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Your data matches 144 different statistics following compositions of up to 3 maps.
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Matching statistic: St000340
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[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]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 2
[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]
=> 1
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 0
[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]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 2
[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]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00102: Dyck paths —rise 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%
Mp00102: Dyck paths —rise 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] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [3,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00102: Dyck paths —rise 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%
Mp00102: Dyck paths —rise 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] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [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 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [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 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [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 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [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 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [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 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00102: Dyck paths —rise 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%
Mp00102: Dyck paths —rise 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] => 1 => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 10011 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 10010 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 10011 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 10011 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 10010 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 3 = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000098
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [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 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [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 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [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 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [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 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [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 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[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]
=> [8] => ([],8)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,1,1,1,2] => ([(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)
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000053
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ?
=> ? = 6
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,1,1,1]
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4,1,1,1,1]
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 3
Description
The number of valleys of the Dyck path.
Matching statistic: St001499
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ?
=> ? = 6
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,1,1,1]
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,1,1,1]
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4,1,1,1,1]
=> ?
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 3
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001068
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ?
=> ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,2,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,2,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,2,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1,1,1]
=> ?
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,1,1,1]
=> ?
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,1,1,1]
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4,1,1,1,1]
=> ?
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000507
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [[1]]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [5,4,6,8,7,3,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [5,4,8,7,6,3,2,1] => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,5,4,8,7,3,2,1] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [3,6,7,5,4,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> [3,6,5,7,4,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,6,7,5,8,4,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,6,5,7,8,4,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,6,7,8,5,4,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,6,5,8,7,4,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,4,6,5,8,7,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,4,7,6,8,5,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,6,8,7,5,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,4,7,8,6,5,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,5,6,4,8,7,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,5,4,6,8,7,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,5,4,8,7,6,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [4,6,7,5,3,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [4,5,7,6,3,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [4,6,5,7,3,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [4,3,6,7,5,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [4,5,3,6,7,8,2,1] => ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [4,6,5,3,7,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [4,3,5,6,8,7,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [4,3,7,6,8,5,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,7,8,6,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [4,3,6,8,7,5,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,3,5,8,7,6,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [4,5,3,7,8,6,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,3,8,7,6,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [4,6,5,3,8,7,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [5,4,3,7,6,8,2,1] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [5,4,6,3,8,7,2,1] => ?
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,5,4,3,8,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [2,5,7,6,4,3,8,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,4,7,6,5,3,8,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,6,3,8,1] => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,4,7,3,8,1] => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [2,6,5,4,7,3,8,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,5,7,4,8,1] => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,6,7,5,4,8,1] => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [2,4,5,3,7,6,8,1] => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [2,5,4,6,3,7,8,1] => ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,6,7,5,4,8,3,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,5,7,6,4,8,3,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,6,5,7,4,8,3,1] => ?
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,6,8,3,1] => ?
=> ? = 4 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000319
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 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] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,1,2,6,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,3,2,4,5,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,5,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,5,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,1,7,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,2,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,2,1,7,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [6,5,3,2,4,1,7,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,2,3,1,7,8] => ?
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,3,4,2,6,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,2,3,6,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,2,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,2,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,1,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,3,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,1,5,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [6,3,2,4,1,5,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,1,5,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,1,2,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,1,2,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,1,2,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,3,1,4,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,1,2,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,1,2,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,1,2,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,1,2,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,1,2,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,1,3,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,1,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [5,6,2,3,1,4,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,1,2,5,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [6,5,3,1,2,4,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,1,2,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,1,3,4,7,8] => ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,1,2,4,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,1,2,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,1,2,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [7,3,2,4,5,6,1,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,4,3,2,5,6,1,8] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [7,3,4,2,5,6,1,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,2,6,1,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,2,3,6,1,8] => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [7,5,2,3,4,6,1,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,2,1,8] => ?
=> ? = 4
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
The following 134 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000147The largest part of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000157The number of descents of a standard tableau. St001581The achromatic number of a graph. St000676The number of odd rises of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. 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. St000925The number of topologically connected components of a set partition. St000024The number of double up and double down steps of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000211The rank of the set partition. St001494The Alon-Tarsi number of a graph. St000234The number of global ascents of a permutation. St000172The Grundy number of a graph. St001029The size of the core of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph 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. St000527The width of the poset. St000159The number of distinct parts of the integer partition. St000071The number of maximal chains in a poset. St000068The number of minimal elements in a poset. St000632The jump number of the poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000245The number of ascents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000167The number of leaves of an ordered tree. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000470The number of runs in a permutation. St000702The number of weak deficiencies of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000809The reduced reflection length of the permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000141The maximum drop size of a permutation. St000007The number of saliances of the permutation. St000308The height of the tree associated to a permutation. St000153The number of adjacent cycles of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000332The positive inversions of an alternating sign matrix. St000164The number of short pairs. St001427The number of descents of a signed permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000065The number of entries equal to -1 in an alternating sign matrix. St000542The number of left-to-right-minima of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000237The number of small exceedances. St000389The number of runs of ones of odd length in a binary word. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000711The number of big exceedences of a permutation. St001330The hat guessing number of a graph. St000619The number of cyclic descents of a permutation. St001152The number of pairs with even minimum in a perfect matching. St000236The number of cyclical small weak excedances. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001864The number of excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001712The number of natural descents of a standard Young tableau. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St000710The number of big deficiencies of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000386The number of factors DDU in a Dyck path.
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