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Your data matches 318 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,3,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,3,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [5,1,2,6,4,3] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,1,5,6,3,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [3,1,5,4,6,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,2,3,5,4] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [5,7,1,2,3,6,4] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,5,6,2,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [6,1,2,7,4,5,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,2,6,4,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,7,4,1,2,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ([(0,1),(0,6),(1,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,3,4,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,4,5,1,7,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 3
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001330
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [1,5] => ([(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,1,5,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [1,5] => ([(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,2,3,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,1,2,5,3,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[]
=> [] => [] => ?
=> ? = 0
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000455
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 60%●distinct values known / distinct values provided: 25%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 60%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => ([],1)
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[]
=> [] => ?
=> ? = 0 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001001
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 2
[]
=> []
=> []
=> []
=> ? = 0 - 2
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001493
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [5,2,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [9,5]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [9,4]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[]
=> []
=> []
=> []
=> ? = 0 - 1
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000091
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> []
=> []
=> [0] => ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [2] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [3] => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [2] => 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [3] => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [2] => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [4,4,3,2,1] => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [3,3,3,2,1] => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [4,4,2,2,1] => ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [2,2,2,2,1] => ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [4,4,2,1,1] => ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [3,3,1,1,1] => ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [5,4,3,2] => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [4,3,3,2] => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [5,4,2,2] => ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [5,3,2,2] => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [3,2,2,2] => ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [4,4,3,1] => ? = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [5,4,3] => ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [5,3,3] => ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [5,3,2] => ? = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [5,2,2] => ? = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [5,4,1] => ? = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [4,4,1] => ? = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [5,4] => ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 0 = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 0 = 2 - 2
[]
=> []
=> []
=> [0] => ? = 0 - 2
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St000232
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> []
=> []
=> {}
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> {{1,2,3},{4,5,6},{7,8,9},{10}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> {{1,2,3,4},{5,6,7,8},{9},{10}}
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12}}
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11}}
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> {{1,2,3,4},{5,6,7},{8,9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> {{1,2,3,4,5},{6,7,8},{9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> {{1,2,3,4,5},{6,7},{8,9}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> {{1,2,3,4,5},{6,7,8,9},{10}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> {{1,2,3,4},{5,6,7,8},{9}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 2 - 2
[]
=> []
=> []
=> {}
=> ? = 0 - 2
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000233
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> []
=> []
=> {}
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> {{1,2,3},{4,5,6},{7,8,9},{10}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> {{1,2,3,4},{5,6,7,8},{9},{10}}
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12}}
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11}}
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> {{1,2,3,4},{5,6,7},{8,9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> {{1,2,3,4,5},{6,7,8},{9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> {{1,2,3,4,5},{6,7},{8,9}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> {{1,2,3,4,5},{6,7,8,9},{10}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> {{1,2,3,4},{5,6,7,8},{9}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 2 - 2
[]
=> []
=> []
=> {}
=> ? = 0 - 2
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000496
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> []
=> []
=> {}
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> {{1,2,3},{4,5,6},{7,8,9},{10}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> {{1,2,3,4},{5,6,7,8},{9},{10}}
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12}}
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11}}
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> {{1,2,3,4},{5,6,7},{8,9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> {{1,2,3,4,5},{6,7,8},{9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> {{1,2,3,4,5},{6,7},{8,9}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> {{1,2,3,4,5},{6,7,8,9},{10}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> {{1,2,3,4},{5,6,7,8},{9}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 2 - 2
[]
=> []
=> []
=> {}
=> ? = 0 - 2
Description
The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St001781
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001781: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001781: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 55%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> []
=> []
=> {}
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> {{1,2,3},{4,5,6},{7,8,9},{10}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> {{1,2,3,4},{5,6,7,8},{9},{10}}
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12}}
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11}}
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> {{1,2,3,4},{5,6,7},{8,9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> {{1,2,3,4,5},{6,7,8},{9,10}}
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> {{1,2,3,4,5},{6,7},{8,9}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> {{1,2,3,4,5},{6,7,8,9},{10}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> {{1,2,3,4},{5,6,7,8},{9}}
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> {{1,2,3,4,5},{6,7,8,9}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 2 - 2
[]
=> []
=> []
=> {}
=> ? = 0 - 2
Description
The interlacing number of a set partition.
Let $\pi$ be a set partition of $\{1,\dots,n\}$ with $k$ blocks. To each block of $\pi$ we add the element $\infty$, which is larger than $n$. Then, an interlacing of $\pi$ is a pair of blocks $B=(B_1 < \dots < B_b < B_{b+1} = \infty)$ and $C=(C_1 < \dots < C_c < C_{c+1} = \infty)$ together with an index $1\leq i\leq \min(b, c)$, such that $B_i < C_i < B_{i+1} < C_{i+1}$.
The following 308 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000842The breadth of a permutation. St000264The girth of a graph, which is not a tree. St000570The Edelman-Greene number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000993The multiplicity of the largest part of an integer partition. St001162The minimum jump of a permutation. 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 $K[x]/(x^n)$. St001344The neighbouring number of a permutation. St001590The crossing number of a perfect matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000623The number of occurrences of the pattern 52341 in a permutation. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001552The number of inversions between excedances and fixed points of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000914The sum of the values of the Möbius function of a poset. St000255The number of reduced Kogan faces with the permutation as type. St001811The Castelnuovo-Mumford regularity of a permutation. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. 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. 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. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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 $eA$. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000781The number of proper colouring schemes of a Ferrers diagram. St000886The number of permutations with the same antidiagonal sums. St001568The smallest positive integer that does not appear twice in the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001625The Möbius invariant of a lattice. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000779The tier of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001545The second Elser number of a connected graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000660The number of rises of length at least 3 of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. 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 $eA$. St000834The number of right outer peaks of a permutation. St001555The order of a signed permutation. St001896The number of right descents of a signed permutations. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000635The number of strictly order preserving maps of a poset into itself. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001128The exponens consonantiae of a partition. St001280The number of parts of an integer partition that are at least two. 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. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000567The sum of the products of all pairs of parts. 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. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. 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. 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. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000630The length of the shortest palindromic decomposition of a binary word. St001644The dimension of a graph. St000633The size of the automorphism group of a poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000850The number of 1/2-balanced pairs in a poset. St001864The number of excedances of a signed permutation. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000647The number of big descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001870The number of positive entries followed by a negative entry in a signed permutation. St000456The monochromatic index of a connected graph. St000670The reversal length of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000764The number of strong records in an integer composition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001875The number of simple modules with projective dimension at most 1. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000920The logarithmic height of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001469The holeyness of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001673The degree of asymmetry of an integer composition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000155The number of exceedances (also excedences) of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000876The number of factors in the Catalan decomposition of a binary word. St000893The number of distinct diagonal sums of an alternating sign matrix. St000903The number of different parts of an integer composition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001642The Prague dimension of a graph. St001569The maximal modular displacement of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000907The number of maximal antichains of minimal length in a poset. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001884The number of borders of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in 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. St001520The number of strict 3-descents. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000701The protection number of a binary tree. St000730The maximal arc length of a set partition. St000758The length of the longest staircase fitting into an integer composition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000808The number of up steps of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St000902 The minimal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000942The number of critical left to right maxima of the parking functions. St000982The length of the longest constant subword. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001096The size of the overlap set of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001415The length of the longest palindromic prefix of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000028The number of stack-sorts needed to sort a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000761The number of ascents in an integer composition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000983The length of the longest alternating subword. St000996The number of exclusive left-to-right maxima of a permutation. 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 $K[x]/(x^n)$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001267The length of the Lyndon factorization of the binary word. St001335The cardinality of a minimal cycle-isolating set of a graph. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001935The number of ascents in a parking function. 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. St000640The rank of the largest boolean interval in a poset. St000717The number of ordinal summands of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. 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. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001597The Frobenius rank of a skew partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001052The length of the exterior of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001812The biclique partition number of a graph. St001854The size of the left Kazhdan-Lusztig cell, St000402Half the size of the symmetry class of a permutation. St000477The weight of a partition according to Alladi. St001857The number of edges in the reduced word graph of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation.
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