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Your data matches 440 different statistics following compositions of up to 3 maps.
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Matching statistic: St000050
(load all 11 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
[[.,.],[.,[.,.]]]
 => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => 3 = 4 - 1
[[.,[.,.]],[.,.]]
 => 3 = 4 - 1
[[[.,.],.],[.,.]]
 => 3 = 4 - 1
[[.,[.,[.,.]]],.]
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => 4 = 5 - 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 10 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
[[.,.],[.,[.,.]]]
 => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => 3 = 4 - 1
[[.,[.,.]],[.,.]]
 => 3 = 4 - 1
[[[.,.],.],[.,.]]
 => 3 = 4 - 1
[[.,[.,[.,.]]],.]
 => 4 = 5 - 1
[[.,[[.,.],.]],.]
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => 3 = 4 - 1
[[[.,[.,.]],.],.]
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => 4 = 5 - 1

Description

The number of distinct nonempty subtrees of a binary tree.

Matching statistic: St001237
(load all 20 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000528
(load all 7 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),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],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),(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

Description

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

Matching statistic: St000863
(load all 99 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
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => 3 = 4 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => 3 = 4 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => 3 = 4 - 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

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

Description

The number of maximal antichains in a poset.

Matching statistic: St001343
(load all 7 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),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],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),(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

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

Description

The largest size of an interval in a poset.

Matching statistic: St000080
(load all 8 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),(1,2),(2,3)],4)
 => 2 = 4 - 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 4 - 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 4 - 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],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),(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

Description

The rank of the poset.

Matching statistic: St000380
(load all 6 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. Put differently, this is the smallest number

$n$ such that the partition fits into the triangular partition

$(n-1,n-2,\dots,1)$ .

The following 430 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000308The height of the tree associated to a permutation. St000384The maximal part of the shifted composition of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring 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. St001389The number of partitions of the same length below the given integer partition. St001616The number of neutral elements in a lattice. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001820The size of the image of the pop stack sorting operator. St000245The number of ascents of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000144The pyramid weight of the Dyck path. St000151The number of facets in the chain polytope of the poset. St000216The absolute length of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000393The number of strictly increasing runs in a binary word. St000533The minimum of the number of parts and the size of the first part of 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

$c_{n-1}$ . 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. 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. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. 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. 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. St001179Number of indecomposable injective modules with projective dimension at most 2 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St000010The length of the partition. 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. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000141The maximum drop size of a permutation. St000160The multiplicity of the smallest part of a partition. St000172The Grundy number of a graph. St000189The number of elements in the poset. St000213The number of weak exceedances (also weak excedences) of a permutation. St000225Difference between largest and smallest parts in a partition. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000470The number of runs in a permutation. St000475The number of parts equal to 1 in a partition. 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. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000806The semiperimeter of the associated bargraph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000907The number of maximal antichains of minimal length in a poset. 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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: St001250The number of parts of a partition that are not congruent 0 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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. 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). St001530The depth of a Dyck path. 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. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001883The mutual visibility number of a graph. St001933The largest multiplicity of a part in an integer partition. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. 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. St000442The maximal area to the right of an up step of a Dyck path. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000778The metric dimension of a graph. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. 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. St001489The maximum of the number of descents and the number of inverse descents. 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. St001613The binary logarithm of the size of the center of a lattice. 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. St001668The number of points of the poset minus the width of the poset. St001812The biclique partition number of a graph. St001935The number of ascents in a parking function. St001962The proper pathwidth of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001651The Frankl number of a lattice. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000702The number of weak deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000906The length of the shortest maximal chain in a poset. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000060The greater neighbor of the maximum. St000083The number of left oriented leafs of a binary tree except the first one. St000327The number of cover relations in a poset. St000354The number of recoils of a permutation. 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. St000922The minimal number such that all substrings of this length are unique. St000932The number of occurrences of the pattern UDU in a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001637The number of (upper) dissectors of a poset. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000568The hook number of a binary tree. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001110The 3-dynamic chromatic number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000171The degree of the graph. St000271The chromatic index of a graph. St000273The domination number of a graph. St000387The matching number of a graph. St000916The packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001621The number of atoms of a lattice. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000521The number of distinct subtrees of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St001058The breadth of the ordered tree. St001875The number of simple modules with projective dimension at most 1. St000386The number of factors DDU in a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000834The number of right outer peaks of a permutation. St000451The length of the longest pattern of the form k 1 2. St000647The number of big descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000981The length of the longest zigzag subpath. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000094The depth of an ordered tree. St000166The depth minus 1 of an ordered tree. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000628The balance of a binary word. 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. St001114The number of odd descents of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001512The minimum rank of a graph. St001517The length of a longest pair of twins in a permutation. St001642The Prague dimension of a graph. St001667The maximal size of a pair of weak twins for a permutation. St001734The lettericity of a graph. 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). St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. 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. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. 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. 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

$eA$ . St001570The minimal number of edges to add to make a graph Hamiltonian. St000307The number of rowmotion orbits of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St000422The energy of a graph, if it is integral. St001117The game chromatic index of a graph. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001649The length of a longest trail in a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001578The minimal number of edges to add or remove to make a graph a line graph. St001964The interval resolution global dimension of a poset. St000374The number of exclusive right-to-left minima of a permutation. St000443The number of long tunnels of a Dyck path. St000482The (zero)-forcing number of a graph. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000831The number of indices that are either descents or recoils. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001352The number of internal nodes in the modular decomposition of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001391The disjunction number of a graph. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001488The number of corners of a skew partition. St001566The length of the longest arithmetic progression in a permutation. St001703The villainy of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001765The number of connected components of the friends and strangers graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000117The number of centered tunnels of a Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000317The cycle descent number of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000363The number of minimal vertex covers of a graph. St000519The largest length of a factor maximising the subword complexity. St000535The rank-width of a graph. St000537The cutwidth of a graph. St000644The number of graphs with given frequency partition. St000739The first entry in the last row of a semistandard tableau. St000742The number of big ascents of a permutation after prepending zero. St000836The number of descents of distance 2 of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000958The number of Bruhat factorizations of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001281The normalized isoperimetric number of a graph. St001316The domatic number of a graph. St001345The Hamming dimension of a graph. St001350Half of the Albertson index of a graph. St001401The number of distinct entries in a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001589The nesting number of a perfect matching. St001665The number of pure excedances of a permutation. St001684The reduced word complexity of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001712The number of natural descents of a standard Young tableau. St001725The harmonious chromatic number of a graph. St001735The number of permutations with the same set of runs. St001739The number of graphs with the same edge polytope as the given graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000039The number of crossings of a permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000223The number of nestings in the permutation. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000241The number of cyclical small excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000637The length of the longest cycle in a graph. St000677The standardized bi-alternating inversion number of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000735The last entry on the main diagonal of a standard tableau. St000837The number of ascents of distance 2 of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001271The competition number of a graph. St001310The number of induced diamond graphs in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001353The number of prime nodes in the modular decomposition of a graph. St001395The number of strictly unfriendly partitions of a graph. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001537The number of cyclic crossings of a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001638The book thickness of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001691The number of kings in a graph. St001783The number of odd automorphisms of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation.