Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000701
(load all 17 compositions to match this statistic)
Mapping
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => 1
[.,[.,.]]
 => 1
[[.,.],.]
 => 1
[.,[.,[.,.]]]
 => 1
[.,[[.,.],.]]
 => 1
[[.,.],[.,.]]
 => 2
[[.,[.,.]],.]
 => 1
[[[.,.],.],.]
 => 1
[.,[.,[.,[.,.]]]]
 => 1
[.,[.,[[.,.],.]]]
 => 1
[.,[[.,.],[.,.]]]
 => 1
[.,[[.,[.,.]],.]]
 => 1
[.,[[[.,.],.],.]]
 => 1
[[.,.],[.,[.,.]]]
 => 2
[[.,.],[[.,.],.]]
 => 2
[[.,[.,.]],[.,.]]
 => 2
[[[.,.],.],[.,.]]
 => 2
[[.,[.,[.,.]]],.]
 => 1
[[.,[[.,.],.]],.]
 => 1
[[[.,.],[.,.]],.]
 => 1
[[[.,[.,.]],.],.]
 => 1
[[[[.,.],.],.],.]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => 1
[.,[[[[.,.],.],.],.]]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => 2
[[.,.],[.,[[.,.],.]]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => 2
[[.,.],[[[.,.],.],.]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => 2
[[[.,.],.],[[.,.],.]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => 2
[[[[.,.],.],.],[.,.]]
 => 2
Description
The protection number of a binary tree. This is the minimal distance from the root to a leaf.
Matching statistic: St000700
(load all 3 compositions to match this statistic)
Mapping
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [[],[]]
 => 1
[.,[.,.]]
 => [[],[[],[]]]
 => 1
[[.,.],.]
 => [[[],[]],[]]
 => 1
[.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 1
[.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 1
[[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2
[[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 1
[[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 1
[.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 1
[.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 1
[.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 1
[.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 1
[.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 1
[[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 2
[[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 2
[[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 2
[[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 2
[[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 1
[[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 1
[[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 1
[[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 1
[[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [[],[[],[[],[[],[[],[]]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [[],[[],[[],[[[],[]],[]]]]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [[],[[],[[[],[]],[[],[]]]]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [[],[[],[[[],[[],[]]],[]]]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [[],[[],[[[[],[]],[]],[]]]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [[],[[[],[]],[[],[[],[]]]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [[],[[[],[]],[[[],[]],[]]]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [[],[[[],[[],[]]],[[],[]]]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [[],[[[[],[]],[]],[[],[]]]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [[],[[[],[[],[[],[]]]],[]]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [[],[[[],[[[],[]],[]]],[]]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [[],[[[[],[]],[[],[]]],[]]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [[],[[[[],[[],[]]],[]],[]]]
 => 1
[.,[[[[.,.],.],.],.]]
 => [[],[[[[[],[]],[]],[]],[]]]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => 2
[[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => 2
[[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [[[[],[]],[[],[]]],[[],[]]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 2
[[[[.,.],.],.],[.,.]]
 => [[[[[],[]],[]],[]],[[],[]]]
 => 2
Description
The protection number of an ordered tree. This is the minimal distance from the root to a leaf.
Matching statistic: St001085
(load all 21 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [1,2] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,3,2,4] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,2,4] => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,2,1,4] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,1,3] => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,4,3,5] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,3,2,4,5] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,3,2,5,4] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,4,2,3,5] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,3,2,5] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,5,3,2,4] => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [2,1,3,4,5] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,1,3,5,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [2,1,4,3,5] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [2,1,5,3,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,5,4,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,1,2,4,5] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,2,5,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2,1,4,5] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,2,1,5,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1,2,3,5] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,3,2,5] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,1,3,5] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [4,3,1,2,5] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [4,3,2,1,5] => 1 = 2 - 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St000553
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000553: 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)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[.,.],[.,.]]
 => ([(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)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(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)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(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)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(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)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(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)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(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)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(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)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(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)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(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)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(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)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(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)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(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)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(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)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(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)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => 2
Description
The number of blocks of a graph. A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St000552
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => ([(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)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => 1 = 2 - 1
Description
The number of cut vertices of a graph. A cut vertex is one whose deletion increases the number of connected components.
Matching statistic: St001086
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => [2,3,1] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [2,3,1] => [3,1,2] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => [3,4,2,1] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,4,2,1] => [4,3,1,2] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,3,4] => [2,3,4,5,1] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,2,3] => [3,4,5,2,1] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,2,3] => [3,4,5,1,2] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,1,4,2,3] => [2,4,5,3,1] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,5,4,2,3] => [1,4,5,3,2] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => [4,5,3,2,1] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => [4,5,3,1,2] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,3,4,1,2] => [4,5,2,3,1] => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => [4,5,2,1,3] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => [4,5,1,2,3] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,1,2,4,3] => [2,3,5,4,1] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,3,5,4,2] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,1,3,2] => [3,5,4,2,1] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,5,1,3,2] => [3,5,4,1,2] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,1,4,3,2] => [2,5,4,3,1] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,3,4,2] => [2,5,3,4,1] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,5,3,2] => [1,5,4,2,3] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,5,3,4,2] => [1,5,3,4,2] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,4,3,5,2] => [1,5,3,2,4] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,3,4,5,2] => [1,5,2,3,4] => 1 = 2 - 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St001878
(load all 3 compositions to match this statistic)
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001570
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => ? = 1 - 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => ? = 1 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(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)
 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => ([(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)
 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(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)
 => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(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)
 => 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
 => ([(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)
 => 0 = 1 - 1
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000456
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[.,.]
 => [1] => ([],1)
 => ([],1)
 => ? = 1
[.,[.,.]]
 => [2,1] => ([],2)
 => ([],2)
 => ? = 1
[[.,.],.]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => ([],3)
 => ([],3)
 => ? = 1
[.,[[.,.],.]]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[[.,.],[.,.]]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[[.,[.,.]],.]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[[[.,.],.],.]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([],4)
 => ([],4)
 => ? = 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 2
[[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([],5)
 => ([],5)
 => ? = 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 2
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 2
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 2
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 2
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => ([],6)
 => ([],6)
 => ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ? = 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [4,5,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[.,[.,[[.,.],[.,.]]]],.]
 => [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[.,[.,[[.,[.,.]],.]]],.]
 => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[.,[.,[[[.,.],.],.]]],.]
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[.,[[.,.],[.,[.,.]]]],.]
 => [5,4,2,3,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[.,[[.,.],[[.,.],.]]],.]
 => [4,5,2,3,1,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[.,[[.,[.,.]],[.,.]]],.]
 => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[.,[[[.,.],.],[.,.]]],.]
 => [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[.,[[.,[.,[.,.]]],.]],.]
 => [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[.,[[.,[[.,.],.]],.]],.]
 => [3,4,2,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[.,[[[.,.],[.,.]],.]],.]
 => [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[.,[[[.,[.,.]],.],.]],.]
 => [3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[.,[[[[.,.],.],.],.]],.]
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
[[[.,.],[.,[.,[.,.]]]],.]
 => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[[.,.],[.,[[.,.],.]]],.]
 => [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[.,.],[[.,.],[.,.]]],.]
 => [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[.,.],[[.,[.,.]],.]],.]
 => [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[[.,.],[[[.,.],.],.]],.]
 => [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
[[[.,[.,.]],[.,[.,.]]],.]
 => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[[.,[.,.]],[[.,.],.]],.]
 => [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[[[.,.],.],[.,[.,.]]],.]
 => [5,4,1,2,3,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[[[.,.],.],[[.,.],.]],.]
 => [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
[[[.,[.,[.,.]]],[.,.]],.]
 => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[[.,[[.,.],.]],[.,.]],.]
 => [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[[.,.],[.,.]],[.,.]],.]
 => [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[[.,[.,.]],.],[.,.]],.]
 => [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[[[[.,.],.],.],[.,.]],.]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001198
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[.,.]
 => [1] => [1] => [1,0]
 => ? = 1 + 1
[.,[.,.]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? = 1 + 1
[[.,.],.]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 1 + 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 1 + 1
[[.,.],[.,.]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2 + 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [6,4,3,5,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[.,[.,[[.,.],.]]]],.]
 => [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[.,[[.,.],[.,.]]]],.]
 => [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[.,[[.,[.,.]],.]]],.]
 => [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[.,[[[.,.],.],.]]],.]
 => [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],[.,[.,.]]]],.]
 => [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,.],[[.,.],.]]],.]
 => [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,[.,.]],[.,.]]],.]
 => [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[[.,.],.],[.,.]]],.]
 => [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,[.,[.,.]]],.]],.]
 => [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[.,[[.,.],.]],.]],.]
 => [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[[.,.],[.,.]],.]],.]
 => [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[[.,[.,.]],.],.]],.]
 => [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[.,[[[[.,.],.],.],.]],.]
 => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[.,[.,[.,.]]]],.]
 => [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[.,[[.,.],.]]],.]
 => [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[[.,.],[.,.]]],.]
 => [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[[.,[.,.]],.]],.]
 => [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,.],[[[.,.],.],.]],.]
 => [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[.,.]],[.,[.,.]]],.]
 => [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[.,.]],[[.,.],.]],.]
 => [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[[.,.],.],[.,[.,.]]],.]
 => [5,4,1,2,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[[.,.],.],[[.,.],.]],.]
 => [4,5,1,2,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[.,[.,.]]],[.,.]],.]
 => [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[.,[[.,.],.]],[.,.]],.]
 => [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[[.,.],[.,.]],[.,.]],.]
 => [5,3,1,2,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[[.,[.,.]],.],[.,.]],.]
 => [5,2,1,3,4,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[[[[[.,.],.],.],[.,.]],.]
 => [5,1,2,3,4,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St000768The number of peaks in an integer composition. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1.