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Your data matches 79 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000786
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 95%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 95%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(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)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(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)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(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)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ([(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)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => ([(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)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => ([],10)
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => ([(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)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
=> ? = 7 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => ([(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => ([(8,9)],10)
=> ? = 8 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000996
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => [] => 0
[1,0,1,0]
=> [1,2] => [1] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [2,1] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,2] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [2,1,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [3,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [2,4,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [2,4,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [3,2,4,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [3,2,4,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,4,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,4,2,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => ? = 5
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => [1,6,5,4,3,7,2] => [2,7,6,5,4,1,3] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,5,7,6,8] => [2,3,4,1,5,7,6] => [3,4,5,2,6,1,7] => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,4,1,5,7,8,6] => [2,3,4,1,5,7,6] => [3,4,5,2,6,1,7] => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,8,7,6] => [2,3,4,1,5,7,6] => [3,4,5,2,6,1,7] => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,5,1,8,7,6] => [2,3,4,5,1,7,6] => [3,4,5,6,2,1,7] => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => [3,4,5,6,7,2,1] => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,6,1,8,7] => [2,3,4,5,6,1,7] => [3,4,5,6,7,2,1] => ? = 5
[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,2,4,5,6,7,1,8] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 4
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [4,5,3,6,7,1,2] => ? = 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [4,5,6,3,7,1,2] => ? = 4
[1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [3,6,5,4,2,1,8,7] => [3,6,5,4,2,1,7] => [4,7,6,5,3,2,1] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7] => [5,4,3,2,6,7,1] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,7] => [5,4,3,2,6,7,1] => ? = 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [5,4,3,6,7,1,2] => ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [4,3,6,5,2,1,8,7] => [4,3,6,5,2,1,7] => [5,4,7,6,3,2,1] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [4,5,3,6,2,1,8,7] => [4,5,3,6,2,1,7] => [5,6,4,7,3,2,1] => ? = 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [4,5,6,3,2,1,8,7] => [4,5,6,3,2,1,7] => [5,6,7,4,3,2,1] => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7] => [6,5,4,3,2,7,1] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,4,3,2,1,6,8,7] => [5,4,3,2,1,6,7] => [6,5,4,3,2,7,1] => ? = 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,4,3,2,6,1,7,8] => [5,4,3,2,6,1,7] => [6,5,4,3,7,2,1] => ? = 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,4,3,2,6,1,8,7] => [5,4,3,2,6,1,7] => [6,5,4,3,7,2,1] => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,4,3,6,2,1,8,7] => [5,4,3,6,2,1,7] => [6,5,4,7,3,2,1] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [5,6,4,3,2,1,8,7] => [5,6,4,3,2,1,7] => [6,7,5,4,3,2,1] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [4,3,5,6,7,8,1,2] => ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => [4,5,3,6,7,8,1,2] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [2,3,4,6,7,8,1,5] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [3,4,5,6,7,8,9,1,2] => ? = 7
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [4,3,5,6,7,8,9,1,2] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => [3,4,5,6,7,8,9,10,1,2] => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [2,4,3,6,5,8,7,1] => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => [5,4,3,2,6,7,8,1] => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => [6,5,4,3,2,7,8,1] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => [4,3,2,5,6,7,8,9,1] => ? = 6
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => [5,4,3,2,6,7,8,9,1] => ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => [6,5,4,3,2,7,8,9,1] => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => [7,6,5,4,3,2,8,9,1] => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9] => [8,7,6,5,4,3,2,9,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => [3,4,5,6,7,8,2,1] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [3,4,5,6,7,8,9,1,2] => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [11,10,9,8,7,6,5,4,3,2,1] => ? = 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000374
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => [] => [] => 0
[1,0,1,0]
=> [2,1] => [1] => [1] => 0
[1,1,0,0]
=> [1,2] => [1] => [1] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => [2,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => [1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ? => ? => ? = 2
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => ? => ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,1,3,4,7,5,6,8] => ? => ? => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [3,4,1,5,6,7,2] => [3,7,1,2,4,5,6] => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,5,6,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,4,1,5,6,2,8,7] => ? => ? => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,5,6,8,2,7] => ? => ? => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,1,5,6,2,7,8] => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,5,1,6,7,8,2] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,6,1,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,7,1,2,8] => ? => ? => ? = 5
[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,4,5,6,7,2,8,3] => ? => ? => ? = 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,5,1,6,7,8,2,3] => ? => ? => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,6,7,4,8] => ? => ? => ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [5,1,6,2,3,7,4,8] => ? => ? => ? = 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ? => ? => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,2,3,4,6,8,5,7] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,4,6,5,7,8] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,2,6,3,4,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,1,2,3,4,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [3,1,4,5,6,7,8,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8,10] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,8,9,2,3] => ? => ? => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,2,3,4,5,9,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,2,3,4,9,5,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,2,3,9,4,5,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,2,9,3,4,5,6,7,8] => ? => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,9,1,5] => [2,3,4,6,7,8,1,5] => [7,1,2,3,8,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => ? => ? => ? = 9
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,8,9,10,2,3] => ? => ? => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,10,11,1,2] => ? => ? => ? = 8
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,3] => [1,2,4,5,6,7,8,3] => [1,2,8,3,4,5,6,7] => ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,5] => [1,2,3,4,6,7,8,5] => [1,2,3,4,8,5,6,7] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,10,3] => [1,2,4,5,6,7,8,9,3] => [1,2,9,3,4,5,6,7,8] => ? = 6
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,9,10,4] => [1,2,3,5,6,7,8,9,4] => [1,2,3,9,4,5,6,7,8] => ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,10,5] => [1,2,3,4,6,7,8,9,5] => [1,2,3,4,9,5,6,7,8] => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,8,9,10,6] => [1,2,3,4,5,7,8,9,6] => [1,2,3,4,5,9,6,7,8] => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,10,7] => [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,9,7,8] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,7,1,8,9,2] => ? => ? => ? = 6
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000703
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => [] => [] => 0
[1,0,1,0]
=> [2,1] => [1] => [1] => 0
[1,1,0,0]
=> [1,2] => [1] => [1] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => [2,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => [1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ? => ? => ? = 2
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => ? => ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,1,3,4,7,5,6,8] => ? => ? => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [3,4,1,5,6,7,2] => [3,7,1,2,4,5,6] => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,5,6,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,4,1,5,6,2,8,7] => ? => ? => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,5,6,8,2,7] => ? => ? => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,1,5,6,2,7,8] => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,5,1,6,7,8,2] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,6,1,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,7,1,2,8] => ? => ? => ? = 5
[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,4,5,6,7,2,8,3] => ? => ? => ? = 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,5,1,6,7,8,2,3] => ? => ? => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,6,7,4,8] => ? => ? => ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [5,1,6,2,3,7,4,8] => ? => ? => ? = 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ? => ? => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,2,3,4,6,8,5,7] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,4,6,5,7,8] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,2,6,3,4,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,1,2,3,4,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [3,1,4,5,6,7,8,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8,10] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,8,9,2,3] => ? => ? => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,2,3,4,5,9,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,2,3,4,9,5,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,2,3,9,4,5,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,2,9,3,4,5,6,7,8] => ? => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,9,1,5] => [2,3,4,6,7,8,1,5] => [7,1,2,3,8,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => ? => ? => ? = 9
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,8,9,10,2,3] => ? => ? => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,10,11,1,2] => ? => ? => ? = 8
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,3] => [1,2,4,5,6,7,8,3] => [1,2,8,3,4,5,6,7] => ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,5] => [1,2,3,4,6,7,8,5] => [1,2,3,4,8,5,6,7] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,10,3] => [1,2,4,5,6,7,8,9,3] => [1,2,9,3,4,5,6,7,8] => ? = 6
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,9,10,4] => [1,2,3,5,6,7,8,9,4] => [1,2,3,9,4,5,6,7,8] => ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,10,5] => [1,2,3,4,6,7,8,9,5] => [1,2,3,4,9,5,6,7,8] => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,8,9,10,6] => [1,2,3,4,5,7,8,9,6] => [1,2,3,4,5,9,6,7,8] => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,10,7] => [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,9,7,8] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,7,1,8,9,2] => ? => ? => ? = 6
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000318
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 92%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 92%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> []
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> [6,5,4,3,2]
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> [6,5,4,2,1]
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,1]
=> ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,2,1]
=> ?
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,1]
=> ?
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,1]
=> [5,5,4,2,1]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> [6,5,3,2,1]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,2,1]
=> [5,5,3,2,1]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> [6,4,3,2,1]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,2,1]
=> [6,4,3,2,1]
=> ? = 5 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> [6,5,4,3]
=> ? = 4 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> [6,5,4]
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4]
=> ?
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> [6,5]
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [6,5]
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,1]
=> [6,5,4,3,1]
=> ? = 5 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2,1]
=> [6,6,5,4,3,2,1]
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,3,2,1]
=> [6,5,4,3,3,2,1]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,6,6,5,4,3,2,1]
=> [6,6,5,4,3,2,1]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,7,6,5,4,3,2,1]
=> [7,7,6,5,4,3,2,1]
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,9,8,7,6,5,4,3,2,1]
=> [9,8,7,6,5,4,3,2,1]
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2]
=> [7,6,5,4,3,2]
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [7,7,5,5,3,3,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2]
=> [7,6,5,4,3,2]
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,6,4,4,2,2]
=> ?
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3]
=> [7,6,5,4,3]
=> ? = 5 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000053
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => []
=> ? = 0
[1,0,1,0]
=> [1,2] => [1] => [1,0]
=> 0
[1,1,0,0]
=> [2,1] => [1] => [1,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
Description
The number of valleys of the Dyck path.
Matching statistic: St001068
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => []
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => [1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => [1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000337
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 91%●distinct values known / distinct values provided: 90%
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 91%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => [] => [] => 0
[1,0,1,0]
=> [2,1] => [1] => [1] => 0
[1,1,0,0]
=> [1,2] => [1] => [1] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => [2,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => [1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ? => ? => ? = 2
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => ? => ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,1,3,4,7,5,6,8] => ? => ? => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [3,4,1,5,6,7,2] => [3,7,1,2,4,5,6] => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,5,6,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,4,1,5,6,2,8,7] => ? => ? => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,5,6,8,2,7] => ? => ? => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,1,5,6,2,7,8] => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,5,1,6,7,8,2] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,6,2,7,8] => ? => ? => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,6,1,7,2,8] => ? => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,7,1,2,8] => ? => ? => ? = 5
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,4,5,6,7,2,8,3] => ? => ? => ? = 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,5,1,6,7,8,2,3] => ? => ? => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,6,7,4,8] => ? => ? => ? = 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,8,3,4] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [1,5,2,3,4,7,6,8] => [1,5,2,3,4,7,6] => [1,3,4,5,2,7,6] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [5,1,6,2,3,7,4,8] => ? => ? => ? = 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ? => ? => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,2,3,4,6,8,5,7] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,4,6,5,7,8] => ? => ? => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,2,3,6,4,7,8,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => ? = 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,2,3,6,4,7,5,8] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,2,6,3,4,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,1,2,3,4,7,5,8] => ? => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [3,1,4,5,6,7,8,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8,10] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,8,9,2,3] => ? => ? => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,2,3,4,5,9,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,2,3,4,9,5,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,2,3,9,4,5,6,7,8] => ? => ? => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,2,9,3,4,5,6,7,8] => ? => ? => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,9,1,5] => [2,3,4,6,7,8,1,5] => [7,1,2,3,8,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => ? => ? => ? = 9
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,8,9,10,2,3] => ? => ? => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,10,11,1,2] => ? => ? => ? = 8
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,3] => [1,2,4,5,6,7,8,3] => [1,2,8,3,4,5,6,7] => ? = 5
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000528The height of a poset. St000912The number of maximal antichains in a poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000308The height of the tree associated to a permutation. St000159The number of distinct parts of the integer partition. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St000542The number of left-to-right-minima of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000482The (zero)-forcing number of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St000021The number of descents of a permutation. St000204The number of internal nodes of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000080The rank of the poset. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000083The number of left oriented leafs of a binary tree except the first one. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000836The number of descents of distance 2 of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000742The number of big ascents of a permutation after prepending zero. St000312The number of leaves in a graph. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000837The number of ascents of distance 2 of a permutation. St000024The number of double up and double down steps of a Dyck path. St000636The hull number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000710The number of big deficiencies of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001737The number of descents of type 2 in a permutation. St001864The number of excedances of a signed permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000236The number of cyclical small weak excedances. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001964The interval resolution global dimension of a poset. St001556The number of inversions of the third entry of a permutation.
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