Your data matches 111 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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
[[],[]]
 => [[.,.],.]
 => 1
[[[]]]
 => [.,[.,.]]
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 2
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 3
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => 2
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 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: 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 = 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
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 1 = 2 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 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
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => 1 = 2 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => 1 = 2 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => 1 = 2 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => 1 = 2 - 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
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 1 = 2 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => 0 = 1 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => 0 = 1 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => 0 = 1 - 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 9 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 - 1
[[],[]]
 => [1,0,1,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 6 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 - 1
[[],[]]
 => [1,0,1,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
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)],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),(1,2),(2,4),(4,3)],5)
 => 2
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(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)
 => 3
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(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),(1,2),(2,3),(3,4)],5)
 => 2
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 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)],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),(1,2),(2,4),(4,3)],5)
 => 2
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(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)
 => 3
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(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),(1,2),(2,3),(3,4)],5)
 => 2
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 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)],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),(1,2),(2,4),(4,3)],5)
 => 2
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(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)
 => 3
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(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),(1,2),(2,3),(3,4)],5)
 => 2
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
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 = 1 - 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 0 = 1 - 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 0 = 1 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1 = 2 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1 = 2 - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1 = 2 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
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 - 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
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1 = 2 - 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
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{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
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 1 = 2 - 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
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 1 = 2 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0 = 1 - 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 $w_i > w_{i+1}$.
Matching statistic: St000010
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => [1]
 => 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => [2]
 => 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => [2]
 => 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 2
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [3]
 => 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 2
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 2
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 2
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 2
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 2
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => 3
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => 2
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => 2
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 1
Description
The length of the partition.
The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000985The number of positive eigenvalues of the adjacency matrix of the graph. 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. St000568The hook number of a binary tree. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000884The number of isolated descents of a permutation. St000470The number of runs in a permutation. St001928The number of non-overlapping descents 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. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000159The number of distinct parts of the integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000647The number of big descents of a permutation. St001732The number of peaks visible from the left. 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. 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. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000325The width of the tree associated to a permutation. St000409The number of pitchforks in a binary tree. 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. St001597The Frobenius rank of a skew partition. 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. 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. St000703The number of deficiencies of a permutation. St000640The rank of the largest boolean interval in a poset. 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. St001877Number of indecomposable injective modules with projective dimension 2. St001729The number of visible descents of a permutation. 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. 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 $3$. 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. 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 $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. 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. 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. St001487The number of inner corners of a skew partition. St000456The monochromatic index of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000761The number of ascents in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph.