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Your data matches 136 different statistics following compositions of up to 3 maps.
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Matching statistic: St000678
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 3
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000382
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => 2
[1,1,0,0]
=> [2] => [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,1,0,0]
=> [2,1] => [2,1] => 2
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => 3
Description
The first part of an integer composition.
Matching statistic: St000010
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000097
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 11 => 2
[1,1,0,0]
=> [1,1,0,0]
=> [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000297
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1] => 11 => 2
[1,1,0,0]
=> [2] => [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 111 => 3
[1,0,1,1,0,0]
=> [1,2] => [2,1] => 101 => 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => 110 => 2
[1,1,0,1,0,0]
=> [2,1] => [1,2] => 110 => 2
[1,1,1,0,0,0]
=> [3] => [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 1011 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => 1101 => 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => 1101 => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1001 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 1110 => 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 1010 => 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1110 => 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => 1110 => 3
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 1010 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1100 => 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 1100 => 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 1100 => 2
[1,1,1,1,0,0,0,0]
=> [4] => [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 10111 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 11011 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 11011 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 10011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => 11001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => 11001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => 11001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 10001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 10010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => 10010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 11100 => 3
Description
The number of leading ones in a binary word.
Matching statistic: St001581
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 88% ●values known / values provided: 98%●distinct values known / distinct values provided: 88%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 88% ●values known / values provided: 98%●distinct values known / distinct values provided: 88%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(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)
=> ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(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
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [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)
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(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)
=> ? = 6
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(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
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [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)
=> ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [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)
=> ? = 6
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => ([(0,7),(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
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => ([(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
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(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
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [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)
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [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)
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,3,1] => ([(0,7),(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
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,3] => ([(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
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [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)
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(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)
=> ? = 7
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ([(0,4),(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
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [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
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [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
[1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,1,3,1] => ([(0,7),(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
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,1,1,3] => ([(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
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => ([(0,3),(0,4),(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)
=> ? = 7
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000326
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 => 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 01 => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 101 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 010 => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0000 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0100 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 0100 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0100 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0100 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0001 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0001 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0001 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0100 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 0100 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 0100 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ? => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [5,6,1,7,2,8,3,4] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,2,6,7,3,4,8,5] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ? => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,5,6,2,3,7,4,8] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,5,6,2,7,3,8,4] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,7,8,2,5,6] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,7,8,2,5,6] => ? => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,7,1,2,8,5,6] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,4,5,1,8,2,6,7] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,8,6,7] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,7,2,8,5,6] => ? => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [5,1,6,8,2,3,4,7] => ? => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,5,7,8,2,3,6] => ? => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,4,2,5,6,3,7,8] => ? => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> [4,1,5,7,2,3,8,6] => ? => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> [4,5,7,1,8,2,3,6] => ? => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [4,5,1,2,7,3,8,6] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5,7,8] => ? => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5,7,8] => ? => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5,7,8] => ? => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,6,8,7] => ? => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,7,1,2,5,6,8] => ? => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,4,7,2,5,6,8] => ? => ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,4,5,1,6,8,2,7] => ? => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,1,4,5,2,6,8,7] => ? => ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,7,2,5,6,8] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,2,6,8,7] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3,7,8,6] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3,6,8,7] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,8,7] => ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ? => ? = 5
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,2,5,6,7,3,4,8] => ? => ? = 5
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,7,1,2,3,6,8] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,6,8,7] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6,8,7] => ? => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ? => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [3,1,4,6,7,2,5,8] => ? => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,4,5,7,2,6,8] => ? => ? = 1
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,7,6,8] => ? => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,4,5,7,2,8,3,6] => ? => ? = 4
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4,8,7] => ? => ? = 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000383
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 88%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 88%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,1,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,1,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,2,1,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,2,1,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,1,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,1,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,1,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,1,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,1,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,2,1,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
Description
The last part of an integer composition.
Matching statistic: St000439
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
The following 126 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000011The number of touch points (or returns) of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000098The chromatic number of a graph. St000925The number of topologically connected components of a set partition. St000068The number of minimal elements in a poset. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St000504The cardinality of the first block of a set partition. St000502The number of successions of a set partitions. St001733The number of weak left to right maxima of a Dyck path. St000617The number of global maxima of a Dyck path. St000025The number of initial rises of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000026The position of the first return of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St000053The number of valleys of the Dyck path. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001316The domatic number of a graph. St000069The number of maximal elements of a poset. St000234The number of global ascents of a permutation. St000286The number of connected components of the complement of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000759The smallest missing part in an integer partition. St000883The number of longest increasing subsequences of a permutation. St000993The multiplicity of the largest part of an integer partition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000989The number of final rises of a permutation. St000007The number of saliances of the permutation. St000237The number of small exceedances. St000054The first entry of the permutation. St000546The number of global descents of a permutation. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000740The last entry of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001330The hat guessing number of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000738The first entry in the last row of a standard tableau. St000843The decomposition number of a perfect matching. St000203The number of external nodes of a binary tree. St000648The number of 2-excedences of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000260The radius of a connected graph. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000374The number of exclusive right-to-left minima of a permutation. St000822The Hadwiger number of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000133The "bounce" of a permutation. St000331The number of upper interactions of a Dyck path. St000918The 2-limited packing number of a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001812The biclique partition number of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000924The number of topologically connected components of a perfect matching. St000259The diameter of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St000338The number of pixed points 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. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001889The size of the connectivity set of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001904The length of the initial strictly increasing segment of a parking function. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
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