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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
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Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2] => 2
[1,1,0,0]
=> [1,2] => [2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => 3
[1,0,1,1,0,0]
=> [2,3,1] => [3] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [3] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [3] => 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,1] => 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => 4
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,1,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,1,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,3,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,2] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,2] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,2] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,2,2] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,2,2] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => 2
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St001352
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 88%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => ([],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,4,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,3,2,1] => [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,6,5,8,4,3,2,1] => [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,4,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,4,3,2,1] => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,8,3,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,8,3,2,1] => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,3,2,1] => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,3,2,1] => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,3,2,1] => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,3,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,4,2,1] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,3,4,2,1] => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,2,1] => [1,2,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,3,4,2,1] => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,3,4,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,6,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,8,2,1] => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,3,8,2,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,2,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,3,8,2,1] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,4,8,2,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,6,4,3,5,8,2,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,5,8,2,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,7,8,2,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,8,2,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,3,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,2,3,1] => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,2,3,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,2,3,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,4,2,3,1] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,2,3,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,2,3,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,2,3,1] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,2,4,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,2,4,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,3,2,4,1] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,3,2,8,1] => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,3,2,8,1] => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,3,2,8,1] => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St000777
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 75%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> ? = 2
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> ? = 2
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> ? = 2
[1,1,0,0,1,0]
=> [3,1,2] => [3] => ([],3)
=> ? = 2
[1,1,0,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> ? = 2
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> ? = 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> ? = 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? = 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> ? = 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 4
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> ? = 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> ? = 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> ? = 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> ? = 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> ? = 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => ([],5)
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => ([],5)
=> ? = 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => ([],5)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> ? = 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5] => ([],5)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [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)
=> 4
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [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)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [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)
=> 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [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)
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [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)
=> 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001488
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 62%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 62%
Values
[1,0]
=> [1] => [1] => [[1],[]]
=> 1
[1,0,1,0]
=> [2,1] => [2] => [[2],[]]
=> 2
[1,1,0,0]
=> [1,2] => [2] => [[2],[]]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [[2,2],[1]]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [3] => [[3],[]]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [3] => [[3],[]]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [3] => [[3],[]]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [[3],[]]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => [[3,3],[2]]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => [[3,3],[2]]
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => [[3,2],[1]]
=> 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => [[4],[]]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => [[3,1],[]]
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => [[4],[]]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => [[3,2],[1]]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => [[3,2],[1]]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => [[4],[]]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => [[4],[]]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => [[4],[]]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => [[4],[]]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [[4],[]]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => [[4,4],[3]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => [[4,4],[3]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,1] => [[3,3,1],[2]]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,2] => [[3,2,1],[1]]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => [[4,3],[2]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => [[4,4],[3]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => [[4,3],[2]]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => [[4,2],[1]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => [[5],[]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,2] => [[3,2,1],[1]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => [[4,3],[2]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => [[4,1],[]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => [[4,1],[]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => [[5],[]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,2] => [[3,2,1],[1]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => [[4,3],[2]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,2,2] => [[3,2,1],[1]]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => [[4,3],[2]]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => [[4,3],[2]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => [[4,3],[2]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => [[4,2],[1]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => [[5],[]]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ? = 4
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ? = 5
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => [3,1,2] => [[4,3,3],[2,2]]
=> ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [4,2] => [[5,4],[3]]
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [4,2] => [[5,4],[3]]
=> ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,4,1] => [[4,4,1],[3]]
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [1,2,3] => [[4,2,1],[1]]
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,3] => [[5,3],[2]]
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [5,1] => [[5,5],[4]]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [4,2] => [[5,4],[3]]
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [3,3] => [[5,3],[2]]
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [2,4] => [[5,2],[1]]
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6] => [[6],[]]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [2,2,2] => [[4,3,2],[2,1]]
=> ? = 6
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [4,2] => [[5,4],[3]]
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [1,3,2] => [[4,3,1],[2]]
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => [4,2] => [[5,4],[3]]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [1,2,3] => [[4,2,1],[1]]
=> ? = 5
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
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