- FindStat

Your data matches 135 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000061
(load all 10 compositions to match this statistic)

Mapping

St000061: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of nodes on the left branch of a binary tree. Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path.

Matching statistic: St000011
(load all 27 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000025
(load all 16 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000056
(load all 114 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The decomposition (or block) number of a permutation. For

$\pi \in \mathcal{S}_n$ , this is given by

$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].

Matching statistic: St000084
(load all 4 compositions to match this statistic)

Mapping

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of subtrees.

Matching statistic: St000991
(load all 92 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].

Matching statistic: St000234
(load all 114 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St000439
(load all 19 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St000989
(load all 27 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of final rises of a permutation. For a permutation

$\pi$ of length

$n$ , this is the maximal

$k$ such that

$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$ Equivalently, this is

$n-1$ minus the position of the last descent [[St000653]].

Matching statistic: St001640
(load all 12 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000273The domination number of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000717The number of ordinal summands of a poset. St000843The decomposition number of a perfect matching. St000906The length of the shortest maximal chain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000133The "bounce" of a permutation. St000203The number of external nodes of a binary tree. St000237The number of small exceedances. St000288The number of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000738The first entry in the last row of a standard tableau. 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. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000504The cardinality of the first block of a set partition. St000553The number of blocks of a graph. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

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

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

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001829The common independence number of a graph. St001963The tree-depth of a graph. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000502The number of successions of a set partitions. St000536The pathwidth of a graph. St000734The last entry in the first row of a standard tableau. St000883The number of longest increasing subsequences of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

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

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000648The number of 2-excedences of a permutation. St000924The number of topologically connected components of a perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000260The radius of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000181The number of connected components of the Hasse diagram for the poset. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000259The diameter of a connected graph. St001889The size of the connectivity set of a signed permutation. St001862The number of crossings of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. 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.