- FindStat

Your data matches 480 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000092
(load all 68 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of outer peaks of a permutation. An outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ or

$n$ if

$w_{n} > w_{n-1}$ . In other words, it is a peak in the word

$[0,w_1,..., w_n,0]$ .

Matching statistic: St000099
(load all 50 compositions to match this statistic)

Mapping

Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [.,.]
 => [1] => 1 = 2 - 1
[.,[.,.]]
 => [.,[.,.]]
 => [2,1] => 1 = 2 - 1
[[.,.],.]
 => [[.,.],.]
 => [1,2] => 1 = 2 - 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1 = 2 - 1
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => [2,3,1] => 2 = 3 - 1
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => [1,3,2] => 2 = 3 - 1
[[[.,.],.],.]
 => [[[.,.],.],.]
 => [1,2,3] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 3 - 1
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 1 = 2 - 1

Description

The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].

Matching statistic: St000201
(load all 13 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [.,.]
 => 1 = 2 - 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => 1 = 2 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => 1 = 2 - 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1

Description

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size

$n$ , at least

$1$ , with exactly one leaf node for is

$2^{n-1}$ , see [2]. The number of binary tree of size

$n$ , at least

$3$ , with exactly two leaf nodes is

$n(n+1)2^{n-2}$ , see [3].

Matching statistic: St000396
(load all 13 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [.,.]
 => 1 = 2 - 1
[.,[.,.]]
 => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[[.,.],.]
 => [1,2] => [.,[.,.]]
 => 1 = 2 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => 1 = 2 - 1
[.,[[.,.],.]]
 => [2,3,1] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[.,.],[.,.]]
 => [1,3,2] => [.,[[.,.],.]]
 => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1

Description

The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree

$[[[.,.],[.,.]],[[.,.],[.,.]]]$ : its register function is 3, whereas the dimension of the associated poset is 2.

Matching statistic: St000522
(load all 14 compositions to match this statistic)

Mapping

Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
Mp00246: Ordered trees —rotate⟶ Ordered trees
St000522: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [[]]
 => [[]]
 => 1 = 2 - 1
[.,[.,.]]
 => [[],[]]
 => [[],[]]
 => 1 = 2 - 1
[[.,.],.]
 => [[[]]]
 => [[[]]]
 => 1 = 2 - 1
[.,[.,[.,.]]]
 => [[],[],[]]
 => [[],[],[]]
 => 1 = 2 - 1
[.,[[.,.],.]]
 => [[],[[]]]
 => [[[]],[]]
 => 2 = 3 - 1
[[.,.],[.,.]]
 => [[[]],[]]
 => [[[],[]]]
 => 1 = 2 - 1
[[.,[.,.]],.]
 => [[[],[]]]
 => [[],[[]]]
 => 2 = 3 - 1
[[[.,.],.],.]
 => [[[[]]]]
 => [[[[]]]]
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [[],[],[],[]]
 => [[],[],[],[]]
 => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [[],[],[[]]]
 => [[],[[]],[]]
 => 2 = 3 - 1
[.,[[.,.],[.,.]]]
 => [[],[[]],[]]
 => [[[]],[],[]]
 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [[],[[],[]]]
 => [[[],[]],[]]
 => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [[],[[[]]]]
 => [[[[]]],[]]
 => 2 = 3 - 1
[[.,.],[.,[.,.]]]
 => [[[]],[],[]]
 => [[[],[],[]]]
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [[[]],[[]]]
 => [[[[]],[]]]
 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [[[],[]],[]]
 => [[],[[],[]]]
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [[[[]]],[]]
 => [[[[],[]]]]
 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [[[],[],[]]]
 => [[],[],[[]]]
 => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [[[],[[]]]]
 => [[[]],[[]]]
 => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [[[[]],[]]]
 => [[[],[[]]]]
 => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [[[[],[]]]]
 => [[],[[[]]]]
 => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [[[[[]]]]]
 => [[[[[]]]]]
 => 1 = 2 - 1

Description

The number of 1-protected nodes of a rooted tree. This is the number of nodes with minimal distance one to a leaf.

Matching statistic: St000862
(load all 62 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts 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 number of parts of the shifted shape.

Matching statistic: St000023
(load all 50 compositions to match this statistic)

Mapping

Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].

Matching statistic: St000196
(load all 13 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by

$2^{n-2}$ .

Matching statistic: St001037
(load all 5 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1,0]
 => 0 = 2 - 2
[.,[.,.]]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0 = 2 - 2
[[.,.],.]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0 = 2 - 2
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0 = 2 - 2
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0 = 2 - 2
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0 = 2 - 2
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 2 - 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 2 - 2
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 2 - 2

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St001712
(load all 4 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of natural descents of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation.

The following 470 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000325The width of the tree associated to a permutation. St000397The Strahler number of a rooted tree. St000470The number of runs in a permutation. St000758The length of the longest staircase fitting into an integer composition. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000216The absolute length of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000884The number of isolated descents of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000920The logarithmic height of a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001487The number of inner corners of a skew partition. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001928The number of non-overlapping descents in a permutation. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000353The number of inner valleys of a permutation. St000386The number of factors DDU in a Dyck path. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001394The genus of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001731The factorization defect of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001801Half the number of preimage-image pairs of different parity in a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001874Lusztig's a-function for the symmetric group. St001931The weak major index of an integer composition regarded as a word. St001960The number of descents of a permutation minus one if its first entry is not one. St000568The hook number of a binary tree. St000291The number of descents of a binary word. St000624The normalized sum of the minimal distances to a greater element. St000875The semilength of the longest Dyck word in the Catalan factorisation 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. St000619The number of cyclic descents of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000354The number of recoils of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000539The number of odd inversions of a permutation. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001569The maximal modular displacement of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000650The number of 3-rises of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. 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. St001160The number of proper blocks (or intervals) of a permutations. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000264The girth of a graph, which is not a tree. St001488The number of corners of a skew partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000455The second largest eigenvalue of a graph if it is integral. St000567The sum of the products of all pairs of parts. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St001399The distinguishing number of a poset. St001545The second Elser number of a connected graph. St000640The rank of the largest boolean interval in a poset. St000680The Grundy value for Hackendot on posets. St000741The Colin de Verdière graph invariant. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001060The distinguishing index of a graph. St001597The Frobenius rank of a skew partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001596The number of two-by-two squares inside a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001518The number of graphs with the same ordinary spectrum as the given graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001828The Euler characteristic of a graph. St001890The maximum magnitude of the Möbius function of a poset. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000315The number of isolated vertices of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. 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. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St000064The number of one-box pattern of a permutation. St000258The burning number of a graph. St000259The diameter of a connected graph. St000272The treewidth of a graph. St000311The number of vertices of odd degree in a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000364The exponent of the automorphism group of a graph. St000469The distinguishing number of a graph. St000536The pathwidth of a graph. St000696The number of cycles in the breakpoint graph of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000746The number of pairs with odd minimum in a perfect matching. St000778The metric dimension of a graph. St000842The breadth of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000918The 2-limited packing number of a graph. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001277The degeneracy of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001358The largest degree of a regular subgraph of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001644The dimension of a graph. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001716The 1-improper chromatic number of a graph. St001792The arboricity of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000037The sign of a permutation. St000056The decomposition (or block) number of a permutation. St000260The radius of a connected graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000458The number of permutations obtained by switching adjacencies or successions. St000544The cop number of a graph. St000546The number of global descents of a permutation. St000553The number of blocks of a graph. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000694The number of affine bounded permutations that project to a given permutation. St000732The number of double deficiencies of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000785The number of distinct colouring schemes of a graph. St000788The number of nesting-similar perfect matchings of a perfect matching. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000883The number of longest increasing subsequences of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000916The packing number of a graph. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St000968We 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

$c_{n−1}$ . St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001081The number of minimal length factorizations of a permutation into star transpositions. St001108The 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001162The minimum jump of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . 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

$eA$ . St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001256Number of simple reflexive modules that are 2-stable reflexive. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001315The dissociation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001342The number of vertices in the center of a graph. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001405The number of bonds in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001479The number of bridges of a graph. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001580The acyclic chromatic number of a graph. St001590The crossing number of a perfect matching. St001645The pebbling number of a connected graph. St001656The monophonic position number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001672The restrained domination number of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001743The discrepancy of a graph. St001765The number of connected components of the friends and strangers graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001829The common independence number of a graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001948The number of augmented double ascents of a permutation. St000022The number of fixed points of a permutation. St000102The charge of a semistandard tableau. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000302The determinant of the distance matrix of a connected graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000359The number of occurrences of the pattern 23-1. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000488The number of cycles of a permutation of length at most 2. St000552The number of cut vertices of a graph. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000731The number of double exceedences of a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001110The 3-dynamic chromatic number of a graph. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001204Call 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001423The number of distinct cubes in a binary word. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001520The number of strict 3-descents. St001536The number of cyclic misalignments of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St001964The interval resolution global dimension of a poset. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001109The number of proper colourings of a graph with as few colours as possible.