
=> [2,5,1,3,4,6] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 0
[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]
=> [2,3,1,4,5,6] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[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]
=> [2,5,1,6,3,4] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1
[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]
=> [2,3,1,6,4,5] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6}
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St000205
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> []
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 0
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1}
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1}
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,2,2,2}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,2,2,2}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,2,2,2}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,2,2,2}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,2,2,2}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> ? ∊ {0,0,0,1,2,2,2,2,2,2,3,3,3,3,5,5,5,6}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
The following 538 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000624The normalized sum of the minimal distances to a greater element. St000750The number of occurrences of the pattern 4213 in a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St001330The hat guessing number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001435The number of missing boxes in the first row. St001964The interval resolution global dimension of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000441The number of successions of a permutation. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000028The number of stack-sorts needed to sort a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000451The length of the longest pattern of the form k 1 2. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001423The number of distinct cubes in a binary word. St001556The number of inversions of the third entry of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001902The number of potential covers of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001846The number of elements which do not have a complement in the lattice. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001720The minimal length of a chain of small intervals in a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001651The Frankl number of a lattice. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001438The number of missing boxes of a skew partition. St001875The number of simple modules with projective dimension at most 1. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001176The size of a partition minus its first part. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000934The 2-degree of an integer partition. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001820The size of the image of the pop stack sorting operator. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St000237The number of small exceedances. St000359The number of occurrences of the pattern 23-1. St000884The number of isolated descents of a permutation. St001394The genus of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001569The maximal modular displacement of a permutation. St001684The reduced word complexity of a permutation. St000045The number of linear extensions of a binary tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000741The Colin de Verdière graph invariant. St000764The number of strong records in an integer composition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000883The number of longest increasing subsequences of a permutation. St000456The monochromatic index of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000455The second largest eigenvalue of a graph if it is integral. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001083The number of boxed occurrences of 132 in a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St000352The Elizalde-Pak rank of a permutation. St000842The breadth of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000369The dinv deficit of a Dyck path. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000496The rcs statistic of a set partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000647The number of big descents of a permutation. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000699The toughness times the least common multiple of 1,. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St001115The number of even descents of a permutation. St000834The number of right outer peaks of a permutation. St001050The number of terminal closers of a set partition. St001621The number of atoms of a lattice. St000408The number of occurrences of the pattern 4231 in a permutation. St001862The number of crossings of a signed permutation. St000068The number of minimal elements in a poset. St000669The number of permutations obtained by switching ascents or descents of size 2. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001889The size of the connectivity set of a signed permutation. St000366The number of double descents of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000001The number of reduced words for a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000635The number of strictly order preserving maps of a poset into itself. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000563The number of overlapping pairs of blocks of a set partition. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000295The length of the border of a binary word. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000153The number of adjacent cycles of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000648The number of 2-excedences of a permutation. St000944The 3-degree of an integer partition. St001153The number of blocks with even minimum in a set partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001625The Möbius invariant of a lattice. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001645The pebbling number of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000145The Dyson rank of a partition. 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. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001098The 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 vertex labelled trees. St001099The 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. St001100The 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 trees. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001867The number of alignments of type EN of a signed permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000422The energy of a graph, if it is integral. St001722The number of minimal chains with small intervals between a binary word and the top element. St000091The descent variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St000017The number of inversions of a standard tableau. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000370The genus of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001434The number of negative sum pairs of a signed permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001691The number of kings in a graph. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001797The number of overfull subgraphs of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000078The number of alternating sign matrices whose left key is the permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000492The rob statistic of a set partition. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000694The number of affine bounded permutations that project to a given permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000805The number of peaks of the associated bargraph. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001151The number of blocks with odd minimum. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001220The width of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001340The cardinality of a minimal non-edge isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001768The number of reduced words of a signed permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001884The number of borders of a binary word. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001946The number of descents in a parking function. St001949The rigidity index of a graph. St000058The order of a permutation. St000236The number of cyclical small weak excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000469The distinguishing number of a graph. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000679The pruning number of an ordered tree. St000758The length of the longest staircase fitting into an integer composition. St000839The largest opener of a set partition. St000906The length of the shortest maximal chain in a poset. St001062The maximal size of a block of a set partition. St001093The detour number of a graph. St001267The length of the Lyndon factorization of the binary word. St001352The number of internal nodes in the modular decomposition of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000230Sum of the minimal elements of the blocks of a set partition. St001487The number of inner corners of a skew partition. St000219The number of occurrences of the pattern 231 in a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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