- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001119

Mapping

St001119: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 0
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => 0
([(2,3)],4)
 => 0
([(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
([(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)
 => 3
([],5)
 => 0
([(3,4)],5)
 => 0
([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => 0
([(1,4),(2,3),(3,4)],5)
 => 0
([(0,1),(2,4),(3,4)],5)
 => 1
([(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(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)
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(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)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(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),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 4

Description

The length of a shortest maximal path in a graph.

Matching statistic: St000667
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 43%

Values

([],1)
 => [1]
 => []
 => ? = 0 + 1
([],2)
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => ? = 1 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 3 + 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]
 => []
 => ? = 3 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 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
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 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 = 0 + 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 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 4 + 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]
 => []
 => ? = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2 = 1 + 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]
 => []
 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 4 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 3 + 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]
 => []
 => ? = 3 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
([(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),(3,4)],5)
 => [5]
 => []
 => ? = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 4 + 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]
 => []
 => ? = 4 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 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 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 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 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 4 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 4 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 3 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 4 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 3 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 3 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 3 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 4 + 1

Description

The greatest common divisor of the parts of the partition.

Matching statistic: St000993

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 43%

Values

([],1)
 => [1]
 => []
 => []
 => ? = 0 + 1
([],2)
 => [1,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => []
 => ? = 1 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 2 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => ? = 3 + 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]
 => []
 => []
 => ? = 3 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 2 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [2,1]
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 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]
 => ? = 0 + 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]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => []
 => ? = 4 + 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]
 => []
 => []
 => ? = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,1]
 => 2 = 1 + 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]
 => []
 => []
 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => ? = 4 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 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]
 => []
 => []
 => ? = 3 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(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),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => []
 => ? = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 4 + 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]
 => []
 => []
 => ? = 4 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => ? = 0 + 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]
 => [3,1]
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [2,1]
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 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]
 => ? = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 3 = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => ? = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [2,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => 1 = 0 + 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1 = 0 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [2,1]
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => [1,1]
 => 2 = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [2]
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [2,1,1]
 => 1 = 0 + 1

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St001038

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 43%

Values

([],1)
 => [1]
 => []
 => []
 => ? = 0 + 1
([],2)
 => [1,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => []
 => ? = 1 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 2 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => ? = 3 + 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]
 => []
 => []
 => ? = 3 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 2 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 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]
 => [1,0]
 => ? = 0 + 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]
 => [1,0]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => []
 => ? = 4 + 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]
 => []
 => []
 => ? = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 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]
 => []
 => []
 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => ? = 4 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 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]
 => []
 => []
 => ? = 3 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(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),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => []
 => ? = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 4 + 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]
 => []
 => []
 => ? = 4 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1,0]
 => ? = 0 + 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]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 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]
 => [1,0]
 => ? = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1,0]
 => ? = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1

Description

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

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: 29%

Values

([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(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]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(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]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(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]
 => []
 => ?
 => ? = 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1
([(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]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(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]
 => []
 => ?
 => ? = 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(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),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(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]
 => []
 => ?
 => ? = 4
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(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]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(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]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1]
 => 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
([(2,6),(3,5),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,6),(4,5),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 0

Description

The largest nonnegative integer which is not a part and is smaller than the largest part 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: 29%

Values

([],1)
 => [1]
 => []
 => ?
 => ? = 0 + 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 3 + 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]
 => []
 => ?
 => ? = 3 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 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 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 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]
 => []
 => ? = 0 + 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]
 => []
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4 + 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]
 => []
 => ?
 => ? = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1 + 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]
 => []
 => ?
 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 4 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 + 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]
 => []
 => ?
 => ? = 3 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(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),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4 + 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]
 => []
 => ?
 => ? = 4 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0 + 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]
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 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]
 => []
 => ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1]
 => 1 = 0 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 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: 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: 29%

Values

([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(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]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => ? = 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(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]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(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]
 => []
 => ?
 => ? = 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1
([(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]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(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]
 => []
 => ?
 => ? = 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(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),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(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]
 => []
 => ?
 => ? = 4
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [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: 29%

Values

([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 2
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([(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]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => ? = 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(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]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(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]
 => []
 => ?
 => ? = 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1
([(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]
 => []
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(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]
 => []
 => ?
 => ? = 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(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),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 4
([(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]
 => []
 => ?
 => ? = 4
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [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

$\lambda$ .