Your data matches 123 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000679
(load all 4 compositions to match this statistic)
Mapping
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 1
[[],[]]
 => 1
[[[]]]
 => 1
[[],[],[]]
 => 1
[[],[[]]]
 => 1
[[[]],[]]
 => 1
[[[],[]]]
 => 2
[[[[]]]]
 => 1
[[],[],[],[]]
 => 1
[[],[],[[]]]
 => 1
[[],[[]],[]]
 => 1
[[],[[],[]]]
 => 2
[[],[[[]]]]
 => 1
[[[]],[],[]]
 => 1
[[[]],[[]]]
 => 1
[[[],[]],[]]
 => 2
[[[[]]],[]]
 => 1
[[[],[],[]]]
 => 2
[[[],[[]]]]
 => 2
[[[[]],[]]]
 => 2
[[[[],[]]]]
 => 2
[[[[[]]]]]
 => 1
[[],[],[],[],[]]
 => 1
[[],[],[],[[]]]
 => 1
[[],[],[[]],[]]
 => 1
[[],[],[[],[]]]
 => 2
[[],[],[[[]]]]
 => 1
[[],[[]],[],[]]
 => 1
[[],[[]],[[]]]
 => 1
[[],[[],[]],[]]
 => 2
[[],[[[]]],[]]
 => 1
[[],[[],[[]]]]
 => 2
[[],[[[]],[]]]
 => 2
[[],[[[[]]]]]
 => 1
[[[]],[],[],[]]
 => 1
[[[]],[],[[]]]
 => 1
[[[]],[[]],[]]
 => 1
[[[]],[[],[]]]
 => 2
[[[]],[[[]]]]
 => 1
[[[],[]],[],[]]
 => 2
[[[[]]],[],[]]
 => 1
[[[],[]],[[]]]
 => 2
[[[[]]],[[]]]
 => 1
[[[],[[]]],[]]
 => 2
[[[[]],[]],[]]
 => 2
[[[[[]]]],[]]
 => 1
[[[],[[]],[]]]
 => 2
[[[],[[[]]]]]
 => 2
[[[[]],[[]]]]
 => 2
[[[[[]]],[]]]
 => 2
Description
The pruning number of an ordered tree. A hanging branch of an ordered tree is a proper factor of the form $[^r]^r$ for some $r\geq 1$. A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch. A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to $[]$.
Matching statistic: St000396
(load all 13 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 1
[[],[]]
 => [.,[.,.]]
 => 1
[[[]]]
 => [[.,.],.]
 => 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 1
[[[],[]]]
 => [[.,.],[.,.]]
 => 2
[[[[]]]]
 => [[[.,.],.],.]
 => 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => 2
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => 2
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => 1
[[[],[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[[[],[[[]]]]]
 => [[.,.],[[[.,.],.],.]]
 => 2
[[[[]],[[]]]]
 => [[[.,.],.],[[.,.],.]]
 => 2
[[[[[]]],[]]]
 => [[[[.,.],.],.],[.,.]]
 => 2
Description
The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000201
(load all 6 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [.,.]
 => 1
[[],[]]
 => [.,[.,.]]
 => [.,[.,.]]
 => 1
[[[]]]
 => [[.,.],.]
 => [[.,.],.]
 => 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [.,[[.,.],.]]
 => 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [[.,[.,.]],.]
 => 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => 2
[[[[]]]]
 => [[[.,.],.],.]
 => [[[.,.],.],.]
 => 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => 2
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => 2
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => 2
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => 2
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => 2
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 2
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => 1
[[[],[[]],[]]]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[[[],[[[]]]]]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => 2
[[[[]],[[]]]]
 => [[[.,.],[[.,.],.]],.]
 => [[[.,.],[[.,.],.]],.]
 => 2
[[[[[]]],[]]]
 => [[[[.,.],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],.]
 => 2
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000298
(load all 5 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(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),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[],[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[],[[[]]]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[[]],[[]]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[[[[[]]],[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000920
(load all 13 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St000196
(load all 6 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [.,.]
 => 0 = 1 - 1
[[],[]]
 => [.,[.,.]]
 => [.,[.,.]]
 => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => [[.,.],.]
 => 0 = 1 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [.,[[.,.],.]]
 => 0 = 1 - 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [[.,[.,.]],.]
 => 0 = 1 - 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [[[.,.],.],.]
 => 0 = 1 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => 0 = 1 - 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => 0 = 1 - 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => 1 = 2 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => 0 = 1 - 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => 0 = 1 - 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => 0 = 1 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => 1 = 2 - 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => 1 = 2 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => 0 = 1 - 1
[[[],[[]],[]]]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => 1 = 2 - 1
[[[],[[[]]]]]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => 1 = 2 - 1
[[[[]],[[]]]]
 => [[[.,.],[[.,.],.]],.]
 => [[[.,.],[[.,.],.]],.]
 => 1 = 2 - 1
[[[[[]]],[]]]
 => [[[[.,.],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],.]
 => 1 = 2 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000386
(load all 27 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001022
(load all 4 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001839
(load all 5 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => {{1}}
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => {{1},{2}}
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => {{1,2}}
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0 = 1 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1 = 2 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0 = 1 - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1 = 2 - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 1 = 2 - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 1 = 2 - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 1 = 2 - 1
Description
The number of excedances of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1 \dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. Let $\bar w$ be the nondecreasing rearrangement of $w$. The word $w$ has an excedance at position $i$ if $w_i > \bar w_i$.
Matching statistic: St001840
(load all 5 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => {{1}}
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => {{1},{2}}
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => {{1,2}}
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0 = 1 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1 = 2 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0 = 1 - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1 = 2 - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 1 = 2 - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 1 = 2 - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 1 = 2 - 1
Description
The number of descents of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.
The following 113 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000159The number of distinct parts of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000527The width of the poset. St000758The length of the longest staircase fitting into an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000862The number of parts of the shifted shape of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001432The order dimension of the partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000010The length of the partition. St000035The number of left outer peaks of a permutation. St000147The largest part of an integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001712The number of natural descents of a standard Young tableau. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001928The number of non-overlapping descents in a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000149The number of cells of the partition whose leg is zero and arm is odd. St000307The number of rowmotion orbits of a poset. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000647The number of big descents of a permutation. St001394The genus of a permutation. St001330The hat guessing number of a graph. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St000779The tier of a permutation. St000710The number of big deficiencies of a permutation. St000353The number of inner valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000097The order of the largest clique of the graph. St000254The nesting number of a set partition. St000098The chromatic number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000023The number of inner peaks of a permutation. St000397The Strahler number of a rooted tree. St000646The number of big ascents of a permutation. St000325The width of the tree associated to a permutation. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001358The largest degree of a regular subgraph of a graph. St001734The lettericity of a graph. St001792The arboricity of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000624The normalized sum of the minimal distances to a greater element. St001029The size of the core of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001494The Alon-Tarsi number of a graph. St001874Lusztig's a-function for the symmetric group. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000640The rank of the largest boolean interval in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. 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 $eA$. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000805The number of peaks of the associated bargraph.