Your data matches 53 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000328
(load all 2 compositions to match this statistic)
Mapping
St000328: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 1
[[],[]]
 => 2
[[[]]]
 => 1
[[],[],[]]
 => 3
[[],[[]]]
 => 2
[[[]],[]]
 => 2
[[[],[]]]
 => 2
[[[[]]]]
 => 1
[[],[],[],[]]
 => 4
[[],[],[[]]]
 => 3
[[],[[]],[]]
 => 3
[[],[[],[]]]
 => 2
[[],[[[]]]]
 => 2
[[[]],[],[]]
 => 3
[[[]],[[]]]
 => 2
[[[],[]],[]]
 => 2
[[[[]]],[]]
 => 2
[[[],[],[]]]
 => 3
[[[],[[]]]]
 => 2
[[[[]],[]]]
 => 2
[[[[],[]]]]
 => 2
[[[[[]]]]]
 => 1
[[],[],[],[],[]]
 => 5
[[],[],[],[[]]]
 => 4
[[],[],[[]],[]]
 => 4
[[],[],[[],[]]]
 => 3
[[],[],[[[]]]]
 => 3
[[],[[]],[],[]]
 => 4
[[],[[]],[[]]]
 => 3
[[],[[],[]],[]]
 => 3
[[],[[[]]],[]]
 => 3
[[],[[],[],[]]]
 => 3
[[],[[],[[]]]]
 => 2
[[],[[[]],[]]]
 => 2
[[],[[[],[]]]]
 => 2
[[],[[[[]]]]]
 => 2
[[[]],[],[],[]]
 => 4
[[[]],[],[[]]]
 => 3
[[[]],[[]],[]]
 => 3
[[[]],[[],[]]]
 => 2
[[[]],[[[]]]]
 => 2
[[[],[]],[],[]]
 => 3
[[[[]]],[],[]]
 => 3
[[[],[]],[[]]]
 => 2
[[[[]]],[[]]]
 => 2
[[[],[],[]],[]]
 => 3
[[[],[[]]],[]]
 => 2
[[[[]],[]],[]]
 => 2
[[[[],[]]],[]]
 => 2
[[[[[]]]],[]]
 => 2
Description
The maximum number of child nodes in a tree.
Matching statistic: St000846
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[[[]]]
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 3
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 3
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 2
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 2
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 2
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 3
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 2
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 2
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St001239
(load all 2 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => 2
[[[]]]
 => [1,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 2
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 3
[[[[]]]]
 => [1,1,1,0,0,0]
 => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 3
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 4
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001192
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 4 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St000308
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => 1
[[],[]]
 => [[.,.],.]
 => [1,2] => 2
[[[]]]
 => [.,[.,.]]
 => [2,1] => 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => 3
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 4
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 3
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 3
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 2
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 3
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 5
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 4
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => 4
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => 3
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => 3
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => 4
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => 3
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => 3
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => 3
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 4
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 3
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 3
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 3
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 3
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 3
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 2
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 2
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 2
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St000381
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => 1
[[[]]]
 => [1,1,0,0]
 => [2] => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 4
Description
The largest part of an integer composition.
Matching statistic: St000392
(load all 5 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 10 => 1
[[],[]]
 => [1,0,1,0]
 => 1010 => 1
[[[]]]
 => [1,1,0,0]
 => 1100 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => 101010 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 101100 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 110010 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 110100 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 111000 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 4
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000845
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(3,1),(3,2)],4)
 => 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 3
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 3
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 2
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 3
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => 5
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => 4
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => 4
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => 3
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => 3
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => 4
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => 3
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => 3
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => 3
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => 3
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
 => 2
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
 => 2
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => 2
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => 4
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => 3
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => 3
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => 3
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => 3
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => 2
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => 3
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
 => 2
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
 => 2
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 2
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St001235
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => 2
[[[]]]
 => [1,1,0,0]
 => [2] => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => 3
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 4
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 3
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 3
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 3
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 3
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 5
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 4
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 3
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 3
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 3
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 3
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 3
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 2
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 4
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => 4
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 4
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 2
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001372
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 10 => 1
[[],[]]
 => [1,0,1,0]
 => 1010 => 1
[[[]]]
 => [1,1,0,0]
 => 1100 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => 101010 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 101100 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 110010 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 110100 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 111000 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 4
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000147The largest part of an integer partition. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000662The staircase size of the code of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000485The length of the longest cycle of a permutation. St001062The maximal size of a block of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000956The maximal displacement of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. 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. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000143The largest repeated part of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001875The number of simple modules with projective dimension at most 1. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.