- FindStat

Your data matches 153 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000522
(load all 11 compositions to match this statistic)

Mapping

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
[[[]],[]]
 => 2 = 3 - 1
[[[],[]]]
 => 1 = 2 - 1
[[[[]]]]
 => 1 = 2 - 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
[[[],[[]]]]
 => 2 = 3 - 1
[[[[]],[]]]
 => 2 = 3 - 1
[[[[],[]]]]
 => 1 = 2 - 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: St000092
(load all 21 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
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
[[],[]]
 => [[.,.],.]
 => [1,2] => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 2 = 3 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 2 = 3 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2 = 3 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 2 = 3 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2 = 3 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 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 15 compositions to match this statistic)

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ 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
[[[]],[]]
 => [[.,.],[.,.]]
 => [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: St000023
(load all 15 compositions to match this statistic)

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ 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
[[[]],[]]
 => [[.,.],[.,.]]
 => [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: St000891

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct diagonal sums of a permutation matrix. For example, the sums of the diagonals of the matrix

$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$ are

$(1,0,1,0,2,0)$ , so the statistic is

$3$ .

Matching statistic: St001200

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St000201

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
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
[[],[]]
 => [[.,.],.]
 => [1,2] => [.,[.,.]]
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [[.,[.,.]],.]
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 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 2 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00223: Permutations —runsort⟶ 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 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [3,1,2] => [1,2,3] => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => [2,3,1] => [1,2,3] => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => 1 = 2 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [1,4,2,3] => 2 = 3 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [1,3,2,4] => 2 = 3 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,2,3,4] => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 1 = 2 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,2,3,4,5] => 1 = 2 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,2,5,3,4] => 2 = 3 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,5,2,4,3] => 2 = 3 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,4,2,3,5] => 2 = 3 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,4,5,2,3] => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,3,4,2,5] => 2 = 3 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,2,4,3,5] => 2 = 3 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,4,2,3,5] => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,3,4,5] => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,3,2,5,4] => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => 1 = 2 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 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 3 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
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
[[],[]]
 => [[.,.],.]
 => [1,2] => [.,[.,.]]
 => 1 = 2 - 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [.,[.,[.,.]]]
 => 1 = 2 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [[.,.],[.,.]]
 => 2 = 3 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [[.,[.,.]],.]
 => 1 = 2 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [[[.,.],.],.]
 => 1 = 2 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 2 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 2 = 3 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 1 = 2 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 2 = 3 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 1 = 2 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 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.

The following 143 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000470The number of runs in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001489The maximum of the number of descents and the number of inverse descents. 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. 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 [[.,.],[.,. St000353The number of inner valleys of a permutation. St000662The staircase size of the code 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. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001665The number of pure excedances of a permutation. 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. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. 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. St000292The number of ascents 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. 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000233The number of nestings of a set partition. St000650The number of 3-rises of a permutation. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001488The number of corners of a skew partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of 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. 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. St000944The 3-degree of an integer 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. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001330The hat guessing number of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000264The girth of a graph, which is not a tree. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001060The distinguishing index of a graph. St001645The pebbling number of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001644The dimension of a graph. St000822The Hadwiger number of the graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001626The number of maximal proper sublattices of a lattice. St001734The lettericity of a graph. St000096The number of spanning trees of a graph. 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. 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. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001828The Euler characteristic of a graph. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. 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. St001556The number of inversions of the third entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. 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. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation.