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Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000907
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => 1
[[]]
 => ([(0,1)],2)
 => 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 2
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 3
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 1
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 1
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 1
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000974
(load all 3 compositions to match this statistic)
Mapping
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => ? = 1 - 1
[[]]
 => 1 = 2 - 1
[[],[]]
 => 0 = 1 - 1
[[[]]]
 => 2 = 3 - 1
[[],[],[]]
 => 0 = 1 - 1
[[],[[]]]
 => 0 = 1 - 1
[[[]],[]]
 => 0 = 1 - 1
[[[],[]]]
 => 1 = 2 - 1
[[[[]]]]
 => 3 = 4 - 1
[[],[],[],[]]
 => 0 = 1 - 1
[[],[],[[]]]
 => 0 = 1 - 1
[[],[[]],[]]
 => 0 = 1 - 1
[[],[[],[]]]
 => 0 = 1 - 1
[[],[[[]]]]
 => 0 = 1 - 1
[[[]],[],[]]
 => 0 = 1 - 1
[[[]],[[]]]
 => 0 = 1 - 1
[[[],[]],[]]
 => 0 = 1 - 1
[[[[]]],[]]
 => 0 = 1 - 1
[[[],[],[]]]
 => 1 = 2 - 1
[[[],[[]]]]
 => 1 = 2 - 1
[[[[]],[]]]
 => 1 = 2 - 1
[[[[],[]]]]
 => 2 = 3 - 1
[[[[[]]]]]
 => 4 = 5 - 1
[[],[],[],[],[]]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => 0 = 1 - 1
[[],[],[[]],[]]
 => 0 = 1 - 1
[[],[],[[],[]]]
 => 0 = 1 - 1
[[],[],[[[]]]]
 => 0 = 1 - 1
[[],[[]],[],[]]
 => 0 = 1 - 1
[[],[[]],[[]]]
 => 0 = 1 - 1
[[],[[],[]],[]]
 => 0 = 1 - 1
[[],[[[]]],[]]
 => 0 = 1 - 1
[[],[[],[],[]]]
 => 0 = 1 - 1
[[],[[],[[]]]]
 => 0 = 1 - 1
[[],[[[]],[]]]
 => 0 = 1 - 1
[[],[[[],[]]]]
 => 0 = 1 - 1
[[],[[[[]]]]]
 => 0 = 1 - 1
[[[]],[],[],[]]
 => 0 = 1 - 1
[[[]],[],[[]]]
 => 0 = 1 - 1
[[[]],[[]],[]]
 => 0 = 1 - 1
[[[]],[[],[]]]
 => 0 = 1 - 1
[[[]],[[[]]]]
 => 0 = 1 - 1
[[[],[]],[],[]]
 => 0 = 1 - 1
[[[[]]],[],[]]
 => 0 = 1 - 1
[[[],[]],[[]]]
 => 0 = 1 - 1
[[[[]]],[[]]]
 => 0 = 1 - 1
[[[],[],[]],[]]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => 0 = 1 - 1
[[[[]],[]],[]]
 => 0 = 1 - 1
[[[[],[]]],[]]
 => 0 = 1 - 1
[[[[[]]]],[]]
 => 0 = 1 - 1
Description
The length of the trunk of an ordered tree. This is the length of the path from the root to the first vertex which has not exactly one child.
Matching statistic: St001107
(load all 17 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[]
 => []
 => []
 => []
 => ? = 1 - 1
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => ? = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 3 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[[],[[],[[],[[],[[],[]]]]]]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[],[[[],[]],[]]]]]
 => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[[],[]],[[],[]]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[[],[[],[]]],[]]]]
 => [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[[[],[]],[]],[]]]]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[],[]],[[],[[],[]]]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[],[]],[[[],[]],[]]]]
 => [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[],[[],[]]],[[],[]]]]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[[],[]],[]],[[],[]]]]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[],[[],[[],[]]]],[]]]
 => [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[],[[[],[]],[]]],[]]]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[[],[]],[[],[]]],[]]]
 => [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[[],[[],[]]],[]],[]]]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[[[[],[]],[]],[]],[]]]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[[[],[]],[[],[[],[[],[]]]]]
 => [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[],[]],[[],[[[],[]],[]]]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[],[]],[[[],[]],[[],[]]]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[],[]],[[[],[[],[]]],[]]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[],[]],[[[[],[]],[]],[]]]
 => [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 - 1
[[[],[[],[]]],[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[],[[],[]]],[[[],[]],[]]]
 => [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[],[]],[]],[[],[[],[]]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[[],[]],[]],[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1 - 1
[[[],[[],[[],[]]]],[[],[]]]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[[],[[[],[]],[]]],[[],[]]]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[],[]],[[],[]]],[[],[]]]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[[],[[],[]]],[]],[[],[]]]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[[],[]],[]],[]],[[],[]]]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[[[],[[],[[],[[],[]]]]],[]]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[[],[[],[[[],[]],[]]]],[]]
 => [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[[],[[[],[]],[[],[]]]],[]]
 => [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[],[[[],[[],[]]],[]]],[]]
 => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[[],[[[[],[]],[]],[]]],[]]
 => [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[],[]],[[],[[],[]]]],[]]
 => [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[[],[]],[[[],[]],[]]],[]]
 => [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[[],[[],[]]],[[],[]]],[]]
 => [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[[],[]],[]],[[],[]]],[]]
 => [1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[[],[[],[[],[]]]],[]],[]]
 => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
[[[[],[[[],[]],[]]],[]],[]]
 => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[[],[]],[[],[]]],[]],[]]
 => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ?
 => ? = 1 - 1
[[[[[],[[],[]]],[]],[]],[]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ?
 => ? = 1 - 1
[[[[[[],[]],[]],[]],[]],[]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
 => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
 => [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
 => [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => ?
 => ? = 1 - 1
[[],[[],[[[],[]],[[],[[],[]]]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,0,0]
 => ?
 => ? = 1 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000160
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => ([],1)
 => [1]
 => 1
[[]]
 => ([(0,1)],2)
 => ([],2)
 => [1,1]
 => 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 3
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
[[],[],[],[],[]]
 => ([(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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[[],[[],[[],[[],[]]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[],[[[],[]],[]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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),(1,6),(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
[[],[[[],[[],[]]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[[[],[]],[]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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
[[[],[[],[[],[]]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[],[[[],[]],[]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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),(1,6),(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
[[[[],[[],[]]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[[[],[]],[]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[],[[],[[],[[],[]]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[],[[[],[]],[]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[],[]],[[],[]]]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[],[[],[]]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[[],[]],[]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[]],[[],[[],[]]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[]],[[[],[]],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[],[]]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[]],[]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[],[[],[]]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[[],[]],[]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[]],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[[],[]]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[[],[]],[]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[],[[],[[],[]]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[],[[[],[]],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[],[]],[[],[]]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[[],[]],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[]]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[]]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[],[]]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[]],[]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[],[]]],[[],[]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[[],[]]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[[],[]],[]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[],[[],[]]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[[],[]],[]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[]],[[],[]]]],[]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[[],[]]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[[],[]],[]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[],[[],[]]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[[],[]],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[[],[]]],[[],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000475
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => ([],1)
 => [1]
 => 1
[[]]
 => ([(0,1)],2)
 => ([],2)
 => [1,1]
 => 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 3
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
[[],[],[],[],[]]
 => ([(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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[[],[[],[[],[[],[]]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[],[[[],[]],[]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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),(1,6),(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
[[],[[[],[[],[]]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[[[],[]],[]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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
[[[],[[],[[],[]]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[],[[[],[]],[]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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),(1,6),(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
[[[[],[[],[]]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[[[],[]],[]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[],[[],[[],[[],[]]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[],[[[],[]],[]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[],[]],[[],[]]]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[],[[],[]]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[[],[]],[]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[]],[[],[[],[]]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[]],[[[],[]],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[],[]]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[]],[]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[],[[],[]]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[[],[]],[]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[]],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[[],[]]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[[],[]],[]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[],[[],[[],[]]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[],[[[],[]],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[],[]],[[],[]]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[[],[]],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[]]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[]]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[],[]]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[]],[]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[],[]]],[[],[]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[[],[]]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[[],[]],[]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[],[[],[]]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[[],[]],[]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[]],[[],[]]]],[]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[[],[]]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[[],[]],[]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[],[[],[]]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[[],[]],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[[],[]]],[[],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001933
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => ([],1)
 => [1]
 => 1
[[]]
 => ([(0,1)],2)
 => ([],2)
 => [1,1]
 => 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 3
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
[[],[],[],[],[]]
 => ([(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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
[[],[[],[[],[[],[]]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[],[[[],[]],[]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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),(1,6),(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
[[],[[[],[[],[]]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[[[],[]],[]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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
[[[],[[],[[],[]]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[],[[[],[]],[]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(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),(1,6),(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
[[[[],[[],[]]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[[[[],[]],[]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
[[],[[],[[],[[],[[],[]]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[],[[[],[]],[]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[],[]],[[],[]]]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[],[[],[]]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[],[[[[],[]],[]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[]],[[],[[],[]]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[]],[[[],[]],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[],[]]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[]],[]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[],[[],[]]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[],[[[],[]],[]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[]],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[],[[],[]]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[],[[[[[],[]],[]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[],[[],[[],[]]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[],[[[],[]],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[],[]],[[],[]]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[]],[[[[],[]],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[]]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[]]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[],[]]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[]],[]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[],[]]],[[],[]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[[],[]]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[[],[]],[]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[],[[],[]]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[],[[[],[]],[]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[]],[[],[]]]],[]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[],[[],[]]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[],[[[[],[]],[]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[],[[],[]]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[]],[[[],[]],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
[[[[],[[],[]]],[[],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001091
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => ([],1)
 => [1]
 => 0 = 1 - 1
[[]]
 => ([(0,1)],2)
 => ([],2)
 => [1,1]
 => 1 = 2 - 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 0 = 1 - 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => 2 = 3 - 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 0 = 1 - 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 0 = 1 - 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 0 = 1 - 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 1 = 2 - 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 3 = 4 - 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 0 = 1 - 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1 = 2 - 1
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1 = 2 - 1
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1 = 2 - 1
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 2 = 3 - 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 4 = 5 - 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 0 = 1 - 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 0 = 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 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)
 => [5,1]
 => 0 = 1 - 1
[[],[[],[[],[[],[]]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
[[],[[],[[[],[]],[]]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 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 - 1
[[],[[[],[[],[]]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
[[],[[[[],[]],[]],[]]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 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 - 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 - 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 - 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 - 1
[[[],[[],[[],[]]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
[[[],[[[],[]],[]]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 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 - 1
[[[[],[[],[]]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
[[[[[],[]],[]],[]],[]]
 => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
 => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
[[],[[],[[],[[],[[],[]]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[],[[],[[[],[]],[]]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[],[[[],[]],[[],[]]]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[],[[[],[[],[]]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[],[[[[],[]],[]],[]]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[],[]],[[],[[],[]]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[],[]],[[[],[]],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[],[[],[]]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[[],[]],[]],[[],[]]]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[],[[],[[],[]]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[],[[[],[]],[]]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[[],[]],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[[],[[],[]]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[],[[[[[],[]],[]],[]],[]]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[]],[[],[[],[[],[]]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[]],[[],[[[],[]],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[]],[[[],[]],[[],[]]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[]],[[[],[[],[]]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[]],[[[[],[]],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[],[]]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[],[]]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[]],[]],[[],[[],[]]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[]],[]],[[[],[]],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
 => ([(1,2),(1,4),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[],[[],[]]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[[],[]],[]]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[]],[[],[]]],[[],[]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
 => ([(1,8),(1,9),(1,10),(2,3),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[[],[]]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[[],[]],[]],[]],[[],[]]]
 => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(1,7),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[],[[],[[],[]]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[],[[[],[]],[]]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[[],[]],[[],[]]]],[]]
 => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,4),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[[],[[],[]]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[],[[[[],[]],[]],[]]],[]]
 => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
 => ([(1,10),(2,9),(2,10),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[]],[[],[[],[]]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[]],[[[],[]],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
[[[[],[[],[]]],[[],[]]],[]]
 => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(1,10),(2,6),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ?
 => ? = 1 - 1
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000221
(load all 16 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => .
 => ? => ? = 1 - 1
[[]]
 => [.,.]
 => [1] => 1 = 2 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => 2 = 3 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 0 = 1 - 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => 0 = 1 - 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0 = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 0 = 1 - 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 0 = 1 - 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 0 = 1 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 0 = 1 - 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 0 = 1 - 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 0 = 1 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 0 = 1 - 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 0 = 1 - 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1 = 2 - 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1 = 2 - 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1 = 2 - 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 2 = 3 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 0 = 1 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => 0 = 1 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 0 = 1 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 0 = 1 - 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => 0 = 1 - 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => 0 = 1 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => 0 = 1 - 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => 0 = 1 - 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 0 = 1 - 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 0 = 1 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => 0 = 1 - 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 0 = 1 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 0 = 1 - 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => 0 = 1 - 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => 0 = 1 - 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 0 = 1 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => 0 = 1 - 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => 0 = 1 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 0 = 1 - 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 0 = 1 - 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => 0 = 1 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 0 = 1 - 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 0 = 1 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 0 = 1 - 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 0 = 1 - 1
[[],[[],[[],[[],[]]]]]
 => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => ? = 1 - 1
[[],[[],[[[],[]],[]]]]
 => [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [6,4,3,5,7,2,8,1] => ? = 1 - 1
[[],[[[],[]],[[],[]]]]
 => [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [6,5,7,3,2,4,8,1] => ? = 1 - 1
[[],[[[],[[],[]]],[]]]
 => [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [7,4,3,5,2,6,8,1] => ? = 1 - 1
[[],[[[[],[]],[]],[]]]
 => [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [7,5,3,2,4,6,8,1] => ? = 1 - 1
[[[],[]],[[],[[],[]]]]
 => [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [6,5,7,4,8,2,1,3] => ? = 1 - 1
[[[],[]],[[[],[]],[]]]
 => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [7,5,4,6,8,2,1,3] => ? = 1 - 1
[[[],[[],[]]],[[],[]]]
 => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
 => [7,6,8,3,2,4,1,5] => ? = 1 - 1
[[[[],[]],[]],[[],[]]]
 => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
 => [7,6,8,4,2,1,3,5] => ? = 1 - 1
[[[],[[],[[],[]]]],[]]
 => [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
 => [8,4,3,5,2,6,1,7] => ? = 1 - 1
[[[],[[[],[]],[]]],[]]
 => [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
 => [8,5,3,2,4,6,1,7] => ? = 1 - 1
[[[[],[]],[[],[]]],[]]
 => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
 => [8,5,4,6,2,1,3,7] => ? = 1 - 1
[[[[],[[],[]]],[]],[]]
 => [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
 => [8,6,3,2,4,1,5,7] => ? = 1 - 1
[[[[[],[]],[]],[]],[]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
 => [8,6,4,2,1,3,5,7] => ? = 1 - 1
[[],[[],[[],[[],[[],[]]]]]]
 => [.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
 => [6,5,7,4,8,3,9,2,10,1] => ? = 1 - 1
[[],[[],[[],[[[],[]],[]]]]]
 => [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
 => [7,5,4,6,8,3,9,2,10,1] => ? = 1 - 1
[[],[[],[[[],[]],[[],[]]]]]
 => [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
 => [7,6,8,4,3,5,9,2,10,1] => ? = 1 - 1
[[],[[],[[[],[[],[]]],[]]]]
 => [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
 => [8,5,4,6,3,7,9,2,10,1] => ? = 1 - 1
[[],[[],[[[[],[]],[]],[]]]]
 => [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
 => [8,6,4,3,5,7,9,2,10,1] => ? = 1 - 1
[[],[[[],[]],[[],[[],[]]]]]
 => [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
 => [7,6,8,5,9,3,2,4,10,1] => ? = 1 - 1
[[],[[[],[]],[[[],[]],[]]]]
 => [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
 => [8,6,5,7,9,3,2,4,10,1] => ? = 1 - 1
[[],[[[],[[],[]]],[[],[]]]]
 => [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
 => [8,7,9,4,3,5,2,6,10,1] => ? = 1 - 1
[[],[[[[],[]],[]],[[],[]]]]
 => [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
 => [8,7,9,5,3,2,4,6,10,1] => ? = 1 - 1
[[],[[[],[[],[[],[]]]],[]]]
 => [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
 => [9,5,4,6,3,7,2,8,10,1] => ? = 1 - 1
[[],[[[],[[[],[]],[]]],[]]]
 => [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
 => [9,6,4,3,5,7,2,8,10,1] => ? = 1 - 1
[[],[[[[],[]],[[],[]]],[]]]
 => [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
 => [9,6,5,7,3,2,4,8,10,1] => ? = 1 - 1
[[],[[[[],[[],[]]],[]],[]]]
 => [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
 => [9,7,4,3,5,2,6,8,10,1] => ? = 1 - 1
[[],[[[[[],[]],[]],[]],[]]]
 => [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
 => [9,7,5,3,2,4,6,8,10,1] => ? = 1 - 1
[[[],[]],[[],[[],[[],[]]]]]
 => [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [7,6,8,5,9,4,10,2,1,3] => ? = 1 - 1
[[[],[]],[[],[[[],[]],[]]]]
 => [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [8,6,5,7,9,4,10,2,1,3] => ? = 1 - 1
[[[],[]],[[[],[]],[[],[]]]]
 => [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [8,7,9,5,4,6,10,2,1,3] => ? = 1 - 1
[[[],[]],[[[],[[],[]]],[]]]
 => [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [9,6,5,7,4,8,10,2,1,3] => ? = 1 - 1
[[[],[]],[[[[],[]],[]],[]]]
 => [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [9,7,5,4,6,8,10,2,1,3] => ? = 1 - 1
[[[],[[],[]]],[[],[[],[]]]]
 => [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
 => [8,7,9,6,10,3,2,4,1,5] => ? = 1 - 1
[[[],[[],[]]],[[[],[]],[]]]
 => [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
 => [9,7,6,8,10,3,2,4,1,5] => ? = 1 - 1
[[[[],[]],[]],[[],[[],[]]]]
 => [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [8,7,9,6,10,4,2,1,3,5] => ? = 1 - 1
[[[[],[]],[]],[[[],[]],[]]]
 => [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [9,7,6,8,10,4,2,1,3,5] => ? = 1 - 1
[[[],[[],[[],[]]]],[[],[]]]
 => [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
 => [9,8,10,4,3,5,2,6,1,7] => ? = 1 - 1
[[[],[[[],[]],[]]],[[],[]]]
 => [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
 => [9,8,10,5,3,2,4,6,1,7] => ? = 1 - 1
[[[[],[]],[[],[]]],[[],[]]]
 => [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
 => [9,8,10,5,4,6,2,1,3,7] => ? = 1 - 1
[[[[],[[],[]]],[]],[[],[]]]
 => [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
 => [9,8,10,6,3,2,4,1,5,7] => ? = 1 - 1
[[[[[],[]],[]],[]],[[],[]]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
 => [9,8,10,6,4,2,1,3,5,7] => ? = 1 - 1
[[[],[[],[[],[[],[]]]]],[]]
 => [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
 => [10,5,4,6,3,7,2,8,1,9] => ? = 1 - 1
[[[],[[],[[[],[]],[]]]],[]]
 => [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
 => [10,6,4,3,5,7,2,8,1,9] => ? = 1 - 1
[[[],[[[],[]],[[],[]]]],[]]
 => [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
 => [10,6,5,7,3,2,4,8,1,9] => ? = 1 - 1
[[[],[[[],[[],[]]],[]]],[]]
 => [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
 => [10,7,4,3,5,2,6,8,1,9] => ? = 1 - 1
[[[],[[[[],[]],[]],[]]],[]]
 => [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
 => [10,7,5,3,2,4,6,8,1,9] => ? = 1 - 1
[[[[],[]],[[],[[],[]]]],[]]
 => [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
 => [10,6,5,7,4,8,2,1,3,9] => ? = 1 - 1
[[[[],[]],[[[],[]],[]]],[]]
 => [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
 => [10,7,5,4,6,8,2,1,3,9] => ? = 1 - 1
Description
The number of strong fixed points of a permutation. $i$ is called a strong fixed point of $\pi$ if 1. $j < i$ implies $\pi_j < \pi_i$, and 2. $j > i$ implies $\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St000315
(load all 4 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000315: Graphs ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => .
 => ? => ?
 => ? = 1 - 1
[[]]
 => [.,.]
 => [1] => ([],1)
 => 1 = 2 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 2 = 3 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 3 = 4 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 1 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 2 = 3 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 4 = 5 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[],[[],[[],[]]]]]
 => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => ([(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)
 => ? = 1 - 1
[[],[[],[[[],[]],[]]]]
 => [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [6,4,3,5,7,2,8,1] => ([(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)
 => ? = 1 - 1
[[],[[[],[]],[[],[]]]]
 => [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [6,5,7,3,2,4,8,1] => ([(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)
 => ? = 1 - 1
[[],[[[],[[],[]]],[]]]
 => [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [7,4,3,5,2,6,8,1] => ([(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)
 => ? = 1 - 1
[[],[[[[],[]],[]],[]]]
 => [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [7,5,3,2,4,6,8,1] => ([(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)
 => ? = 1 - 1
[[[],[]],[[],[[],[]]]]
 => [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [6,5,7,4,8,2,1,3] => ([(0,4),(0,6),(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),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[[[],[]],[[[],[]],[]]]
 => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [7,5,4,6,8,2,1,3] => ?
 => ? = 1 - 1
[[[],[[],[]]],[[],[]]]
 => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
 => [7,6,8,3,2,4,1,5] => ?
 => ? = 1 - 1
[[[[],[]],[]],[[],[]]]
 => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
 => [7,6,8,4,2,1,3,5] => ?
 => ? = 1 - 1
[[[],[[],[[],[]]]],[]]
 => [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
 => [8,4,3,5,2,6,1,7] => ([(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)
 => ? = 1 - 1
[[[],[[[],[]],[]]],[]]
 => [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
 => [8,5,3,2,4,6,1,7] => ([(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)
 => ? = 1 - 1
[[[[],[]],[[],[]]],[]]
 => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
 => [8,5,4,6,2,1,3,7] => ?
 => ? = 1 - 1
[[[[],[[],[]]],[]],[]]
 => [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
 => [8,6,3,2,4,1,5,7] => ([(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)
 => ? = 1 - 1
[[[[[],[]],[]],[]],[]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
 => [8,6,4,2,1,3,5,7] => ([(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)
 => ? = 1 - 1
[[],[[],[[],[[],[[],[]]]]]]
 => [.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
 => [6,5,7,4,8,3,9,2,10,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1 - 1
[[],[[],[[],[[[],[]],[]]]]]
 => [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
 => [7,5,4,6,8,3,9,2,10,1] => ?
 => ? = 1 - 1
[[],[[],[[[],[]],[[],[]]]]]
 => [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
 => [7,6,8,4,3,5,9,2,10,1] => ?
 => ? = 1 - 1
[[],[[],[[[],[[],[]]],[]]]]
 => [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
 => [8,5,4,6,3,7,9,2,10,1] => ?
 => ? = 1 - 1
[[],[[],[[[[],[]],[]],[]]]]
 => [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
 => [8,6,4,3,5,7,9,2,10,1] => ?
 => ? = 1 - 1
[[],[[[],[]],[[],[[],[]]]]]
 => [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
 => [7,6,8,5,9,3,2,4,10,1] => ?
 => ? = 1 - 1
[[],[[[],[]],[[[],[]],[]]]]
 => [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
 => [8,6,5,7,9,3,2,4,10,1] => ?
 => ? = 1 - 1
[[],[[[],[[],[]]],[[],[]]]]
 => [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
 => [8,7,9,4,3,5,2,6,10,1] => ?
 => ? = 1 - 1
[[],[[[[],[]],[]],[[],[]]]]
 => [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
 => [8,7,9,5,3,2,4,6,10,1] => ?
 => ? = 1 - 1
[[],[[[],[[],[[],[]]]],[]]]
 => [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
 => [9,5,4,6,3,7,2,8,10,1] => ?
 => ? = 1 - 1
[[],[[[],[[[],[]],[]]],[]]]
 => [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
 => [9,6,4,3,5,7,2,8,10,1] => ?
 => ? = 1 - 1
[[],[[[[],[]],[[],[]]],[]]]
 => [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
 => [9,6,5,7,3,2,4,8,10,1] => ?
 => ? = 1 - 1
[[],[[[[],[[],[]]],[]],[]]]
 => [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
 => [9,7,4,3,5,2,6,8,10,1] => ?
 => ? = 1 - 1
[[],[[[[[],[]],[]],[]],[]]]
 => [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
 => [9,7,5,3,2,4,6,8,10,1] => ?
 => ? = 1 - 1
[[[],[]],[[],[[],[[],[]]]]]
 => [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [7,6,8,5,9,4,10,2,1,3] => ?
 => ? = 1 - 1
[[[],[]],[[],[[[],[]],[]]]]
 => [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [8,6,5,7,9,4,10,2,1,3] => ?
 => ? = 1 - 1
[[[],[]],[[[],[]],[[],[]]]]
 => [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [8,7,9,5,4,6,10,2,1,3] => ?
 => ? = 1 - 1
[[[],[]],[[[],[[],[]]],[]]]
 => [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [9,6,5,7,4,8,10,2,1,3] => ?
 => ? = 1 - 1
[[[],[]],[[[[],[]],[]],[]]]
 => [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [9,7,5,4,6,8,10,2,1,3] => ?
 => ? = 1 - 1
[[[],[[],[]]],[[],[[],[]]]]
 => [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
 => [8,7,9,6,10,3,2,4,1,5] => ?
 => ? = 1 - 1
[[[],[[],[]]],[[[],[]],[]]]
 => [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
 => [9,7,6,8,10,3,2,4,1,5] => ?
 => ? = 1 - 1
[[[[],[]],[]],[[],[[],[]]]]
 => [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [8,7,9,6,10,4,2,1,3,5] => ?
 => ? = 1 - 1
[[[[],[]],[]],[[[],[]],[]]]
 => [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [9,7,6,8,10,4,2,1,3,5] => ?
 => ? = 1 - 1
[[[],[[],[[],[]]]],[[],[]]]
 => [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
 => [9,8,10,4,3,5,2,6,1,7] => ?
 => ? = 1 - 1
[[[],[[[],[]],[]]],[[],[]]]
 => [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
 => [9,8,10,5,3,2,4,6,1,7] => ?
 => ? = 1 - 1
[[[[],[]],[[],[]]],[[],[]]]
 => [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
 => [9,8,10,5,4,6,2,1,3,7] => ?
 => ? = 1 - 1
[[[[],[[],[]]],[]],[[],[]]]
 => [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
 => [9,8,10,6,3,2,4,1,5,7] => ?
 => ? = 1 - 1
[[[[[],[]],[]],[]],[[],[]]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
 => [9,8,10,6,4,2,1,3,5,7] => ?
 => ? = 1 - 1
[[[],[[],[[],[[],[]]]]],[]]
 => [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
 => [10,5,4,6,3,7,2,8,1,9] => ?
 => ? = 1 - 1
[[[],[[],[[[],[]],[]]]],[]]
 => [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
 => [10,6,4,3,5,7,2,8,1,9] => ?
 => ? = 1 - 1
[[[],[[[],[]],[[],[]]]],[]]
 => [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
 => [10,6,5,7,3,2,4,8,1,9] => ?
 => ? = 1 - 1
[[[],[[[],[[],[]]],[]]],[]]
 => [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
 => [10,7,4,3,5,2,6,8,1,9] => ?
 => ? = 1 - 1
[[[],[[[[],[]],[]],[]]],[]]
 => [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
 => [10,7,5,3,2,4,6,8,1,9] => ?
 => ? = 1 - 1
[[[[],[]],[[],[[],[]]]],[]]
 => [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
 => [10,6,5,7,4,8,2,1,3,9] => ?
 => ? = 1 - 1
[[[[],[]],[[[],[]],[]]],[]]
 => [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
 => [10,7,5,4,6,8,2,1,3,9] => ?
 => ? = 1 - 1
Description
The number of isolated vertices of a graph.
Matching statistic: St000461
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[]
 => .
 => ? => ? => ? = 1 - 1
[[]]
 => [.,.]
 => [1] => [1] => ? = 2 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => [1,2] => 2 = 3 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 0 = 1 - 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => [3,2,1] => 0 = 1 - 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,3,2,1] => 0 = 1 - 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,2,3,1] => 0 = 1 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => 0 = 1 - 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,3,2,1] => 0 = 1 - 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,3,2,1] => 0 = 1 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [4,3,2,1] => 0 = 1 - 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,2,3,1] => 0 = 1 - 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,2,1,4] => 1 = 2 - 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,4,3,2,1] => 0 = 1 - 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,2,4,1] => 0 = 1 - 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,3,2,4,1] => 0 = 1 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,3,2,4,1] => 0 = 1 - 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,2,3,4,1] => 0 = 1 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => 0 = 1 - 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,4,3,2,1] => 0 = 1 - 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,4,3,2,1] => 0 = 1 - 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,4,3,2,1] => 0 = 1 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,3,2,1] => 0 = 1 - 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,4,3,2,1] => 0 = 1 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,4,3,2,1] => 0 = 1 - 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [5,4,3,2,1] => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,4,3,2,1] => 0 = 1 - 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,4,3,2,1] => 0 = 1 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [5,4,3,2,1] => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [5,4,3,2,1] => 0 = 1 - 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [5,4,3,2,1] => 0 = 1 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [5,3,2,4,1] => 0 = 1 - 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,2,3,4,1] => 0 = 1 - 1
[[[],[],[],[]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[[],[[],[[],[[],[]]]]]
 => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => [8,6,4,3,5,2,7,1] => ? = 1 - 1
[[],[[],[[[],[]],[]]]]
 => [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [6,4,3,5,7,2,8,1] => [8,6,4,3,5,2,7,1] => ? = 1 - 1
[[],[[[],[]],[[],[]]]]
 => [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [6,5,7,3,2,4,8,1] => [8,6,5,4,3,2,7,1] => ? = 1 - 1
[[],[[[],[[],[]]],[]]]
 => [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [7,4,3,5,2,6,8,1] => [8,6,5,4,3,2,7,1] => ? = 1 - 1
[[],[[[[],[]],[]],[]]]
 => [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [7,5,3,2,4,6,8,1] => [8,6,5,4,3,2,7,1] => ? = 1 - 1
[[[],[]],[[],[[],[]]]]
 => [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [6,5,7,4,8,2,1,3] => [8,7,6,4,5,3,2,1] => ? = 1 - 1
[[[],[]],[[[],[]],[]]]
 => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [7,5,4,6,8,2,1,3] => [8,7,6,4,5,3,2,1] => ? = 1 - 1
[[[],[[],[]]],[[],[]]]
 => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
 => [7,6,8,3,2,4,1,5] => ? => ? = 1 - 1
[[[[],[]],[]],[[],[]]]
 => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
 => [7,6,8,4,2,1,3,5] => ? => ? = 1 - 1
[[[],[[],[[],[]]]],[]]
 => [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
 => [8,4,3,5,2,6,1,7] => [8,7,5,4,3,6,2,1] => ? = 1 - 1
[[[],[[[],[]],[]]],[]]
 => [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
 => [8,5,3,2,4,6,1,7] => [8,7,5,4,3,6,2,1] => ? = 1 - 1
[[[[],[]],[[],[]]],[]]
 => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
 => [8,5,4,6,2,1,3,7] => ? => ? = 1 - 1
[[[[],[[],[]]],[]],[]]
 => [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
 => [8,6,3,2,4,1,5,7] => [8,7,6,4,5,3,2,1] => ? = 1 - 1
[[[[[],[]],[]],[]],[]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
 => [8,6,4,2,1,3,5,7] => [8,7,6,5,4,3,2,1] => ? = 1 - 1
[[],[[],[[],[[],[[],[]]]]]]
 => [.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
 => [6,5,7,4,8,3,9,2,10,1] => [10,8,6,4,5,3,7,2,9,1] => ? = 1 - 1
[[],[[],[[],[[[],[]],[]]]]]
 => [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
 => [7,5,4,6,8,3,9,2,10,1] => ? => ? = 1 - 1
[[],[[],[[[],[]],[[],[]]]]]
 => [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
 => [7,6,8,4,3,5,9,2,10,1] => ? => ? = 1 - 1
[[],[[],[[[],[[],[]]],[]]]]
 => [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
 => [8,5,4,6,3,7,9,2,10,1] => ? => ? = 1 - 1
[[],[[],[[[[],[]],[]],[]]]]
 => [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
 => [8,6,4,3,5,7,9,2,10,1] => ? => ? = 1 - 1
[[],[[[],[]],[[],[[],[]]]]]
 => [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
 => [7,6,8,5,9,3,2,4,10,1] => ? => ? = 1 - 1
[[],[[[],[]],[[[],[]],[]]]]
 => [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
 => [8,6,5,7,9,3,2,4,10,1] => ? => ? = 1 - 1
[[],[[[],[[],[]]],[[],[]]]]
 => [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
 => [8,7,9,4,3,5,2,6,10,1] => ? => ? = 1 - 1
[[],[[[[],[]],[]],[[],[]]]]
 => [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
 => [8,7,9,5,3,2,4,6,10,1] => ? => ? = 1 - 1
[[],[[[],[[],[[],[]]]],[]]]
 => [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
 => [9,5,4,6,3,7,2,8,10,1] => ? => ? = 1 - 1
[[],[[[],[[[],[]],[]]],[]]]
 => [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
 => [9,6,4,3,5,7,2,8,10,1] => ? => ? = 1 - 1
[[],[[[[],[]],[[],[]]],[]]]
 => [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
 => [9,6,5,7,3,2,4,8,10,1] => ? => ? = 1 - 1
[[],[[[[],[[],[]]],[]],[]]]
 => [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
 => [9,7,4,3,5,2,6,8,10,1] => ? => ? = 1 - 1
[[],[[[[[],[]],[]],[]],[]]]
 => [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
 => [9,7,5,3,2,4,6,8,10,1] => ? => ? = 1 - 1
[[[],[]],[[],[[],[[],[]]]]]
 => [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [7,6,8,5,9,4,10,2,1,3] => ? => ? = 1 - 1
[[[],[]],[[],[[[],[]],[]]]]
 => [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [8,6,5,7,9,4,10,2,1,3] => ? => ? = 1 - 1
[[[],[]],[[[],[]],[[],[]]]]
 => [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
 => [8,7,9,5,4,6,10,2,1,3] => ? => ? = 1 - 1
[[[],[]],[[[],[[],[]]],[]]]
 => [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
 => [9,6,5,7,4,8,10,2,1,3] => ? => ? = 1 - 1
[[[],[]],[[[[],[]],[]],[]]]
 => [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
 => [9,7,5,4,6,8,10,2,1,3] => ? => ? = 1 - 1
[[[],[[],[]]],[[],[[],[]]]]
 => [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
 => [8,7,9,6,10,3,2,4,1,5] => ? => ? = 1 - 1
[[[],[[],[]]],[[[],[]],[]]]
 => [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
 => [9,7,6,8,10,3,2,4,1,5] => ? => ? = 1 - 1
[[[[],[]],[]],[[],[[],[]]]]
 => [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
 => [8,7,9,6,10,4,2,1,3,5] => ? => ? = 1 - 1
[[[[],[]],[]],[[[],[]],[]]]
 => [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
 => [9,7,6,8,10,4,2,1,3,5] => ? => ? = 1 - 1
[[[],[[],[[],[]]]],[[],[]]]
 => [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
 => [9,8,10,4,3,5,2,6,1,7] => ? => ? = 1 - 1
[[[],[[[],[]],[]]],[[],[]]]
 => [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
 => [9,8,10,5,3,2,4,6,1,7] => ? => ? = 1 - 1
[[[[],[]],[[],[]]],[[],[]]]
 => [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
 => [9,8,10,5,4,6,2,1,3,7] => ? => ? = 1 - 1
[[[[],[[],[]]],[]],[[],[]]]
 => [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
 => [9,8,10,6,3,2,4,1,5,7] => ? => ? = 1 - 1
[[[[[],[]],[]],[]],[[],[]]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
 => [9,8,10,6,4,2,1,3,5,7] => ? => ? = 1 - 1
[[[],[[],[[],[[],[]]]]],[]]
 => [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
 => [10,5,4,6,3,7,2,8,1,9] => ? => ? = 1 - 1
[[[],[[],[[[],[]],[]]]],[]]
 => [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
 => [10,6,4,3,5,7,2,8,1,9] => ? => ? = 1 - 1
[[[],[[[],[]],[[],[]]]],[]]
 => [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
 => [10,6,5,7,3,2,4,8,1,9] => ? => ? = 1 - 1
[[[],[[[],[[],[]]],[]]],[]]
 => [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
 => [10,7,4,3,5,2,6,8,1,9] => ? => ? = 1 - 1
[[[],[[[[],[]],[]],[]]],[]]
 => [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
 => [10,7,5,3,2,4,6,8,1,9] => ? => ? = 1 - 1
[[[[],[]],[[],[[],[]]]],[]]
 => [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
 => [10,6,5,7,4,8,2,1,3,9] => ? => ? = 1 - 1
Description
The rix statistic of a permutation. This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then $rix(w) := 0$ if $i = 1 < k$, $rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and $rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000674The number of hills of a Dyck path. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000993The multiplicity of the largest part of an integer partition. St000264The girth of a graph, which is not a tree. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001479The number of bridges of a graph.