- FindStat

Your data matches 74 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000183

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The side length of the Durfee square of an integer partition. Given a partition

$\lambda = (\lambda_1,\ldots,\lambda_n)$ , the Durfee square is the largest partition

$(s^s)$ whose diagram fits inside the diagram of

$\lambda$ . In symbols,

$s = \max\{ i \mid \lambda_i \geq i \}$ . This is also known as the Frobenius rank.

Matching statistic: St001031

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of the bicoloured Motzkin path associated with the Dyck path.

Matching statistic: St001175

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 97%●distinct values known / distinct values provided: 67%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001393
(load all 3 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%

Values

[[]]
 => ([(0,1)],2)
 => ([],2)
 => ([],2)
 => 0
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],3)
 => 0
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],4)
 => 0
[[],[],[],[]]
 => ([(0,4),(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)
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],5)
 => 0
[[],[],[],[],[]]
 => ([(0,5),(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)
 => 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(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)
 => ([(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)
 => ? = 1
[[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
 => ? = 2
[[[],[],[],[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? = 1
[[[],[],[[]],[]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? = 1
[[[],[[]],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? = 1
[[[[]],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)
 => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? = 1
[[[[[[[[]]]]]]]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ?
 => ? = 0
[[],[[[],[]],[[],[]]]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
 => ([(1,8),(2,3),(2,6),(2,7),(2,8),(3,4),(3,5),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[],[]],[[],[[],[]]]]
 => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
 => ([(1,5),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[],[]],[[[],[]],[]]]
 => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
 => ([(1,5),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[],[[],[]]],[[],[]]]
 => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
 => ([(1,5),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[[],[]],[]],[[],[]]]
 => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
 => ([(1,5),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[[],[]],[[],[]]],[]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
 => ([(1,8),(2,3),(2,6),(2,7),(2,8),(3,4),(3,5),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1

Description

The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.

Matching statistic: St001333
(load all 2 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%

Values

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

Description

The cardinality of a minimal edge-isolating set of a graph. Let

$\mathcal F$ be a set of graphs. A set of vertices

$S$ is

$\mathcal F$ -isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of

$S$ does not contain any graph in

$\mathcal F$ . This statistic returns the cardinality of the smallest isolating set when

$\mathcal F$ contains only the graph with one edge.

Matching statistic: St001261
(load all 2 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%

Values

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

Description

The Castelnuovo-Mumford regularity of a graph.

Matching statistic: St000260
(load all 5 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%

Values

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

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000455
(load all 5 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 55%●distinct values known / distinct values provided: 33%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001695

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%

Values

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

Description

The natural comajor index of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation. The natural comajor index of a tableau of size

$n$ with natural descent set

$D$ is then

$\sum_{d\in D} n-d$ .

Matching statistic: St001698

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%

Values

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

Description

The comajor index of a standard tableau minus the weighted size of its shape.

The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000862The number of parts of the shifted shape of a permutation. St001665The number of pure excedances of a permutation. St001624The breadth of a lattice. St001741The largest integer such that all patterns of this size are contained in the permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001271The competition number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000374The number of exclusive right-to-left minima of a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001093The detour number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001734The lettericity of a graph. St000451The length of the longest pattern of the form k 1 2. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000929The constant term of the character polynomial of an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001488The number of corners of a skew partition.