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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000228

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
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000738

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St000319
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

([],2)
 => [1,1]
 => [2]
 => 1 = 0 + 1
([],3)
 => [1,1,1]
 => [3]
 => 2 = 1 + 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1 = 0 + 1
([],4)
 => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
([],5)
 => [1,1,1,1,1]
 => [5]
 => 4 = 3 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 3 = 2 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([],6)
 => [1,1,1,1,1,1]
 => [6]
 => 5 = 4 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [5,1]
 => 4 = 3 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,2]
 => 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,2,1]
 => 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [3,3]
 => 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,2,1]
 => 2 = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
 => [14,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
 => [10,1,1]
 => [3,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
 => [14,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [4,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [5,1,1,1,1,1,1,1,1,1,1]
 => ? = 3 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => [9,1,1]
 => [3,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
 => [11,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
 => [13,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [4,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => [6,1,1,1,1,1]
 => [6,1,1,1,1,1]
 => ? = 4 + 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [12,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [4,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => [10,1,1,1,1]
 => [5,1,1,1,1,1,1,1,1,1]
 => ? = 3 + 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [12,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
 => [15,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => [11,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
 => [13,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
 => [10,1,1]
 => [3,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

([],2)
 => [1,1]
 => [2]
 => 1 = 0 + 1
([],3)
 => [1,1,1]
 => [3]
 => 2 = 1 + 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1 = 0 + 1
([],4)
 => [1,1,1,1]
 => [4]
 => 3 = 2 + 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
([],5)
 => [1,1,1,1,1]
 => [5]
 => 4 = 3 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 3 = 2 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
([],6)
 => [1,1,1,1,1,1]
 => [6]
 => 5 = 4 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [5,1]
 => 4 = 3 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,2]
 => 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,2,1]
 => 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [3,3]
 => 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,2,1]
 => 2 = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 1 = 0 + 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
 => [14,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
 => [10,1,1]
 => [3,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
 => [14,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [4,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [5,1,1,1,1,1,1,1,1,1,1]
 => ? = 3 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => [9,1,1]
 => [3,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
 => [11,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
 => [13,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [4,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => [6,1,1,1,1,1]
 => [6,1,1,1,1,1]
 => ? = 4 + 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [12,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [4,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => [10,1,1,1,1]
 => [5,1,1,1,1,1,1,1,1,1]
 => ? = 3 + 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [12,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
 => [15,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => [11,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [4,1,1,1,1,1,1,1,1,1]
 => ? = 2 + 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
 => [13,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
 => [10,1,1]
 => [3,1,1,1,1,1,1,1,1,1]
 => ? = 1 + 1

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St000371

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

([],2)
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
([(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 3
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 2
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 0
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 4
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => 3
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => 2
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 0
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
 => [14,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
 => ? => ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
 => [10,1,1]
 => [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
 => ? => ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
 => [14,1]
 => [[1,3,4,5,6,7,8,9,10,11,12,13,14,15],[2]]
 => ? => ? = 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
 => ? => ? = 2
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
 => ? => ? = 2
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
 => ? => ? = 3
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
 => ? => ? = 2
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
 => ? => ? = 2
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => [9,1,1]
 => [[1,4,5,6,7,8,9,10,11],[2],[3]]
 => ? => ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
 => [11,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
 => ? => ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
 => [13,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
 => ? => ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ? => ? = 2
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
 => ? => ? = 2
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
 => ? => ? = 2
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ? => ? = 4
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [12,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
 => ? => ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ? => ? = 2
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
 => ? => ? = 2
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => [10,1,1,1,1]
 => [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
 => ? => ? = 3
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [12,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
 => ? => ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
 => [15,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[2],[3]]
 => ? => ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => [11,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
 => ? => ? = 2
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
 => ? => ? = 2
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
 => [13,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
 => ? => ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
 => [10,1,1]
 => [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
 => ? => ? = 1

Description

The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) > \pi(j) > \pi(k)$ . In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

Matching statistic: St001336

Mapping

Mp00147: Graphs —square⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 71%

Values

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

Description

The minimal number of vertices in a graph whose complement is triangle-free.

Matching statistic: St001232
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001330
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%

Values

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

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St000454

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%

Values

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

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001645The pebbling number of a connected graph. St000567The sum of the products of all pairs of parts. St000699The toughness times the least common multiple of 1,. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001281The normalized isoperimetric number of a graph. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001651The Frankl number of a lattice. St001812The biclique partition number of a graph. St000264The girth of a graph, which is not a tree. St000287The number of connected components of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000286The number of connected components of the complement of a graph. St000553The number of blocks of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001060The distinguishing index of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra.