Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001316
(load all 3 compositions to match this statistic)
Mapping
St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => 1
([(3,4)],5)
 => 1
([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => 1
([(1,4),(2,3),(3,4)],5)
 => 1
([(0,1),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
Description
The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St000667
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 2
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 3
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 4
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 5
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 2
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1]
 => 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St001571
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 2
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 3
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 4
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 5
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 2
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1]
 => 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001392
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 1 - 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 2 - 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 3 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 5 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 2 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1]
 => 0 = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000478
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 1 - 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 2 - 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 3 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 5 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
Description
Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St000934
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 1 - 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 2 - 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 3 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 5 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
Description
The 2-degree of an integer partition. For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.