Processing math: 40%

Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000260
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => 0
[[],[]]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 3
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[[],[[[],[[]]]]]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[],[[[[[]]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[[[]],[[[]],[]]]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[[]],[[[],[]]]]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[[]],[[[[]]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 3
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[[],[]],[],[[]]]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2
[[[[]]],[],[[]]]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St001597
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[[]]
 => [1,0]
 => []
 => [[],[]]
 => ? = 0
[[],[]]
 => [1,0,1,0]
 => [1]
 => [[1],[]]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ? = 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ? = 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? = 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [[4,3,3,2,1],[]]
 => ? = 3
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [[3,3,3,2,1],[]]
 => ? = 2
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [[4,2,2,2,1],[]]
 => ? = 2
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [[3,2,2,2,1],[]]
 => ? = 2
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [[2,2,2,2,1],[]]
 => ? = 2
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[4,4,3,1,1],[]]
 => ? = 2
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[4,3,3,1,1],[]]
 => ? = 2
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ? = 2
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[4,4,1,1,1],[]]
 => ? = 2
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ? = 2
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ? = 2
[[],[[[],[[]]]]]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => 2
[[],[[[[[]]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [[5,3,3,2],[]]
 => ? = 2
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [[4,3,3,2],[]]
 => ? = 2
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [[3,3,3,2],[]]
 => ? = 2
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[5,2,2,2],[]]
 => ? = 2
[[[]],[[[]],[]]]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [[4,2,2,2],[]]
 => ? = 2
[[[]],[[[],[]]]]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [[3,2,2,2],[]]
 => ? = 2
[[[]],[[[[]]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[2,2,2,2],[]]
 => ? = 2
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [[5,4,3,1],[]]
 => ? = 3
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [[5,4,3],[]]
 => ? = 2
[[[],[]],[],[[]]]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [[4,4,3,1],[]]
 => ? = 2
[[[[]]],[],[[]]]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => [[4,4,3],[]]
 => ? = 2
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [[5,3,3,1],[]]
 => ? = 2
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => [[5,3,3],[]]
 => ? = 2
[[[],[]],[[],[]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [[4,3,3,1],[]]
 => ? = 2
[[[],[]],[[[]]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1]
 => [[3,3,3,1],[]]
 => ? = 2
[[[[]]],[[],[]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,3,3]
 => [[4,3,3],[]]
 => ? = 2
[[[[]]],[[[]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [[3,3,3],[]]
 => ? = 2
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [[5,4,1,1],[]]
 => ? = 2
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [5,4,1]
 => [[5,4,1],[]]
 => ? = 2
[[[[[]]]],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,4]
 => [[5,4],[]]
 => ? = 2
[[[],[[]]],[[]]]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1]
 => [[4,4,1,1],[]]
 => ? = 2
[[[[],[]]],[[]]]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [4,4,1]
 => [[4,4,1],[]]
 => ? = 2
[[[[[]]]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [[4,4],[]]
 => ? = 2
[[[[]],[[]]],[]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => [[5,2,2],[]]
 => ? = 2
[[[[],[]],[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [5,3,1]
 => [[5,3,1],[]]
 => ? = 2
[[[[[]]],[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => [[5,3],[]]
 => ? = 2
[[[[[]],[]]],[]]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => [[5,2],[]]
 => 2
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => [[5,4,3,3,2,1],[]]
 => ? = 3
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [[4,4,3,3,2,1],[]]
 => ? = 2
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => [[5,3,3,3,2,1],[]]
 => ? = 3
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => [[4,3,3,3,2,1],[]]
 => ? = 2
[[],[],[],[[[[]]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [[3,3,3,3,2,1],[]]
 => ? = 2
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => [[5,4,4,2,2,1],[]]
 => ? = 3
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [[4,4,4,2,2,1],[]]
 => ? = 2
[[],[[[[[[]]]]]]]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => 1
[[[[[[[]]]]]],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => [[6],[]]
 => 1
Description
The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St001269
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 75%
Values
[[]]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,4,1,2,6] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,2,3,6] => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,5,1,2,3,6] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,2,4,6] => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,4,6] => 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,1,7] => ? = 3
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,1,7] => ? = 2
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,1,7] => ? = 2
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => ? = 2
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,1,7] => ? = 2
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [5,6,4,2,3,1,7] => ? = 2
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [5,4,6,2,3,1,7] => ? = 2
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [4,5,6,2,3,1,7] => ? = 2
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [5,6,2,3,4,1,7] => ? = 2
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,1,7] => ? = 2
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,1,7] => ? = 2
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,1,7] => ? = 2
[[],[[[],[[]]]]]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => ? = 2
[[],[[[[[]]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => ? = 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,1,2,7] => ? = 2
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,1,2,7] => ? = 2
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,1,2,7] => ? = 2
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,1,2,7] => ? = 2
[[[]],[[[]],[]]]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,1,2,7] => ? = 2
[[[]],[[[],[]]]]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,1,2,7] => ? = 2
[[[]],[[[[]]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,1,2,7] => ? = 2
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2,1,3,7] => ? = 3
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => ? = 2
[[[],[]],[],[[]]]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [5,6,4,2,1,3,7] => ? = 2
[[[[]]],[],[[]]]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [5,6,4,1,2,3,7] => ? = 2
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [6,4,5,2,1,3,7] => ? = 2
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [6,4,5,1,2,3,7] => ? = 2
[[[],[]],[[],[]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [5,4,6,2,1,3,7] => ? = 2
[[[],[]],[[[]]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [4,5,6,2,1,3,7] => ? = 2
[[[[]]],[[],[]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [5,4,6,1,2,3,7] => ? = 2
[[[[]]],[[[]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [4,5,6,1,2,3,7] => ? = 2
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [6,5,2,3,1,4,7] => ? = 2
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2,1,3,4,7] => ? = 2
[[[[[]]]],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? = 2
[[[],[[]]],[[]]]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [5,6,2,3,1,4,7] => ? = 2
[[[[],[]]],[[]]]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [5,6,2,1,3,4,7] => ? = 2
[[[[[]]]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [5,6,1,2,3,4,7] => ? = 2
[[[[]],[[]]],[]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [6,3,4,1,2,5,7] => ? = 2
[[[[],[]],[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [6,4,2,1,3,5,7] => ? = 2
[[[[[]]],[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [6,4,1,2,3,5,7] => ? = 2
[[[[[]],[]]],[]]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => ? = 2
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => ? = 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,3,2,1,8] => ? = 3
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [5,6,4,7,3,2,1,8] => ? = 2
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,3,2,1,8] => ? = 3
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1,8] => ? = 2
[[],[],[],[[[[]]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,3,2,1,8] => ? = 2
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [6,5,7,3,4,2,1,8] => ? = 3
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [5,6,7,3,4,2,1,8] => ? = 2
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [6,4,5,3,7,2,1,8] => ? = 3
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001874
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001874: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 75%
Values
[[]]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,3,2] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,3,5,2,6,4] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,3,5,6,2,4] => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,3,6,2,4,5] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,3,4,6,2,5] => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,3,4,5,6,2] => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,5,6,3] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,4,5,2,6,3] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,4,5,6,2,3] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,2,6,3,5] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,2,6,3,4] => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,4,6,2,3,5] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,5,6,2,3,4] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,5,2,3,6,4] => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,5] => 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,3,5,2,7,4,6] => ? = 3
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,3,5,7,2,4,6] => ? = 2
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,3,5,2,6,7,4] => ? = 2
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,3,5,6,2,7,4] => ? = 2
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,3,5,6,7,2,4] => ? = 2
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,3,6,2,4,7,5] => ? = 2
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,3,6,2,7,4,5] => ? = 2
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5] => ? = 2
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,7,2,4,5,6] => ? = 2
[[],[[],[[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,4,6,2,7,5] => ? = 2
[[],[[],[[[]]]]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,3,4,6,7,2,5] => ? = 2
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,3,4,7,2,5,6] => ? = 2
[[],[[[],[[]]]]]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,3,4,5,7,2,6] => ? = 2
[[],[[[[[]]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,3,4,5,6,7,2] => ? = 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,4,2,5,7,3,6] => ? = 2
[[[]],[],[[],[]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,4,5,2,7,3,6] => ? = 2
[[[]],[],[[[]]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,4,5,7,2,3,6] => ? = 2
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,4,2,5,6,7,3] => ? = 2
[[[]],[[[]],[]]]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,4,5,2,6,7,3] => ? = 2
[[[]],[[[],[]]]]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,4,5,6,2,7,3] => ? = 2
[[[]],[[[[]]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,4,5,6,7,2,3] => ? = 2
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,6,3,7,5] => ? = 3
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,5,2,6,3,7,4] => ? = 2
[[[],[]],[],[[]]]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,4,6,2,3,7,5] => ? = 2
[[[[]]],[],[[]]]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [1,5,6,2,3,7,4] => ? = 2
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,6,7,3,5] => ? = 2
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,5,2,6,7,3,4] => ? = 2
[[[],[]],[[],[]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [1,4,6,2,7,3,5] => ? = 2
[[[],[]],[[[]]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [1,4,6,7,2,3,5] => ? = 2
[[[[]]],[[],[]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [1,5,6,2,7,3,4] => ? = 2
[[[[]]],[[[]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,5,6,7,2,3,4] => ? = 2
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,4,2,7,3,5,6] => ? = 2
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,5,2,7,3,4,6] => ? = 2
[[[[[]]]],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,6,2,7,3,4,5] => ? = 2
[[[],[[]]],[[]]]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [1,4,7,2,3,5,6] => ? = 2
[[[[],[]]],[[]]]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [1,5,7,2,3,4,6] => ? = 2
[[[[[]]]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,6,7,2,3,4,5] => ? = 2
[[[[]],[[]]],[]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,7,4] => ? = 2
[[[[],[]],[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,5,2,3,7,4,6] => ? = 2
[[[[[]]],[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [1,6,2,3,7,4,5] => ? = 2
[[[[[]],[]]],[]]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,6,2,3,4,7,5] => ? = 2
[[[[[[]]]]],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,7,2,3,4,5,6] => ? = 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,5,2,7,4,8,6] => ? = 3
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,3,5,7,2,4,8,6] => ? = 2
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,3,5,2,7,8,4,6] => ? = 3
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,3,5,7,2,8,4,6] => ? = 2
[[],[],[],[[[[]]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,3,5,7,8,2,4,6] => ? = 2
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,3,5,2,8,4,6,7] => ? = 3
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,3,5,8,2,4,6,7] => ? = 2
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,5,2,6,8,4,7] => ? = 3
Description
Lusztig's a-function for the symmetric group. Let x be a permutation corresponding to the pair of tableaux (P(x),Q(x)) by the Robinson-Schensted correspondence and shape(Q(x)′)=(λ1,...,λk) where Q(x)′ is the transposed tableau. Then a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}. See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.
Matching statistic: St000387
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000387: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 75%
Values
[[]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(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)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(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)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[[],[],[[[]]]]
 => ([(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)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 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)
 => ([(0,6),(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)
 => 3 = 2 + 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)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 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)
 => ([(0,6),(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)
 => 3 = 2 + 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)
 => ([(0,6),(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)
 => 3 = 2 + 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)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 2 + 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)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[[[[]]],[],[]]
 => ([(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)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 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)
 => ([(0,6),(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)
 => 3 = 2 + 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)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 2 + 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)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[],[[[],[]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[[],[[],[]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[[[],[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 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)
 => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[]],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[]],[[[[]]]]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[],[]]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 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)
 => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[],[]]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[[]]]],[[]]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[]],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[],[]],[]],[]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[[]]],[]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[[]],[]]],[]]
 => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
 => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[[],[],[],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 3 + 1
[[],[],[],[[],[[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 2 + 1
[[],[],[],[[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 3 + 1
[[],[],[],[[[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 2 + 1
[[],[],[],[[[[]]]]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 2 + 1
[[],[],[[]],[[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 3 + 1
[[],[],[[]],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 2 + 1
[[],[],[[],[[]],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ?
 => ?
 => ? = 3 + 1
Description
The matching number of a graph. For a graph G, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in G.
Matching statistic: St001569
(load all 48 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
[[]]
 => [.,.]
 => [1] => ? = 0
[[],[]]
 => [.,[.,.]]
 => [2,1] => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 2
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 2
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 2
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 1
[[],[],[],[[],[]]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => ? = 3
[[],[],[],[[[]]]]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => ? = 2
[[],[],[[[]],[]]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [6,3,4,5,2,1] => ? = 2
[[],[],[[[],[]]]]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => [5,3,4,6,2,1] => ? = 2
[[],[],[[[[]]]]]
 => [.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => ? = 2
[[],[[]],[],[[]]]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => [4,5,3,2,6,1] => ? = 2
[[],[[]],[[],[]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [6,4,5,2,3,1] => ? = 2
[[],[[]],[[[]]]]
 => [.,[[.,[[[.,.],.],.]],.]]
 => [3,4,5,2,6,1] => ? = 2
[[],[[[]]],[[]]]
 => [.,[[[.,[[.,.],.]],.],.]]
 => [3,4,2,5,6,1] => ? = 2
[[],[[],[[],[]]]]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => [5,3,4,2,6,1] => ? = 2
[[],[[],[[[]]]]]
 => [.,[[.,.],[[[.,.],.],.]]]
 => [4,5,6,2,3,1] => ? = 2
[[],[[[]],[[]]]]
 => [.,[[[.,.],.],[[.,.],.]]]
 => [5,6,2,3,4,1] => ? = 2
[[],[[[],[[]]]]]
 => [.,[[[.,.],[[.,.],.]],.]]
 => [4,5,2,3,6,1] => ? = 2
[[],[[[[[]]]]]]
 => [.,[[[[[.,.],.],.],.],.]]
 => [2,3,4,5,6,1] => ? = 1
[[[]],[],[[]],[]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => [4,3,5,2,1,6] => ? = 2
[[[]],[],[[],[]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [6,4,5,3,1,2] => ? = 2
[[[]],[],[[[]]]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => [3,4,5,2,1,6] => ? = 2
[[[]],[[[]]],[]]
 => [[.,[[[.,[.,.]],.],.]],.]
 => [3,2,4,5,1,6] => ? = 2
[[[]],[[[]],[]]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => [6,3,4,5,1,2] => ? = 2
[[[]],[[[],[]]]]
 => [[.,.],[[[.,.],[.,.]],.]]
 => [5,3,4,6,1,2] => ? = 2
[[[]],[[[[]]]]]
 => [[.,[[[[.,.],.],.],.]],.]
 => [2,3,4,5,1,6] => ? = 2
[[[],[]],[],[],[]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => [6,5,4,2,1,3] => ? = 3
[[[[]]],[],[],[]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => [4,3,2,1,5,6] => ? = 2
[[[],[]],[],[[]]]
 => [[.,[.,.]],[.,[[.,.],.]]]
 => [5,6,4,2,1,3] => ? = 2
[[[[]]],[],[[]]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => [3,4,2,1,5,6] => ? = 2
[[[],[]],[[]],[]]
 => [[.,[.,.]],[[.,[.,.]],.]]
 => [5,4,6,2,1,3] => ? = 2
[[[[]]],[[]],[]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => [3,2,4,1,5,6] => ? = 2
[[[],[]],[[],[]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => [6,4,5,2,1,3] => ? = 2
[[[],[]],[[[]]]]
 => [[.,[.,.]],[[[.,.],.],.]]
 => [4,5,6,2,1,3] => ? = 2
[[[[]]],[[],[]]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => [6,4,5,1,2,3] => ? = 2
[[[[]]],[[[]]]]
 => [[[.,[[[.,.],.],.]],.],.]
 => [2,3,4,1,5,6] => ? = 2
[[[],[[]]],[],[]]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => [6,5,2,3,1,4] => ? = 2
[[[[],[]]],[],[]]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => [5,4,2,1,3,6] => ? = 2
[[[[[]]]],[],[]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => [3,2,1,4,5,6] => ? = 2
[[[],[[]]],[[]]]
 => [[.,[[.,.],.]],[[.,.],.]]
 => [5,6,2,3,1,4] => ? = 2
[[[[],[]]],[[]]]
 => [[[.,[.,.]],[[.,.],.]],.]
 => [4,5,2,1,3,6] => ? = 2
[[[[[]]]],[[]]]
 => [[[[.,[[.,.],.]],.],.],.]
 => [2,3,1,4,5,6] => ? = 2
[[[[]],[[]]],[]]
 => [[[.,[[.,.],.]],.],[.,.]]
 => [6,2,3,1,4,5] => ? = 2
[[[[],[]],[]],[]]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => [6,4,2,1,3,5] => ? = 2
[[[[[]]],[]],[]]
 => [[[[.,[.,.]],.],.],[.,.]]
 => [6,2,1,3,4,5] => ? = 2
[[[[[]],[]]],[]]
 => [[[[.,[.,.]],.],[.,.]],.]
 => [5,2,1,3,4,6] => ? = 2
[[[[[[]]]]],[]]
 => [[[[[.,[.,.]],.],.],.],.]
 => [2,1,3,4,5,6] => ? = 1
[[],[],[],[[],[],[]]]
 => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [7,6,4,5,3,2,1] => ? = 3
[[],[],[],[[],[[]]]]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [6,7,4,5,3,2,1] => ? = 2
[[],[],[],[[[]],[]]]
 => [.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [7,4,5,6,3,2,1] => ? = 3
[[],[],[],[[[],[]]]]
 => [.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [6,4,5,7,3,2,1] => ? = 2
[[],[],[],[[[[]]]]]
 => [.,[.,[.,[[[[.,.],.],.],.]]]]
 => [4,5,6,7,3,2,1] => ? = 2
[[],[],[[]],[[],[]]]
 => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [7,5,6,3,4,2,1] => ? = 3
[[],[],[[]],[[[]]]]
 => [.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => ? = 2
Description
The maximal modular displacement of a permutation. This is \max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right) for a permutation \pi of \{1,\dots,n\}.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 75%
Values
[[]]
 => [[]]
 => ([(0,1)],2)
 => -1 = 0 - 1
[[],[]]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[],[[]]]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[]],[]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 1
[[],[],[[]]]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 - 1
[[],[[[]]]]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[],[]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[[]],[[]]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 - 1
[[[[]]],[]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 - 1
[[],[],[[],[]]]
 => [[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 - 1
[[],[[]],[[]]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 - 1
[[[],[]],[],[]]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 - 1
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2 - 1
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2 - 1
[[[[]]],[[]]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 1 - 1
[[],[],[],[[],[]]]
 => [[[[[[]],[]]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 3 - 1
[[],[],[],[[[]]]]
 => [[[[[],[],[]]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[],[[[]],[]]]
 => [[[[],[[]],[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[],[],[[[],[]]]]
 => [[[[[]],[],[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[],[],[[[[]]]]]
 => [[[[],[],[],[]]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[[]],[],[[]]]
 => [[[],[[[],[]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? = 2 - 1
[[],[[]],[[],[]]]
 => [[[],[[[]],[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[[]],[[[]]]]
 => [[[],[[],[],[]]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[[[]]],[[]]]
 => [[[],[],[[],[]]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[[],[[],[]]]]
 => [[[[[]],[]],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[[],[[[]]]]]
 => [[[[],[],[]],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[[[]],[[]]]]
 => [[[],[[],[]],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[[[],[[]]]]]
 => [[[[],[]],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[],[[[[[]]]]]]
 => [[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[[]],[],[[]],[]]
 => [[],[[[],[[]]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[]],[],[[],[]]]
 => [[],[[[[]],[]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[]],[],[[[]]]]
 => [[],[[[],[],[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[]],[[[]]],[]]
 => [[],[[],[],[[]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[]],[[[]],[]]]
 => [[],[[],[[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[]],[[[],[]]]]
 => [[],[[[]],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[]],[[[[]]]]]
 => [[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[],[]],[],[],[]]
 => [[[]],[[[[]]]]]
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 3 - 1
[[[[]]],[],[],[]]
 => [[],[],[[[[]]]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ? = 2 - 1
[[[],[]],[],[[]]]
 => [[[]],[[[],[]]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ? = 2 - 1
[[[[]]],[],[[]]]
 => [[],[],[[[],[]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? = 2 - 1
[[[],[]],[[]],[]]
 => [[[]],[[],[[]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[[]]],[[]],[]]
 => [[],[],[[],[[]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 - 1
[[[],[]],[[],[]]]
 => [[[]],[[[]],[]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[],[]],[[[]]]]
 => [[[]],[[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[[]]],[[],[]]]
 => [[],[],[[[]],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 - 1
[[[[]]],[[[]]]]
 => [[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[],[[]]],[],[]]
 => [[[],[]],[[[]]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ? = 2 - 1
[[[[],[]]],[],[]]
 => [[[]],[],[[[]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[[[]]]],[],[]]
 => [[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[],[[]]],[[]]]
 => [[[],[]],[[],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? = 2 - 1
[[[[],[]]],[[]]]
 => [[[]],[],[[],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 - 1
[[[[[]]]],[[]]]
 => [[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 - 1
[[[[]],[[]]],[]]
 => [[],[[],[]],[[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 - 1
[[[[],[]],[]],[]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[[]]],[]],[]]
 => [[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[[]],[]]],[]]
 => [[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[[[]]]]],[]]
 => [[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[],[],[]]]
 => [[[[[[[]]],[]]]]]
 => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
 => ? = 3 - 1
[[],[],[],[[],[[]]]]
 => [[[[[[],[]],[]]]]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => ? = 2 - 1
[[],[],[],[[[]],[]]]
 => [[[[[],[[]],[]]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[[],[],[],[[[],[]]]]
 => [[[[[[]],[],[]]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 - 1
[[],[],[],[[[[]]]]]
 => [[[[[],[],[],[]]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 2 - 1
[[],[],[[]],[[],[]]]
 => [[[[],[[[]],[]]]]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[[],[],[[]],[[[]]]]
 => [[[[],[[],[],[]]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 2 - 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.