- FindStat

Your data matches 696 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000092
(load all 232 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000092: 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] => 1 = 2 - 1
[[.,.],[.,.]]
 => [3,1,2] => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => 2 = 3 - 1
[[[.,.],.],.]
 => [1,2,3] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2 = 3 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [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 243 compositions to match this statistic)

Mapping

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
[[.,.],[.,.]]
 => [1,3,2] => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => 1 = 2 - 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] => 2 = 3 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => 1 = 2 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => 2 = 3 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => 1 = 2 - 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: St000522
(load all 4 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ 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
[.,[[.,.],.]]
 => [[[],[]]]
 => 1 = 2 - 1
[[.,.],[.,.]]
 => [[],[[]]]
 => 2 = 3 - 1
[[.,[.,.]],.]
 => [[[]],[]]
 => 2 = 3 - 1
[[[.,.],.],.]
 => [[],[],[]]
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [[[[[]]]]]
 => 1 = 2 - 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
[[.,[.,[.,.]]],.]
 => [[[[]]],[]]
 => 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: St000023
(load all 243 compositions to match this statistic)

Mapping

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
[[.,.],[.,.]]
 => [1,3,2] => 1 = 3 - 2
[[.,[.,.]],.]
 => [2,1,3] => 0 = 2 - 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] => 1 = 3 - 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => 1 = 3 - 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => 1 = 3 - 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => 1 = 3 - 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0 = 2 - 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => 1 = 3 - 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => 1 = 3 - 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => 0 = 2 - 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: St000201
(load all 22 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ 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] => [[.,[.,.]],.]
 => 1 = 2 - 1
[[.,.],[.,.]]
 => [3,1,2] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 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: St000243
(load all 13 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclic valleys and cyclic peaks of a permutation. This is given by the number of indices

$i$ such that

$\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices

$i$ such that

$\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.

Matching statistic: St000325
(load all 13 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.

Matching statistic: St000396
(load all 22 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ 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] => [[.,[.,.]],.]
 => 1 = 2 - 1
[[.,.],[.,.]]
 => [3,1,2] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 1 = 2 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 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: St000470
(load all 13 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

Matching statistic: St000537
(load all 3 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000537: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cutwidth of a graph. This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering

$\sigma$ is the maximum, among all the prefixes of

$\sigma$ , of the number of edges that have exactly one vertex in a prefix.

The following 686 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000758The length of the longest staircase fitting into an integer composition. St000862The number of parts of the shifted shape of a permutation. St000920The logarithmic height of a Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001270The bandwidth of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001644The dimension of a graph. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001962The proper pathwidth of a graph. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001037The number of inner corners of the upper path 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001638The book thickness of a graph. St001665The number of pure excedances of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001712The number of natural descents of a standard Young tableau. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001928The number of non-overlapping descents in a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000397The Strahler number of a rooted tree. St000759The smallest missing part in an integer partition. St000891The number of distinct diagonal sums of a permutation matrix. 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

$eA$ . St001471The magnitude of a Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000010The length of the partition. St000054The first entry of the permutation. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000216The absolute length of a permutation. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000628The balance of a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000701The protection number of a binary tree. St000703The number of deficiencies of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000935The number of ordered refinements of an integer partition. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001029The size of the core of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001116The game chromatic number of a graph. St001151The number of blocks with odd minimum. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001432The order dimension of the partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001484The number of singletons of an integer partition. St001486The number of corners of the ribbon associated with an integer composition. St001487The number of inner corners of a skew partition. St001494The Alon-Tarsi number of a graph. 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. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001597The Frobenius rank of a skew partition. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. 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). St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001963The tree-depth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000142The number of even parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000154The sum of the descent bottoms of a permutation. St000157The number of descents of a standard tableau. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000245The number of ascents of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000336The leg major index of a standard tableau. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000523The number of 2-protected nodes of a rooted tree. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000632The jump number of the poset. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000742The number of big ascents of a permutation after prepending zero. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of 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

$K[x]/(x^n)$ . St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001394The genus of a permutation. St001423The number of distinct cubes in a binary word. St001427The number of descents of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001470The cyclic holeyness of a permutation. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001728The number of invisible descents of a permutation. St001731The factorization defect of a permutation. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001823The Stasinski-Voll length of a signed permutation. St001896The number of right descents of a signed permutations. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000568The hook number of a binary tree. St000624The normalized sum of the minimal distances to a greater element. St000619The number of cyclic descents of a permutation. St000291The number of descents of a binary word. 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. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001424The number of distinct squares in a binary word. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000886The number of permutations with the same antidiagonal sums. St000933The number of multipartitions of sizes given by an integer partition. St000982The length of the longest constant subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001128The exponens consonantiae of a partition. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001642The Prague dimension of a graph. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000462The major index minus the number of excedences of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000565The major index of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000646The number of big ascents of a permutation. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000809The reduced reflection length of the permutation. St000845The maximal number of elements covered by an element in a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000919The number of maximal left branches of a binary tree. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001485The modular major index of a binary word. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001592The maximal number of simple paths between any two different vertices of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000640The rank of the largest boolean interval in a poset. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000454The largest eigenvalue of a graph if it is integral. St000444The length of the maximal rise of a Dyck path. St001060The distinguishing index of a graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000741The Colin de Verdière graph invariant. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000445The number of rises of length 1 of a Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000013The height of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000264The girth of a graph, which is not a tree. St000617The number of global maxima of a Dyck path. 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. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000871The number of very big ascents of a permutation. 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. St000252The number of nodes of degree 3 of a binary tree. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000025The number of initial rises of a Dyck path. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000312The number of leaves in a graph. St000636The hull number of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. 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

$eA$ . 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: 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

$eA$ . St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000482The (zero)-forcing number of a graph. St000778The metric dimension of a graph. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension 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. 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001199The dominant 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001946The number of descents in a parking function. St000091The descent variation of a composition. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000233The number of nestings of a set partition. St000365The number of double ascents of a permutation. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001160The number of proper blocks (or intervals) of a permutations. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001307The number of induced stars on four vertices in a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001578The minimal number of edges to add or remove to make a graph a line graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001715The number of non-records in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001964The interval resolution global dimension of a poset. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001488The number of corners of a skew partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001490The number of connected components of a skew partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001399The distinguishing number of a poset. St001545The second Elser number of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000680The Grundy value for Hackendot on posets. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function 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. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. 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. 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. 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. 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. 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. 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. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001692The number of vertices with higher degree than the average degree in 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. 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. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. 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. 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. 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. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001108The 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001162The minimum jump of a permutation. 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. 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. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001363The Euler characteristic of a graph according to Knill. 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. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001590The crossing number of a perfect matching. St001645The pebbling number of a connected 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. 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. 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. 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. 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. 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. St001084The number of occurrences of the vincular pattern |1-23 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. 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. 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. 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. 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. 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. 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. 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. 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.