searching the database
Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000259
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(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)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,6,4,3,2,1] => ([(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)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,6,5,3,2,1] => ([(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)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => ([(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)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => ([(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)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,6,5,4,2,1] => ([(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)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,4,3,6,2,1] => ([(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)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [5,2,6,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,2,6,5,3,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [5,3,2,6,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000455
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 50%●distinct values known / distinct values provided: 17%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 50%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 0 - 2
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,2] => [1,2] => ([(1,2)],3)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [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]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [1,1,4] => ([(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] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [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
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [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
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [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
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [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
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,5] => [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
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => [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
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => [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
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [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
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [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
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,5] => [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
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [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
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5] => [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
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => [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
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,5] => [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
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [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
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5] => [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
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5] => [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
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5] => [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
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(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]
=> [2,3,1] => [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)
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [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)
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [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
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 3 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 4 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [7] => ([],7)
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 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: St001204
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0 - 2
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 4 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 2 - 2
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra.
The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St001568
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> []
=> ? = 0 - 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 - 1
[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 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 1 = 2 - 1
[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]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> 1 = 2 - 1
[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 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? = 3 - 1
[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]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [5,4,4]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [6,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [5,3,3]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> ? = 2 - 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001060
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 24%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 24%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,7] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,6,4,7] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,2,6,7,4] => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,2,4,7] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,2,4,5,7] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,4,5] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,2,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,2,5,3,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,2,5,6,3,7] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,2,6,3,5,7] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,6,7,3,5] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001487
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1,0]
=> []
=> [[],[]]
=> ? = 0 - 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[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]
=> [[3],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [[3],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [4,1]
=> [[4,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> [[4],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 2 - 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1,0]
=> []
=> [[],[]]
=> ? = 0 - 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[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]
=> [[3],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [[3],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [4,1]
=> [[4,1],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> [[4],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 2 - 1
Description
The number of connected components of a skew partition.
Matching statistic: St001435
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1,0]
=> []
=> [[],[]]
=> ? = 0 - 2
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 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]
=> [[3],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [[3],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> [[4],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0 = 2 - 2
Description
The number of missing boxes in the first row.
Matching statistic: St001438
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1,0]
=> []
=> [[],[]]
=> ? = 0 - 2
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 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]
=> [[3],[]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 2 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [[3],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> [[4],[]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0 = 2 - 2
Description
The number of missing boxes of a skew partition.
Matching statistic: St001346
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [2,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,5,4,1,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [2,6,5,4,1,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [2,6,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,6,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [2,6,5,4,1,3] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => [2,7,6,5,4,1,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [2,7,6,1,5,4,3] => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [2,7,6,5,1,4,3] => ? = 2 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [2,7,6,5,4,1,3] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => ? = 3 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => ? = 3 + 1
Description
The number of parking functions that give the same permutation.
A '''parking function''' $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!