searching the database
Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000383
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00074: Posets —to 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)
=> ([],2)
=> [2] => 2
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => 1
([],3)
=> ([],3)
=> [3] => 3
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> ([],4)
=> [4] => 4
([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
([],5)
=> ([],5)
=> [5] => 5
([(3,4)],5)
=> ([(3,4)],5)
=> [1,4] => 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00074: Posets —to 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)
=> ([],2)
=> [2] => [2] => 2
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 1
([],3)
=> ([],3)
=> [3] => [3] => 3
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => [2,1] => 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([],4)
=> ([],4)
=> [4] => [4] => 4
([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => [3,1] => 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([],5)
=> ([],5)
=> [5] => [5] => 5
([(3,4)],5)
=> ([(3,4)],5)
=> [1,4] => [4,1] => 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 1
([],2)
=> ([],2)
=> [1,1]
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 3
([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,6),(1,5),(2,7),(7,3),(7,4)],8)
=> ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,4),(1,5),(1,6),(5,3),(5,7),(6,2),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 2
([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,6),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,4),(1,3),(1,7),(5,6),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,6),(1,2),(1,4),(3,7),(4,7),(5,3),(6,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,6),(2,4),(3,7),(5,3),(6,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1
([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,6),(1,7),(2,5),(3,4),(3,5),(4,6),(4,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(1,3),(1,6),(1,7),(4,2),(5,4),(7,5)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(4,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(1,5),(2,4),(2,7),(7,3)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,3),(1,5),(3,7),(4,6),(5,2),(5,7),(7,4)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,6),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(1,5),(3,7),(4,6),(5,2),(5,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(1,4),(2,3),(2,6),(2,7),(4,5),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(1,3),(2,4),(2,6),(4,5),(4,7),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 2
([(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(6,3),(7,3),(7,4),(7,5)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1
Description
The length of the partition.
Matching statistic: St000147
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(2,7),(7,3),(7,4)],8)
=> ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,4),(1,5),(1,6),(5,3),(5,7),(6,2),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2
([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,6),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(1,3),(1,7),(5,6),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,2),(1,4),(3,7),(4,7),(5,3),(6,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(2,4),(3,7),(5,3),(6,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(1,7),(2,5),(3,4),(3,5),(4,6),(4,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(1,3),(1,6),(1,7),(4,2),(5,4),(7,5)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(4,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(2,4),(2,7),(7,3)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,6),(1,3),(1,5),(3,7),(4,6),(5,2),(5,7),(7,4)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,6),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(1,5),(3,7),(4,6),(5,2),(5,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(1,4),(2,3),(2,6),(2,7),(4,5),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,3),(2,4),(2,6),(4,5),(4,7),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2
([(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(6,3),(7,3),(7,4),(7,5)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
Description
The largest part of an integer partition.
Matching statistic: St000378
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [2,1]
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [3]
=> 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [3,1]
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [3,2]
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(2,7),(7,3),(7,4)],8)
=> ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,4),(1,5),(1,6),(5,3),(5,7),(6,2),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2
([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,6),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(1,3),(1,7),(5,6),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,2),(1,4),(3,7),(4,7),(5,3),(6,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(2,4),(3,7),(5,3),(6,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(1,7),(2,5),(3,4),(3,5),(4,6),(4,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(1,3),(1,6),(1,7),(4,2),(5,4),(7,5)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(4,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(2,4),(2,7),(7,3)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,6),(1,3),(1,5),(3,7),(4,6),(5,2),(5,7),(7,4)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,6),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(1,5),(3,7),(4,6),(5,2),(5,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(1,4),(2,3),(2,6),(2,7),(4,5),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,3),(2,4),(2,6),(4,5),(4,7),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2
([(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(6,3),(7,3),(7,4),(7,5)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 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: St000733
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [[1]]
=> 1
([],2)
=> ([],2)
=> [1,1]
=> [[1],[2]]
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(2,7),(7,3),(7,4)],8)
=> ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,4),(1,5),(1,6),(5,3),(5,7),(6,2),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2
([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,6),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(1,3),(1,7),(5,6),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,2),(1,4),(3,7),(4,7),(5,3),(6,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(2,4),(3,7),(5,3),(6,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,6),(1,7),(2,5),(3,4),(3,5),(4,6),(4,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(1,3),(1,6),(1,7),(4,2),(5,4),(7,5)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(4,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(2,4),(2,7),(7,3)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3
([(0,6),(1,3),(1,5),(3,7),(4,6),(5,2),(5,7),(7,4)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,6),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(1,5),(3,7),(4,6),(5,2),(5,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(0,7),(1,4),(2,3),(2,6),(2,7),(4,5),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1
([(0,6),(1,3),(2,4),(2,6),(4,5),(4,7),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2
([(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(6,3),(7,3),(7,4),(7,5)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 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)
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],2)
=> ([],2)
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([],3)
=> ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3 - 1
([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(1,5),(2,7),(7,3),(7,4)],8)
=> ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 3 - 1
([(0,4),(1,5),(1,6),(5,3),(5,7),(6,2),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2 - 1
([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,3),(1,6),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,4),(1,3),(1,7),(5,6),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(1,2),(1,4),(3,7),(4,7),(5,3),(6,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,6),(2,4),(3,7),(5,3),(6,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,6),(1,7),(2,5),(3,4),(3,5),(4,6),(4,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(0,7),(1,3),(1,6),(1,7),(4,2),(5,4),(7,5)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(4,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(1,5),(2,4),(2,7),(7,3)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 3 - 1
([(0,6),(1,3),(1,5),(3,7),(4,6),(5,2),(5,7),(7,4)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,6),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,7),(1,3),(1,4),(1,5),(3,7),(4,6),(5,2),(5,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(0,7),(1,4),(2,3),(2,6),(2,7),(4,5),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 1
([(0,6),(1,3),(2,4),(2,6),(4,5),(4,7),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ?
=> ? = 2 - 1
([(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(6,3),(7,3),(7,4),(7,5)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 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: St000288
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 10 => 1
([],2)
=> ([],2)
=> [1,1]
=> 110 => 2
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 100 => 1
([],3)
=> ([],3)
=> [1,1,1]
=> 1110 => 3
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 11110 => 4
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 10110 => 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 2
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 111110 => 5
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 100000 => 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> 1000000000000 => ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ? => ? = 3
([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(1,5),(2,7),(7,3),(7,4)],8)
=> ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ?
=> ? => ? = 3
([(0,4),(1,5),(1,6),(5,3),(5,7),(6,2),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? => ? = 2
([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ? => ? = 2
([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,3),(1,6),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,4),(1,3),(1,7),(5,6),(6,2),(7,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ? => ? = 2
([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(1,2),(1,4),(3,7),(4,7),(5,3),(6,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,6),(2,4),(3,7),(5,3),(6,5)],8)
=> ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ?
=> ? => ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,6),(1,7),(2,5),(3,4),(3,5),(4,6),(4,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(0,7),(1,3),(1,6),(1,7),(4,2),(5,4),(7,5)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(4,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(1,5),(2,4),(2,7),(7,3)],8)
=> ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ?
=> ? => ? = 3
([(0,6),(1,3),(1,5),(3,7),(4,6),(5,2),(5,7),(7,4)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,6),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,7),(1,3),(1,4),(1,5),(3,7),(4,6),(5,2),(5,6)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7),(6,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(0,7),(1,4),(2,3),(2,6),(2,7),(4,5),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,5),(1,3),(2,4),(2,5),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
=> ?
=> ? => ? = 1
([(0,6),(1,3),(2,4),(2,6),(4,5),(4,7),(6,7)],8)
=> ([(0,1),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? => ? = 2
([(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(2,5),(6,3),(7,3),(7,4),(7,5)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? => ? = 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000544
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],2)
=> ([],2)
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(3,4),(3,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(3,4),(4,6),(6,5)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(0,6),(1,6),(2,3),(2,4),(4,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,5),(2,6),(3,4),(3,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(2,5),(3,4),(4,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(3,6),(4,5),(4,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(0,4),(1,5),(1,6),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(0,6),(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(2,3),(2,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,3),(1,6),(2,6),(3,5),(4,7),(5,4),(6,7)],8)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,6),(2,5),(3,7),(4,7),(5,3),(6,4)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,3),(0,4),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(4,6),(5,6),(6,7)],8)
=> ([(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1
([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ([(0,1),(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ?
=> ? = 1
([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ?
=> ? = 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,7),(2,7),(3,4),(4,2),(5,3),(6,1)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,1),(0,2),(1,2),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> ([(0,2),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,11),(5,6),(5,7),(5,8),(5,9),(5,11),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ?
=> ? = 1
([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ([(0,5),(1,2),(1,9),(2,6),(3,7),(3,9),(4,6),(4,8),(5,7),(6,9),(7,8),(8,9)],10)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,5),(3,6),(3,7),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9)],10)
=> ?
=> ? = 1
([(0,5),(1,6),(2,9),(3,6),(3,9),(4,7),(5,1),(5,2),(5,3),(6,8),(8,7),(9,4),(9,8)],10)
=> ([(0,9),(1,3),(1,8),(2,8),(2,9),(3,5),(4,7),(4,9),(5,7),(5,8),(6,7),(6,8),(6,9)],10)
=> ([(0,1),(0,3),(0,4),(0,7),(0,8),(1,4),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,1),(0,2),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,7),(6,2),(6,7),(7,3),(7,4)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,6),(3,4),(3,6),(4,7),(5,6),(6,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,7),(3,5),(3,6),(5,4),(6,2),(6,4),(7,5),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,1),(0,2),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 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: St001363
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],2)
=> ([],2)
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 2
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> 4
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> 3
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> 5
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> 4
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(3,4),(3,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(3,4),(4,6),(6,5)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(0,6),(1,6),(2,3),(2,4),(4,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,5),(2,6),(3,4),(3,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(2,5),(3,4),(4,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(3,6),(4,5),(4,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(0,4),(1,5),(1,6),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 4
([(0,6),(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(2,3),(2,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 2
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,3),(1,6),(2,6),(3,5),(4,7),(5,4),(6,7)],8)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,6),(2,5),(3,7),(4,7),(5,3),(6,4)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,3),(0,4),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(4,6),(5,6),(6,7)],8)
=> ([(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1
([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ([(0,1),(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ?
=> ? = 1
([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ?
=> ? = 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,7),(2,7),(3,4),(4,2),(5,3),(6,1)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,1),(0,2),(1,2),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> ([(0,2),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,11),(5,6),(5,7),(5,8),(5,9),(5,11),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ?
=> ? = 1
([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ([(0,5),(1,2),(1,9),(2,6),(3,7),(3,9),(4,6),(4,8),(5,7),(6,9),(7,8),(8,9)],10)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,5),(3,6),(3,7),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9)],10)
=> ?
=> ? = 1
([(0,5),(1,6),(2,9),(3,6),(3,9),(4,7),(5,1),(5,2),(5,3),(6,8),(8,7),(9,4),(9,8)],10)
=> ([(0,9),(1,3),(1,8),(2,8),(2,9),(3,5),(4,7),(4,9),(5,7),(5,8),(6,7),(6,8),(6,9)],10)
=> ([(0,1),(0,3),(0,4),(0,7),(0,8),(1,4),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 1
([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,1),(0,2),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,7),(6,2),(6,7),(7,3),(7,4)],8)
=> ([(0,5),(0,7),(1,5),(1,7),(2,4),(2,6),(3,4),(3,6),(4,7),(5,6),(6,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 1
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(0,7),(1,3),(1,7),(3,5),(3,6),(5,4),(6,2),(6,4),(7,5),(7,6)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,1),(0,2),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,7),(1,7),(2,7),(7,3),(7,4),(7,5),(7,6)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
Description
The Euler characteristic of a graph according to Knill.
This is $$\sum_{k\geq 1} (-1)^{k-1} c_k,$$
where $c_k$ is the number of cliques with $k$ vertices.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000273The domination number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000553The number of blocks of a graph. St000286The number of connected components of the complement of a graph. St000456The monochromatic index of a connected graph. St001322The size of a minimal independent dominating set in a graph. 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. St001339The irredundance number of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St000264The girth of a graph, which is not a tree. 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. St000181The number of connected components of the Hasse diagram for the poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000379The number of Hamiltonian cycles in a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!