- FindStat

Your data matches 218 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000805
(load all 9 compositions to match this statistic)

Mapping

St000805: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of peaks of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.

Matching statistic: St000807
(load all 9 compositions to match this statistic)

Mapping

St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the heights of the valleys of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.

Matching statistic: St000128
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000128: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[.,[.,.]],.]]]}}} in a binary tree. [[oeis:A159769]] counts binary trees avoiding this pattern.

Matching statistic: St000129
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000129: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[[.,.],.],.]]]}}} in a binary tree. [[oeis:A159770]] counts binary trees avoiding this pattern.

Matching statistic: St000132
(load all 6 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,[[.,.],.]]]}}} in a binary tree. [[oeis:A159773]] counts binary trees avoiding this pattern.

Matching statistic: St000785
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000785: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => ([],1)
 => 1
[1,1] => [2] => [1,1] => ([(0,1)],2)
 => 1
[2] => [1] => [1] => ([],1)
 => 1
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2] => [1,1] => [2] => ([],2)
 => 1
[2,1] => [1,1] => [2] => ([],2)
 => 1
[3] => [1] => [1] => ([],1)
 => 1
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
 => 1
[1,2,1] => [1,1,1] => [3] => ([],3)
 => 1
[1,3] => [1,1] => [2] => ([],2)
 => 1
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,2] => [2] => [1,1] => ([(0,1)],2)
 => 1
[3,1] => [1,1] => [2] => ([],2)
 => 1
[4] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,2] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,1,2,1] => [2,1,1] => [1,3] => ([(2,3)],4)
 => 2
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
 => 1
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,3,1] => [1,1,1] => [3] => ([],3)
 => 1
[1,4] => [1,1] => [2] => ([],2)
 => 1
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,2] => [1,1,1] => [3] => ([],3)
 => 1
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
 => 1
[2,3] => [1,1] => [2] => ([],2)
 => 1
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2] => [1,1] => [2] => ([],2)
 => 1
[4,1] => [1,1] => [2] => ([],2)
 => 1
[5] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,2] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,2,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,3] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,3,1] => [2,1,1] => [1,3] => ([(2,3)],4)
 => 2
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
 => 1
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,1,2] => [1,1,1,1] => [4] => ([],4)
 => 1
[1,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,2,3] => [1,1,1] => [3] => ([],3)
 => 1
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,2] => [1,1,1] => [3] => ([],3)
 => 1
[1,4,1] => [1,1,1] => [3] => ([],3)
 => 1
[1,5] => [1,1] => [2] => ([],2)
 => 1
[2,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,2,1] => [1,1,1,1] => [4] => ([],4)
 => 1

Description

The number of distinct colouring schemes of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ .

Matching statistic: St000127
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
St000127: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[[.,.],.]]]]}}} in a binary tree. [[oeis:A159768]] counts binary trees avoiding this pattern.

Matching statistic: St000131
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000131: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[[[[.,.],.],.],.]]}}} in a binary tree. [[oeis:A159772]] counts binary trees avoiding this pattern.

Matching statistic: St000966
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of peaks minus the global dimension of the corresponding LNakayama algebra.

Matching statistic: St000243
(load all 2 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => [1] => ? = 1
[1,1] => [2] => [1,1,0,0]
 => [1,2] => 1
[2] => [1] => [1,0]
 => [1] => ? = 1
[1,1,1] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 1
[1,2] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[3] => [1] => [1,0]
 => [1] => ? = 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,3] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[2,2] => [2] => [1,1,0,0]
 => [1,2] => 1
[3,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[4] => [1] => [1,0]
 => [1] => ? = 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,4] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[2,3] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[3,2] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[4,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[5] => [1] => [1,0]
 => [1] => ? = 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 2
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,5] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2
[2,2,2] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 1
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[2,4] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[6] => [1] => [1,0]
 => [1] => ? = 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => ? ∊ {2,2}
[7] => [1] => [1,0]
 => [1] => ? ∊ {2,2}

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.

The following 208 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000649The number of 3-excedences of a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000183The side length of the Durfee square of an integer partition. St000630The length of the shortest palindromic decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St001732The number of peaks visible from the left. 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

$K[x]/(x^n)$ . St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000666The number of right tethers of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001571The Cartan determinant of the integer partition. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001165Number of simple modules with even projective dimension 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001890The maximum magnitude of the Möbius function of a poset. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St000664The number of right ropes of a permutation. St001715The number of non-records in a permutation. St001513The number of nested exceedences of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001344The neighbouring number of a permutation. St001389The number of partitions of the same length below the given integer partition. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. 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. St000570The Edelman-Greene number of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000879The number of long braid edges in the graph of braid moves of a permutation. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000405The number of occurrences of the pattern 1324 in a permutation. St001282The number of graphs with the same chromatic polynomial. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000286The number of connected components of the complement of a graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St000914The sum of the values of the Möbius function of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000366The number of double descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001964The interval resolution global dimension of a poset. St001665The number of pure excedances of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001877Number of indecomposable injective modules with projective dimension 2. St000260The radius of a connected graph. St000516The number of stretching pairs of a permutation. St000456The monochromatic index of a connected graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. 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

$c_{n−1}$ . St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000889The number of alternating sign matrices with the same antidiagonal sums. St001520The number of strict 3-descents. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000326The position of the first one in a binary word after appending a 1 at the end. St000068The number of minimal elements in a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000297The number of leading ones in a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000878The number of ones minus the number of zeros of a binary word. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000657The smallest part of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001260The permanent of an alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000629The defect of a binary word. St000894The trace of an alternating sign matrix. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the 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. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001429The number of negative entries in a signed permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching.