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Your data matches 173 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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(load all 3 compositions to match this statistic)
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ 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)
=> ([(0,1)],2)
=> [2]
=> 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
Description
The length of the partition.
Matching statistic: St000383
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ 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
([],2)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,1)],2)
=> ([],2)
=> [2] => 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
([(0,2),(2,1)],3)
=> ([],3)
=> [3] => 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [4] => 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1,4] => 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
([(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
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000147
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ 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)
=> [2]
=> [1,1]
=> 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> [2]
=> 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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)
=> [5]
=> [1,1,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000288
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —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)
=> ([],1)
=> [1]
=> 10 => 1
([],2)
=> ([(0,1)],2)
=> [2]
=> 100 => 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> 110 => 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 2
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 1110 => 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 10110 => 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 10110 => 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 10110 => 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 11110 => 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ 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)
=> [2]
=> [1,1]
=> 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> [2]
=> 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [3]
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> [2,1]
=> 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [3]
=> 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> [3,1]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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)
=> [5]
=> [1,1,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ 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] => 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)
=> [2,1] => [1,2] => 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 2
([(0,2),(2,1)],3)
=> ([],3)
=> [3] => [3] => 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => [3,1] => 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => [3,1] => 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => [3,1] => 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [4] => [4] => 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [1,4] => 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 1
([(2,3),(2,4)],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] => 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1,4] => [4,1] => 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => 3
([(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] => 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000733
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ 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)
=> [2]
=> [[1,2]]
=> 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> [[1],[2]]
=> 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000273
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)
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> [3] => ([],3)
=> 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(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)
=> 1
([(1,2),(1,3)],4)
=> ([(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
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(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
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [4] => ([],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(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)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4)],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)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(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
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(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)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(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)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],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
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(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
([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(1,6),(2,5),(2,6),(5,4),(6,3)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,3),(1,6),(4,2),(5,4),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(4,3),(4,6),(5,2),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000544
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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)
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> [3] => ([],3)
=> 3
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(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)
=> 1
([(1,2),(1,3)],4)
=> ([(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
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 3
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(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
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [4] => ([],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(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)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4)],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)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(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
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(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)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(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)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],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
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(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
([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(1,6),(2,5),(2,6),(5,4),(6,3)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,4),(2,3),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,3),(1,6),(4,2),(5,4),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(4,3),(4,6),(5,2),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
The following 163 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000916The packing number of a graph. St001829The common independence number of a graph. St001363The Euler characteristic of a graph according to Knill. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000286The number of connected components of the complement of a graph. St000553The number of blocks of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000068The number of minimal elements in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000181The number of connected components of the Hasse diagram for the poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000456The monochromatic index of a connected graph. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001322The size of a minimal independent dominating set in a graph. St000717The number of ordinal summands of a poset. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000100The number of linear extensions of a poset. St000914The sum of the values of the Möbius function of a poset. St001339The irredundance number of a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001890The maximum magnitude of the Möbius function of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001933The largest multiplicity of a part in an integer partition. St001571The Cartan determinant of the integer partition. St000454The largest eigenvalue of a graph if it is integral. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001280The number of parts of an integer partition that are at least two. St001060The distinguishing index of a graph. St000706The product of the factorials of the multiplicities of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000455The second largest eigenvalue of a graph if it is integral. St000770The major index of an integer partition when read from bottom to top. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000117The number of centered tunnels of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000015The number of peaks of a Dyck path. St000331The number of upper interactions of a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000264The girth of a graph, which is not a tree. St001621The number of atoms of a lattice. St001597The Frobenius rank of a skew partition. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000537The cutwidth of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001271The competition number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000469The distinguishing number of a graph. St000552The number of cut vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001883The mutual visibility number of a graph.
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