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Your data matches 175 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
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%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [1,1]
=> 2
([(0,1)],2)
=> [2]
=> 1
([],3)
=> [1,1,1]
=> 3
([(1,2)],3)
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> [1,1,1,1]
=> 4
([(2,3)],4)
=> [2,1,1]
=> 3
([(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> [1,1,1,1,1]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> 4
([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
Description
The length of the partition.
Matching statistic: St000383
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1
([],2)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [1,2] => 2
([(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1
([],4)
=> [4] => 4
([(2,3)],4)
=> [1,3] => 3
([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1
([],5)
=> [5] => 5
([(3,4)],5)
=> [1,4] => 4
([(2,4),(3,4)],5)
=> [1,1,3] => 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
([(1,4),(2,3)],5)
=> [2,3] => 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000553
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> ([],5)
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(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),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(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),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The number of blocks of a graph.
A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St000147
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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%
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
([],2)
=> [1,1]
=> [2]
=> 2
([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> [1,1,1]
=> [3]
=> 3
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([],4)
=> [1,1,1,1]
=> [4]
=> 4
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 4
([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000273
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],2)
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],4)
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> ([],5)
=> ([],5)
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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: St000288
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
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%
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]
=> 10 => 1
([],2)
=> [1,1]
=> 110 => 2
([(0,1)],2)
=> [2]
=> 100 => 1
([],3)
=> [1,1,1]
=> 1110 => 3
([(1,2)],3)
=> [2,1]
=> 1010 => 2
([(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([],4)
=> [1,1,1,1]
=> 11110 => 4
([(2,3)],4)
=> [2,1,1]
=> 10110 => 3
([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([],5)
=> [1,1,1,1,1]
=> 111110 => 5
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 4
([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 100000 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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
([],2)
=> [1,1]
=> [2]
=> 2
([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> [1,1,1]
=> [2,1]
=> 3
([(1,2)],3)
=> [2,1]
=> [3]
=> 2
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([],4)
=> [1,1,1,1]
=> [3,1]
=> 4
([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 3
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [3,2]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 4
([(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ 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)
=> [2] => [2] => 2
([(0,1)],2)
=> [1,1] => [1,1] => 1
([],3)
=> [3] => [3] => 3
([(1,2)],3)
=> [1,2] => [2,1] => 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([],4)
=> [4] => [4] => 4
([(2,3)],4)
=> [1,3] => [3,1] => 3
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,2] => 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,2] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2,1] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 1
([],5)
=> [5] => [5] => 5
([(3,4)],5)
=> [1,4] => [4,1] => 4
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 3
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,1,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,3] => 1
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,1,1,2] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,1,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(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),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,1,1,2] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,2,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,2,1,1] => 1
([(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000544
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],2)
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],4)
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> ([],5)
=> ([],5)
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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].
Matching statistic: St000733
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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%
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
([],2)
=> [1,1]
=> [[1],[2]]
=> 2
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 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)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
Description
The row containing the largest entry of a standard tableau.
The following 165 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. St000157The number of descents of a standard tableau. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000105The number of blocks in the set partition. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000363The number of minimal vertex covers of a graph. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001363The Euler characteristic of a graph according to Knill. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001733The number of weak left to right maxima of a Dyck path. St000053The number of valleys of the Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St001176The size of a partition minus its first part. St001372The length of a longest cyclic run of ones of a binary word. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000806The semiperimeter of the associated bargraph. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000809The reduced reflection length of the permutation. St001316The domatic number of a graph. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000668The least common multiple of the parts of the partition. St000286The number of connected components of the complement of a graph. St000617The number of global maxima of a Dyck path. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000456The monochromatic index of a connected graph. St001330The hat guessing number of a graph. St001545The second Elser number of a connected graph. St000989The number of final rises of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000093The cardinality of a maximal independent set of vertices of a graph. St000271The chromatic index of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000778The metric dimension of a graph. St001323The independence gap of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001949The rigidity index of a graph. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000090The variation of a composition. St000155The number of exceedances (also excedences) of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001742The difference of the maximal and the minimal degree in a graph. St000654The first descent of a permutation. St000216The absolute length of a permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001812The biclique partition number of a graph. St000264The girth of a graph, which is not a tree. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000160The multiplicity of the smallest part of a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000006The dinv of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001118The acyclic chromatic index of a graph. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001152The number of pairs with even minimum in a perfect matching. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001060The distinguishing index of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
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