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Your data matches 115 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [1,1]
=> 1
([(0,1)],2)
=> [2]
=> 2
([],3)
=> [1,1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
([],5)
=> [1,1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 1
([],2)
=> [1,1]
=> [2]
=> 1
([(0,1)],2)
=> [2]
=> [1,1]
=> 2
([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [2,2,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [2,2,2]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [3,3]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [3,2,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [5,4]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [3,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [6,6]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [4,4,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [2,2,2,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [4,4,2]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [4,4,4]
=> 3
Description
The length of the partition.
Matching statistic: St000381
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00111: Graphs —complement⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => 1
([],2)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,1)],2)
=> ([],2)
=> [2] => 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> [3] => 3
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [3,1] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> [4] => 4
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> [3,2] => 3
Description
The largest part of an integer composition.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00111: Graphs —complement⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => 1
([],2)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,1)],2)
=> ([],2)
=> [2] => 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> [3] => 3
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [3,1] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> [4] => 4
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> [3,2] => 3
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00111: Graphs —complement⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => [1] => 1
([],2)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([(0,1)],2)
=> ([],2)
=> [2] => [2] => 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> [3] => [3] => 3
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => [1,3] => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [3,1] => [1,3] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> [4] => [4] => 4
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => [1,4] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> [3,2] => [2,3] => 3
Description
The last part of an integer composition.
Matching statistic: St000723
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([(0,1)],2)
=> ([],2)
=> [2] => ([],2)
=> 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> [3] => ([],3)
=> 3
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> [4] => ([],4)
=> 4
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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)
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph.
The set of so called mating graphs, for which this statistic equals 1, is enumerated by [1].
Matching statistic: St000013
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3
([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,2,1]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,6),(1,3),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,3,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 4
([(0,3),(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4
([(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(0,6),(1,2),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
([(0,5),(0,6),(1,2),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,2),(0,3),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,4),(0,6),(1,3),(1,5),(2,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,1),(0,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,3),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,1),(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [4,2,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> [4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,3,2,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,2),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,2,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,4,3,2,2]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 4
([(0,3),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,3),(0,6),(1,3),(1,5),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(0,6),(1,3),(1,6),(2,3),(2,4),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
Description
The height of a Dyck path.
The height of a Dyck path D of semilength n is defined as the maximal height of a peak of D. The height of D at position i is the number of up-steps minus the number of down-steps before position i.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 01 => 1
([],2)
=> [1,1]
=> 110 => 001 => 1
([(0,1)],2)
=> [2]
=> 100 => 011 => 2
([],3)
=> [1,1,1]
=> 1110 => 0001 => 1
([(1,2)],3)
=> [2,1]
=> 1010 => 0101 => 2
([(0,2),(1,2)],3)
=> [2,2]
=> 1100 => 0011 => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 0111 => 3
([],4)
=> [1,1,1,1]
=> 11110 => 00001 => 1
([(2,3)],4)
=> [2,1,1]
=> 10110 => 01001 => 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> 11010 => 00101 => 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 11100 => 00011 => 2
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 0011 => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 11100 => 00011 => 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01101 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 10100 => 01011 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 111100 => 000011 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => 00111 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 01111 => 4
([],5)
=> [1,1,1,1,1]
=> 111110 => 000001 => 1
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 010001 => 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 110110 => 001001 => 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 111010 => 000101 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 111100 => 000011 => 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 00101 => 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 111010 => 000101 => 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 11100 => 00011 => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 011001 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 111100 => 000011 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 101010 => 010101 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 101100 => 010011 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 1111010 => 0000101 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 1111100 => 0000011 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 110010 => 001101 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 101100 => 010011 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 110100 => 001011 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 11111100 => 00000011 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 111000 => 000111 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 111100 => 000011 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 01011 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> 101100 => 010011 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => 00111 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> 1111100 => 0000011 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 1011100 => 0100011 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 111000 => 000111 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 110100 => 001011 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 011101 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 100100 => 011011 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 101000 => 010111 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 1101100 => 0010011 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> 1111000 => 0000111 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> 1101111010 => 0010000101 => ? = 3
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,1),(0,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> 1101111010 => 0010000101 => ? = 3
([(0,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,5),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,3),(0,6),(1,3),(1,6),(2,4),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,2,1]
=> 1110111010 => 0001000101 => ? = 3
([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,5),(0,6),(1,3),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> 1101111100 => 0010000011 => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 0000000011 => ? = 2
([(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,4),(0,6),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(0,6),(1,2),(1,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,1]
=> 100111010 => 011000101 => ? = 4
([(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,2]
=> 100111100 => 011000011 => ? = 4
([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2]
=> 110111100 => 001000011 => ? = 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000439
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3 + 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3 + 1
([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,2,1]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
([(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,3,3,2,2,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,6),(1,3),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,3,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 4 + 1
([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,3,2]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 + 1
([(0,3),(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
([(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,5),(0,6),(1,2),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,6),(1,2),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,3,2]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 + 1
([(0,5),(0,6),(1,2),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,2),(0,3),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,4),(0,6),(1,3),(1,5),(2,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,1),(0,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
([(0,3),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000734
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [[1]]
=> 1
([],2)
=> [1,1]
=> [[1],[2]]
=> 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(1,2)],3)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(1,3),(2,3)],4)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13],[14]]
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 2
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 3
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 2
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14]]
=> ? = 3
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 3
([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 2
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13],[14]]
=> ? = 3
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14],[15]]
=> ? = 3
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 2
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 3
([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 3
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 3
([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14]]
=> ? = 3
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16]]
=> ? = 3
([(0,5),(1,4),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(0,4),(0,6),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(0,3),(0,6),(1,3),(1,6),(2,4),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14],[15]]
=> ? = 3
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14]]
=> ? = 3
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14,15],[16]]
=> ? = 3
([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(0,5),(0,6),(1,3),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [3,3,2,2,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 3
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13]]
=> ? = 4
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14]]
=> ? = 4
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? = 4
([(0,5),(1,3),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
([(0,1),(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [3,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15]]
=> ? = 3
Description
The last entry in the first row of a standard tableau.
The following 105 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000025The number of initial rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000741The Colin de Verdière graph invariant. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001480The number of simple summands of the module J^2/J^3. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001316The domatic number of a graph. St001656The monophonic position number of a graph. St000273The domination number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001654The monophonic hull number of a graph. St001829The common independence number of a graph. St001949The rigidity index of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001655The general position number of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000148The number of odd parts of a partition. St000286The number of connected components of the complement of a graph. St000548The number of different non-empty partial sums of an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000808The number of up steps of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000918The 2-limited packing number of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001933The largest multiplicity of a part in an integer partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001323The independence gap of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001674The number of vertices of the largest induced star graph in the graph. St001777The number of weak descents in an integer composition. St001330The hat guessing number of a graph. St000667The greatest common divisor of the parts of the partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000160The multiplicity of the smallest part of a partition. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000454The largest eigenvalue of a graph if it is integral. St000479The Ramsey number of a graph. St001645The pebbling number of a connected graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001812The biclique partition number of a graph. St001834The number of non-isomorphic minors of a graph. St001651The Frankl number of a lattice. St000738The first entry in the last row of a standard tableau. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000474Dyson's crank of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001890The maximum magnitude of the Möbius function of a poset.
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