- FindStat

Your data matches 320 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000050
(load all 14 compositions to match this statistic)

Mapping

St000050: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => 1 = 2 - 1
[.,[.,.]]
 => 2 = 3 - 1
[[.,.],.]
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => 3 = 4 - 1
[.,[[.,.],.]]
 => 3 = 4 - 1
[[.,.],[.,.]]
 => 2 = 3 - 1
[[.,[.,.]],.]
 => 3 = 4 - 1
[[[.,.],.],.]
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => 6 = 7 - 1

Description

The depth or height of a binary tree. The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices. The '''height''' of a vertex is the number of edges on the longest path between that node and a leaf. The '''depth''' of a vertex is the number of edges from the vertex to the root. See [1] and [2] for this terminology. The depth (or height) of a tree

$T$ can be recursively defined:

$\operatorname{depth}(T) = 0$ if

$T$ is empty and

$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$ if

$T$ is nonempty with left and right subtrees

$L$ and

$R$ , respectively. The upper and lower bounds on the depth of a binary tree

$T$ of size

$n$ are

$log_2(n) \leq \operatorname{depth}(T) \leq n$ .

Matching statistic: St001554
(load all 14 compositions to match this statistic)

Mapping

St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => 1 = 2 - 1
[.,[.,.]]
 => 2 = 3 - 1
[[.,.],.]
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => 3 = 4 - 1
[.,[[.,.],.]]
 => 3 = 4 - 1
[[.,.],[.,.]]
 => 2 = 3 - 1
[[.,[.,.]],.]
 => 3 = 4 - 1
[[[.,.],.],.]
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => 6 = 7 - 1

Description

The number of distinct nonempty subtrees of a binary tree.

Matching statistic: St001237
(load all 49 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => 2
[.,[.,.]]
 => [1,0,1,0]
 => 3
[[.,.],.]
 => [1,1,0,0]
 => 3
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 4
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 4
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 3
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 4
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 4
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 5
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 5
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 4
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 5
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 5
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 5
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 5
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 4
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 5
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 5
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 6
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 6
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 6
[[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 6
[[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6
[[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 7
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 7
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 7
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 7
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 7
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 7
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 7
[.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 7
[.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 7
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 7
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 7
[.,[[[.,[[.,.],.]],.],.]]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 7
[.,[[[[.,[.,.]],.],.],.]]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 7
[.,[[[[[.,.],.],.],.],.]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 7

Description

The number of simple modules with injective dimension at most one or dominant dimension at least one.

Matching statistic: St000528
(load all 5 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => 1 = 2 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1

Description

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

Matching statistic: St000863
(load all 27 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => 1 = 2 - 1
[.,[.,.]]
 => [2,1] => 2 = 3 - 1
[[.,.],.]
 => [1,2] => 2 = 3 - 1
[.,[.,[.,.]]]
 => [3,2,1] => 3 = 4 - 1
[.,[[.,.],.]]
 => [2,3,1] => 3 = 4 - 1
[[.,.],[.,.]]
 => [3,1,2] => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => 3 = 4 - 1
[[[.,.],.],.]
 => [1,2,3] => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => 4 = 5 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => [5,4,3,2,6,1] => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => [4,5,3,2,6,1] => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [4,3,5,2,6,1] => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => [3,4,5,2,6,1] => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => [4,3,2,5,6,1] => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => [3,4,2,5,6,1] => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => [3,2,4,5,6,1] => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => [2,3,4,5,6,1] => 6 = 7 - 1

Description

The length of the first row of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the length of the first row of

$P$ and

$Q$ .

Matching statistic: St000912
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => 1 = 2 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1

Description

The number of maximal antichains in a poset.

Matching statistic: St001343
(load all 5 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => 1 = 2 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1

Description

The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.

Matching statistic: St001717
(load all 4 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => 1 = 2 - 1
[.,[.,.]]
 => ([(0,1)],2)
 => 2 = 3 - 1
[[.,.],.]
 => ([(0,1)],2)
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1

Description

The largest size of an interval in a poset.

Matching statistic: St000080
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => 0 = 2 - 2
[.,[.,.]]
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,.],.]
 => ([(0,1)],2)
 => 1 = 3 - 2
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 4 - 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 4 - 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2

Description

The rank of the poset.

Matching statistic: St000147
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => ([],1)
 => [2]
 => 2
[.,[.,.]]
 => ([(0,1)],2)
 => [3]
 => 3
[[.,.],.]
 => ([(0,1)],2)
 => [3]
 => 3
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [3,2]
 => 3
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [4]
 => 4
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 4
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 4
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 6
[.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7
[.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => 7

Description

The largest part of an integer partition.

The following 310 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000474Dyson's crank of a partition. St000784The maximum of the length and the largest part of the integer partition. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001619The number of non-isomorphic sublattices of a lattice. St001626The number of maximal proper sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001720The minimal length of a chain of small intervals in a lattice. St000013The height of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000144The pyramid weight of the Dyck path. St000184The size of the centralizer of any permutation of given cycle type. St000203The number of external nodes of a binary tree. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000907The number of maximal antichains of minimal length in a poset. St000916The packing number of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001463The number of distinct columns in the nullspace of a graph. St001616The number of neutral elements in a lattice. St001820The size of the image of the pop stack sorting operator. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000778The metric dimension of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000010The length of the partition. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000393The number of strictly increasing runs in a binary word. St000459The hook length of the base cell of a partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001664The number of non-isomorphic subposets of a poset. St001674The number of vertices of the largest induced star graph in the graph. St001782The order of rowmotion on the set of order ideals of a poset. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000189The number of elements in the poset. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000507The number of ascents of a standard tableau. St000636The hull number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000806The semiperimeter of the associated bargraph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000911The number of maximal antichains of maximal size in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001530The depth of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001883The mutual visibility number of a graph. St001933The largest multiplicity of a part in an integer partition. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000536The pathwidth of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001323The independence gap of a graph. St001358The largest degree of a regular subgraph of a graph. St001405The number of bonds in a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001644The dimension of a graph. St001651The Frankl number of a lattice. St001962The proper pathwidth of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000060The greater neighbor of the maximum. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000656The number of cuts of a poset. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000619The number of cyclic descents of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001246The maximal difference between two consecutive entries of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001118The acyclic chromatic index of a graph. St001645The pebbling number of a connected graph. St000653The last descent of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000299The number of nonisomorphic vertex-induced subtrees. St000533The minimum of the number of parts and the size of the first part of an integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St000160The multiplicity of the smallest part of a partition. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St000475The number of parts equal to 1 in a partition. St000956The maximal displacement of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001812The biclique partition number of a graph. St001668The number of points of the poset minus the width of the poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001948The number of augmented double ascents of a permutation. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St000225Difference between largest and smallest parts in a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000442The maximal area to the right of an up step of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001613The binary logarithm of the size of the center of a lattice. St001545The second Elser number of a connected graph. St000521The number of distinct subtrees of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St000386The number of factors DDU in a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000834The number of right outer peaks of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001110The 3-dynamic chromatic number of a graph. St000171The degree of the graph. St000271The chromatic index of a graph. St000387The matching number of a graph. St000568The hook number of a binary tree. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001112The 3-weak dynamic number of a graph. St000647The number of big descents of a permutation. St001621The number of atoms of a lattice. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000662The staircase size of the code of a permutation. St000094The depth of an ordered tree. St000166The depth minus 1 of an ordered tree. St000981The length of the longest zigzag subpath. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St000628The balance of a binary word. St000699The toughness times the least common multiple of 1,. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000785The number of distinct colouring schemes of a graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001642The Prague dimension of a graph. St001667The maximal size of a pair of weak twins for a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000646The number of big ascents of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001469The holeyness of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001060The distinguishing index of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. 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

$eA$ . 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. St001877Number of indecomposable injective modules with projective dimension 2.