Your data matches 98 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001581
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
[[]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 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)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[],[],[],[[]]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[]],[]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[]],[],[]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[],[],[]]]
 => ([(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)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[[]],[],[],[]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[],[],[]],[]]
 => ([(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)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
Description
The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St001670
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001670: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
[[]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 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)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[],[],[],[[]]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[]],[]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[]],[],[]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[],[],[]]]
 => ([(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)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[[]],[],[],[]]
 => ([(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)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[],[],[]],[]]
 => ([(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)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
Description
The connected partition number of a graph. This is the maximal number of blocks of a set partition $P$ of the set of vertices of a graph such that contracting each block of $P$ to a single vertex yields a clique. Also called the pseudoachromatic number of a graph. This is the largest $n$ such that there exists a (not necessarily proper) $n$-coloring of the graph so that every two distinct colors are adjacent.
Matching statistic: St000521
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
 => []
 => [1,0]
 => [[]]
 => 2 = 1 + 1
[[]]
 => [1,0]
 => [1,1,0,0]
 => [[[]]]
 => 3 = 2 + 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 3 = 2 + 1
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 2 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 4 = 3 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 4 = 3 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 3 = 2 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => 4 = 3 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => 4 = 3 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => 4 = 3 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => 5 = 4 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => 4 = 3 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[[]],[[]]]]
 => 4 = 3 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => 4 = 3 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[[]]],[]]]
 => 5 = 4 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => 4 = 3 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[[],[[]]]]]
 => 5 = 4 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[[]],[]]]]
 => 5 = 4 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => 5 = 4 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => 6 = 5 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[]]]
 => 3 = 2 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[[],[],[],[[]]]]
 => 4 = 3 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[[],[],[[]],[]]]
 => 4 = 3 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[[],[],[[],[]]]]
 => 4 = 3 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[[],[],[[[]]]]]
 => 5 = 4 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[[],[[]],[],[]]]
 => 4 = 3 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[[],[[]],[[]]]]
 => 4 = 3 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[[],[[],[]],[]]]
 => 4 = 3 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[[],[[[]]],[]]]
 => 5 = 4 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[[],[[],[],[]]]]
 => 4 = 3 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[[],[[],[[]]]]]
 => 5 = 4 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[[],[[[]],[]]]]
 => 5 = 4 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[[],[[[],[]]]]]
 => 5 = 4 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[[],[[[[]]]]]]
 => 6 = 5 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[[[]],[],[],[]]]
 => 4 = 3 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[[[]],[],[[]]]]
 => 4 = 3 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[[[]],[[]],[]]]
 => 4 = 3 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[[[]],[[],[]]]]
 => 5 = 4 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[[[]],[[[]]]]]
 => 5 = 4 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[[[],[]],[],[]]]
 => 4 = 3 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[[[[]]],[],[]]]
 => 5 = 4 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[[[],[]],[[]]]]
 => 5 = 4 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [[[[[]]],[[]]]]
 => 5 = 4 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[[[],[],[]],[]]]
 => 4 = 3 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[[[],[[]]],[]]]
 => 5 = 4 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [[[[[]],[]],[]]]
 => 5 = 4 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[[[[],[]]],[]]]
 => 5 = 4 + 1
Description
The number of distinct subtrees of an ordered tree. A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree $[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.
Matching statistic: St000062
(load all 7 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => .
 => ? => ? => ? = 1 - 1
[[]]
 => [.,.]
 => [1] => [1] => 1 = 2 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 2 = 3 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => [2,1] => 1 = 2 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 3 = 4 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 2 = 3 - 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [1,3,2] => [2,1,3] => 2 = 3 - 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => [3,2,1] => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [2,3,1] => 2 = 3 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 4 = 5 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,2,3] => 3 = 4 - 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,1,2,4] => 3 = 4 - 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => 2 = 3 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => 2 = 3 - 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,3,4] => 3 = 4 - 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,4,1,3] => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [3,2,1,4] => 2 = 3 - 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,3,1,4] => 3 = 4 - 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,3,2,1] => 1 = 2 - 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,4,2,1] => 2 = 3 - 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,2,4,1] => 2 = 3 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 5 = 6 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,3,4] => 4 = 5 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [4,1,2,3,5] => 4 = 5 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,2,3] => 3 = 4 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,2,3] => 3 = 4 - 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,1,2,4,5] => 4 = 5 - 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,5,1,2,4] => 3 = 4 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [4,3,1,2,5] => 3 = 4 - 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,4,1,2,5] => 3 = 4 - 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => 2 = 3 - 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => 2 = 3 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [3,5,4,1,2] => 2 = 3 - 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => 2 = 3 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => 3 = 4 - 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,1,3,4,5] => 4 = 5 - 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [2,5,1,3,4] => 3 = 4 - 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [2,4,1,3,5] => 3 = 4 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [2,5,4,1,3] => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,4,5,1,3] => 3 = 4 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,2,1,4,5] => 3 = 4 - 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [2,3,1,4,5] => 4 = 5 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2,5,1,4] => 2 = 3 - 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [2,3,5,1,4] => 3 = 4 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [4,3,2,1,5] => 2 = 3 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,4,2,1,5] => 3 = 4 - 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,4,3,1,5] => 3 = 4 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [3,2,4,1,5] => 3 = 4 - 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,3,4,1,5] => 4 = 5 - 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000213
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => []
 => []
 => [] => ? = 1 - 1
[[]]
 => [1,0]
 => [1,0]
 => [1] => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,2] => 2 = 3 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 2 = 3 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3 = 4 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2 = 3 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3 = 4 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 3 = 4 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 3 = 4 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 3 = 4 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 3 = 4 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3 = 4 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2 = 3 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 3 = 4 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4 = 5 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3 = 4 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3 = 4 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4 = 5 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 3 = 4 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 3 = 4 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 3 = 4 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 3 = 4 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3 = 4 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2 = 3 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 3 = 4 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4 = 5 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 3 = 4 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4 = 5 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1 = 2 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as $$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000314
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => []
 => []
 => [] => ? = 1 - 1
[[]]
 => [1,0]
 => [1,0]
 => [1] => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,2] => 2 = 3 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,1,3] => 2 = 3 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 3 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2 = 3 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 3 = 4 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2 = 3 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2 = 3 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 3 = 4 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 3 = 4 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 3 = 4 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 3 = 4 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 3 = 4 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 3 = 4 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 2 = 3 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 3 = 4 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 4 = 5 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 2 = 3 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 3 = 4 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 3 = 4 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4 = 5 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 3 = 4 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 3 = 4 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 3 = 4 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 3 = 4 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3 = 4 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 2 = 3 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 3 = 4 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 4 = 5 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 3 = 4 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 4 = 5 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1 = 2 - 1
Description
The number of left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$. This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000971
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => .
 => ? => ?
 => ? = 1 - 1
[[]]
 => [.,.]
 => [1] => {{1}}
 => 1 = 2 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => {{1,2}}
 => 2 = 3 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => {{1},{2}}
 => 1 = 2 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => {{1,3},{2}}
 => 2 = 3 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => {{1,2,3}}
 => 3 = 4 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => {{1,2},{3}}
 => 2 = 3 - 1
[[[],[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => {{1,3},{2}}
 => 2 = 3 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => {{1},{2},{3}}
 => 1 = 2 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => {{1,4},{2,3}}
 => 3 = 4 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => {{1,3},{2,4}}
 => 3 = 4 - 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => {{1,3,4},{2}}
 => 2 = 3 - 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => {{1,4},{2},{3}}
 => 2 = 3 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => {{1,2,3,4}}
 => 4 = 5 - 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => {{1,3},{2},{4}}
 => 2 = 3 - 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 3 = 4 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => {{1,4},{2},{3}}
 => 2 = 3 - 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2 = 3 - 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => {{1,4},{2,3}}
 => 3 = 4 - 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 3 = 4 - 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 2 = 3 - 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 2 = 3 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 1 = 2 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 3 = 4 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => {{1,4},{2,5},{3}}
 => 3 = 4 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => {{1,4},{2,3,5}}
 => 4 = 5 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => {{1,5},{2,3,4}}
 => 4 = 5 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => {{1,3,5},{2,4}}
 => 4 = 5 - 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => {{1,4,5},{2,3}}
 => 3 = 4 - 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => {{1,3},{2,4,5}}
 => 3 = 4 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => {{1,3,4,5},{2}}
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => {{1,5},{2,4},{3}}
 => 3 = 4 - 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => {{1,4},{2,5},{3}}
 => 3 = 4 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => {{1,4,5},{2},{3}}
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 5 = 6 - 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => {{1,4},{2,3},{5}}
 => 3 = 4 - 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => {{1,3},{2,4},{5}}
 => 3 = 4 - 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => {{1,3,4},{2},{5}}
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => 4 = 5 - 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => 4 = 5 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => {{1,5},{2,4},{3}}
 => 3 = 4 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => {{1,4},{2,5},{3}}
 => 3 = 4 - 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 3 = 4 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 2 = 3 - 1
Description
The smallest closer of a set partition. A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers. In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St001497
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => []
 => []
 => [] => ? = 1 - 1
[[]]
 => [1,0]
 => [1,0]
 => [1] => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,2] => 2 = 3 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2 = 3 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 2 = 3 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3 = 4 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 3 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2 = 3 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3 = 4 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 3 = 4 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 3 = 4 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 3 = 4 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 3 = 4 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3 = 4 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2 = 3 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 3 = 4 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4 = 5 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3 = 4 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3 = 4 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4 = 5 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 3 = 4 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 3 = 4 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 3 = 4 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 3 = 4 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3 = 4 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2 = 3 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 3 = 4 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4 = 5 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 3 = 4 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4 = 5 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1 = 2 - 1
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000333
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000333: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => []
 => [] => [] => ? = 1 - 2
[[]]
 => [1,0]
 => [1] => [1] => 0 = 2 - 2
[[],[]]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0 = 2 - 2
[[[]]]
 => [1,1,0,0]
 => [2,1] => [2,1] => 1 = 3 - 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0 = 2 - 2
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1 = 3 - 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2 = 4 - 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 1 = 3 - 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [3,2,1] => 1 = 3 - 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 1 = 3 - 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 1 = 3 - 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,3,2] => 1 = 3 - 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2 = 4 - 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => 2 = 4 - 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,2,1,4] => 2 = 4 - 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => 1 = 3 - 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,4,1,2] => 1 = 3 - 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,2,3,1] => 1 = 3 - 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,3,2,1] => 3 = 5 - 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,2,3,1] => 1 = 3 - 2
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 2 - 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 3 - 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 4 - 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => 1 = 3 - 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,5,4,3] => 1 = 3 - 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 4 - 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 4 - 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => 2 = 4 - 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,3,2,5] => 2 = 4 - 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => 1 = 3 - 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,5,2,3] => 1 = 3 - 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,5,3,4,2] => 1 = 3 - 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,5,4,3,2] => 3 = 5 - 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,5,3,4,2] => 1 = 3 - 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 4 - 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 5 - 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 3 = 5 - 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 2 = 4 - 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => 2 = 4 - 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => 2 = 4 - 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,2,1,4,5] => 2 = 4 - 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => 2 = 4 - 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [3,2,1,5,4] => 2 = 4 - 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => 2 = 4 - 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [3,4,1,2,5] => 2 = 4 - 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [4,2,3,1,5] => 2 = 4 - 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => 4 = 6 - 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [4,2,3,1,5] => 2 = 4 - 2
Description
The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
Matching statistic: St000337
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[]
 => .
 => ? => ? => ? = 1 - 2
[[]]
 => [.,.]
 => [1] => [1] => 0 = 2 - 2
[[],[]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 0 = 2 - 2
[[[]]]
 => [[.,.],.]
 => [1,2] => [2,1] => 1 = 3 - 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [2,1,3] => 1 = 3 - 2
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 0 = 2 - 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 1 = 3 - 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => 2 = 4 - 2
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [2,3,1] => 1 = 3 - 2
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1,4] => 1 = 3 - 2
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,1,2,4] => 2 = 4 - 2
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,1,3,4] => 1 = 3 - 2
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,1,4] => 1 = 3 - 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 0 = 2 - 2
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,1,4,3] => 2 = 4 - 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,4,3] => 1 = 3 - 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => 2 = 4 - 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => 1 = 3 - 2
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,2,1,3] => 1 = 3 - 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,2,3] => 3 = 5 - 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,4,1,2] => 2 = 4 - 2
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,4,2] => 2 = 4 - 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => 1 = 3 - 2
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1,5] => 2 = 4 - 2
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,1,2,5] => 2 = 4 - 2
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,2,1,3,5] => 1 = 3 - 2
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [4,2,3,1,5] => 2 = 4 - 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,1,2,3,5] => 3 = 5 - 2
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2,1,4,5] => 1 = 3 - 2
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,1,2,4,5] => 2 = 4 - 2
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,1,5] => 2 = 4 - 2
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,1,3,4,5] => 1 = 3 - 2
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [3,4,2,1,5] => 1 = 3 - 2
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [3,4,1,2,5] => 2 = 4 - 2
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,1,5] => 1 = 3 - 2
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,3,1,4,5] => 1 = 3 - 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0 = 2 - 2
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [3,2,1,5,4] => 2 = 4 - 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,1,2,5,4] => 3 = 5 - 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [2,1,3,5,4] => 2 = 4 - 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,2,3,1,4] => 2 = 4 - 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [1,2,3,5,4] => 1 = 3 - 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,5,2,1,4] => 1 = 3 - 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,1,4,5,3] => 2 = 4 - 2
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,5,1,2,4] => 3 = 5 - 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [1,2,4,5,3] => 1 = 3 - 2
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,5,1,4] => 3 = 5 - 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,5,1,4] => 2 = 4 - 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,4,5,1,3] => 2 = 4 - 2
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [2,4,1,5,3] => 2 = 4 - 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [1,3,4,5,2] => 1 = 3 - 2
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000702The number of weak deficiencies of a permutation. St000354The number of recoils of a permutation. St000392The length of the longest run of ones in a binary word. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000264The girth of a graph, which is not a tree. St000381The largest part of an integer composition. St001621The number of atoms of a lattice. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000299The number of nonisomorphic vertex-induced subtrees. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000983The length of the longest alternating subword. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001712The number of natural descents of a standard Young tableau. St001960The number of descents of a permutation minus one if its first entry is not one. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000932The number of occurrences of the pattern UDU in a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001626The number of maximal proper sublattices of a lattice. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000366The number of double descents of a permutation. St000871The number of very big ascents of a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000824The sum of the number of descents and the number of recoils of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000254The nesting number of a set partition. St000308The height of the tree associated to a permutation. St000502The number of successions of a set partitions. St000864The number of circled entries of the shifted recording tableau of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000907The number of maximal antichains of minimal length in a poset. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001737The number of descents of type 2 in a permutation. St000023The number of inner peaks of a permutation. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.