- FindStat

Your data matches 96 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000147
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000228
(load all 3 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition. This statistic is the constant statistic of the level sets.

Matching statistic: St000288
(load all 3 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 10 => 1
([],2)
 => [1,1]
 => 110 => 2
([(0,1)],2)
 => [2]
 => 100 => 1
([],3)
 => [1,1,1]
 => 1110 => 3
([(1,2)],3)
 => [2,1]
 => 1010 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => 1100 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
([],4)
 => [1,1,1,1]
 => 11110 => 4
([(2,3)],4)
 => [2,1,1]
 => 10110 => 3
([(1,3),(2,3)],4)
 => [2,2,1]
 => 11010 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 11100 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 11100 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 10100 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 111100 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
([],5)
 => [1,1,1,1,1]
 => 111110 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 4
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 110110 => 4
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 3
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 4
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 11100 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 101010 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 110010 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 110100 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 111100 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 110100 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 100100 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 101000 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 110000 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([],6)
 => [1,1,1,1,1,1]
 => 1111110 => 6
([(4,5)],6)
 => [2,1,1,1,1]
 => 1011110 => 5
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => 1101110 => 5
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => 1110110 => 5
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => 110110 => 4
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => 1110110 => 5
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => 111010 => 4

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000378

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [1]
 => 1
([],2)
 => [1,1]
 => [2]
 => 2
([(0,1)],2)
 => [2]
 => [1,1]
 => 1
([],3)
 => [1,1,1]
 => [2,1]
 => 3
([(1,2)],3)
 => [2,1]
 => [3]
 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => [4]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([],4)
 => [1,1,1,1]
 => [3,1]
 => 4
([(2,3)],4)
 => [2,1,1]
 => [2,2]
 => 3
([(1,3),(2,3)],4)
 => [2,2,1]
 => [2,2,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [4]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [2,2,2]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [5]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [5,1,1,1]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [6]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([],5)
 => [1,1,1,1,1]
 => [3,2]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 4
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,1,1]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [4,1,1,1]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [5,1,1,1]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,2,1]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [4,1,1,1]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [2,2,2]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [5,1,1,1]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [5,1]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2,2,1]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [6,1]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2,2,1]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [2,2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [5,1,1,1]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2,2,1]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [6]
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [2,2,2,2]
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [7]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [8]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([],6)
 => [1,1,1,1,1,1]
 => [3,2,1]
 => 6
([(4,5)],6)
 => [2,1,1,1,1]
 => [4,2]
 => 5
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [4,3]
 => 5
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => 5
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,1,1]
 => 4
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => 5
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [4,1,1,1]
 => 4

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

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [[1],[2]]
 => 2
([(0,1)],2)
 => [2]
 => [[1,2]]
 => 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
([(1,3),(2,3)],4)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => 5
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => 5
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => 5
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 4
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => 5
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 4

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St000157
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [[1]]
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
([(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 4 = 5 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3 = 4 - 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3 = 4 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3 = 4 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3 = 4 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 5 = 6 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 4 = 5 - 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 4 = 5 - 1
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 4 = 5 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3 = 4 - 1
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 4 = 5 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 3 = 4 - 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: St000507
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [[1]]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
([(0,1)],2)
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 4
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 3
([(1,3),(2,3)],4)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 4
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[1,4,6],[2,5],[3]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [[1,4,6],[2,5,7],[3]]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [[1,4,7],[2,5],[3,6]]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [[1,4,6],[2,5,7],[3]]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [[1,4,7],[2,5,8],[3,6]]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [[1,4,6],[2,5,7],[3]]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [[1,4,7],[2,5,8],[3,6]]
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [[1,5],[2,6],[3],[4]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => [[1,5],[2,6],[3,7],[4]]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 6
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 5
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [[1,3,5,6,7],[2,4]]
 => 5
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [[1,3,5,7,8],[2,4,6]]
 => 5
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 4
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [[1,3,5,7,8],[2,4,6]]
 => 5
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => 4

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000676

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000734

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [1]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [2]
 => [[1,2]]
 => 2
([(0,1)],2)
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
([],3)
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3
([(1,2)],3)
 => [2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
([(0,2),(1,2)],3)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([],4)
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 4
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 3
([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([],5)
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([],6)
 => [1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 6
([(4,5)],6)
 => [2,1,1,1,1]
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 5
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 5
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 5
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 4

Description

The last entry in the first row of a standard tableau.

The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001462The number of factors of a standard tableaux under concatenation. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001304The number of maximally independent sets of vertices of a graph. 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. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001644The dimension of a graph. St000741The Colin de Verdière graph invariant. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000146The Andrews-Garvan crank of a partition. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001812The biclique partition number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2.