Your data matches 223 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000196
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 0
[[],[]]
 => [[.,.],.]
 => 0
[[[]]]
 => [.,[.,.]]
 => 0
[[],[],[]]
 => [[[.,.],.],.]
 => 0
[[],[[]]]
 => [[.,.],[.,.]]
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 0
[[[],[]]]
 => [.,[[.,.],.]]
 => 0
[[[[]]]]
 => [.,[.,[.,.]]]
 => 0
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 0
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 0
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 0
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 0
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 0
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 0
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 0
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 0
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => 0
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 2
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => 0
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => 0
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 0
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 0
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 0
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 0
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 0
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 10 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 0
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 0
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 8 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 0
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 0
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000201
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 1 = 0 + 1
[[],[]]
 => [[.,.],.]
 => 1 = 0 + 1
[[[]]]
 => [.,[.,.]]
 => 1 = 0 + 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1 = 0 + 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1 = 0 + 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 1 + 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 1 + 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 1 + 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 3 = 2 + 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 1 + 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 1 + 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 0 + 1
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: St000632
(load all 3 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 0
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 0
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 0
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
Description
The jump number of the poset. A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001840
(load all 3 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
[[],[]]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[[[]]]
 => [1,1,0,0]
 => {{1,2}}
 => 0
[[],[],[]]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 0
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 0
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 0
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 0
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 0
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 0
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
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 $w_i > w_{i+1}$.
Matching statistic: St000068
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000071
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000527
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000985
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => ([],1)
 => 0
[[],[]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[[]]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[],[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[[]]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(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)
 => 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
The following 213 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001354The number of series nodes in the modular decomposition of a graph. St001712The number of natural descents of a standard Young tableau. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000010The length of the partition. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001304The number of maximally independent sets of vertices of a graph. St001581The achromatic number of a graph. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000568The hook number of a binary tree. St000884The number of isolated descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001092The number of distinct even parts of a partition. St000659The number of rises of length at least 2 of a Dyck path. St000647The number of big descents of a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000035The number of left outer peaks of a permutation. St000779The tier of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000710The number of big deficiencies of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000646The number of big ascents of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000919The number of maximal left branches of a binary tree. St000251The number of nonsingleton blocks of a set partition. St000360The number of occurrences of the pattern 32-1. St000023The number of inner peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000021The number of descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000325The width of the tree associated to a permutation. St000409The number of pitchforks in a binary tree. St001597The Frobenius rank of a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000834The number of right outer peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000703The number of deficiencies of a permutation. St000640The rank of the largest boolean interval in a poset. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000482The (zero)-forcing number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001057The Grundy value of the game of creating an independent set in a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001111The weak 2-dynamic chromatic number of a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001277The degeneracy of a graph. St001323The independence gap of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001792The arboricity of a graph. St001828The Euler characteristic of a graph. St001949The rigidity index of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St001972The a-number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001820The size of the image of the pop stack sorting operator. St000095The number of triangles of a graph. St000268The number of strongly connected orientations of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000449The number of pairs of vertices of a graph with distance 4. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001353The number of prime nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001638The book thickness of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001871The number of triconnected components of a graph. St001970The signature of a graph. St001729The number of visible descents of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000481The number of upper covers of a partition in dominance order. St001960The number of descents of a permutation minus one if its first entry is not one. St000480The number of lower covers of a partition in dominance order. St000455The second largest eigenvalue of a graph if it is integral. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001846The number of elements which do not have a complement in the lattice. St000454The largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001626The number of maximal proper sublattices of a lattice. St000897The number of different multiplicities of parts of an integer partition. St000031The number of cycles in the cycle decomposition of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001487The number of inner corners of a skew partition. St000761The number of ascents in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001335The cardinality of a minimal cycle-isolating set of a graph.