- FindStat

Your data matches 61 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000659
(load all 13 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: 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]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [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]
 => 2
[.,[[.,[.,.]],.]]
 => [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]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [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]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [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]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [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]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [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]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [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]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [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]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1

Description

The number of rises of length at least 2 of a Dyck path.

Matching statistic: St000919
(load all 12 compositions to match this statistic)

Mapping

Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000919: Binary trees ⟶ 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
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => [[[.,.],[.,.]],.]
 => 1
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => [[.,[.,[.,.]]],.]
 => 1
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => [.,[[[.,.],.],.]]
 => 1
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => [.,[[.,[.,.]],.]]
 => 1
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => [[.,.],[[.,.],.]]
 => 2
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => [.,[.,[[.,.],.]]]
 => 1
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => 1
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => [[.,[.,.]],[.,.]]
 => 1
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => [.,[[.,.],[.,.]]]
 => 1
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => [[.,.],[.,[.,.]]]
 => 1
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => [.,[.,[.,[.,.]]]]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [[[[[.,.],.],.],.],.]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [[[[.,[.,.]],.],.],.]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [[[.,[[.,.],.]],.],.]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [[[[.,.],[.,.]],.],.]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [[.,[[[.,.],.],.]],.]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => [[.,[[.,[.,.]],.]],.]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => [[[.,.],[[.,.],.]],.]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => [[.,[.,[[.,.],.]]],.]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => [[[[.,.],.],[.,.]],.]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => [[[.,[.,.]],[.,.]],.]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => [[[.,.],[.,[.,.]]],.]
 => 1
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [.,[[[[.,.],.],.],.]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [.,[[.,[[.,.],.]],.]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => [.,[[[.,.],[.,.]],.]]
 => 1
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[.,.],[[[.,.],.],.]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[.,.],.],[[.,.],.]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[[.,.],.],.],[.,.]]
 => 1

Description

The number of maximal left branches of a binary tree. A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.

Matching statistic: St000291

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,1,0,0]
 => [2] => 10 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1] => 11 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3] => 100 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,1] => 101 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,2] => 110 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1] => 101 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1] => 111 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1000 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 1001 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 1010 => 2
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 1001 => 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 1011 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 1101 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 1011 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => 10000 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => 10001 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => 10010 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => 10001 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => 10011 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 10100 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 10101 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => 10010 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 10110 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => 10001 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => 10011 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 10101 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => 10011 => 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 10111 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 11001 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 11010 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 11001 => 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 11011 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 10101 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 11101 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 10010 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 10110 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 10001 => 1

Description

The number of descents of a binary word.

Matching statistic: St001280
(load all 2 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000288
(load all 3 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,3,1] => 10 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => 10 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,3,2] => 01 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3,1,2] => 10 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 100 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 100 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 101 => 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 100 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 100 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 010 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 010 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 110 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 001 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 100 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 100 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 100 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 1000 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 1000 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 1000 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 1000 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 1010 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1010 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => 1010 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 1000 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 1000 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 1010 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 1000 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 0100 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 0100 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0101 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => 0100 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0100 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => 1100 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => 1100 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 0010 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0010 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => 1100 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => 0110 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 1010 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 1000 => 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,8,5,7] => ? => ? = 2
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,6,5,7,8] => ? => ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,6,2,7,8,5] => ? => ? = 2
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,2,8,5,7] => ? => ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,4,2,7,5,6,8] => ? => ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,4,7,2,8,5,6] => ? => ? = 2
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,5,6,8,4,7] => ? => ? = 2
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,5,4,6,8,7] => ? => ? = 3
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => ? => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,8,7] => ? => ? = 3
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,3,2,7,4,5,8,6] => ? => ? = 3
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,2,4,7,5,6,8] => ? => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ? => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5,8] => ? => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ? => ? = 2
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,6,8,2,4,5,7] => ? => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,3,2,7,8,4,5,6] => ? => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,2,6,8,7] => ? => ? = 3
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [3,1,4,5,7,2,6,8] => ? => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [3,1,4,5,2,8,6,7] => ? => ? = 3
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5,8] => ? => ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,2,5,7,6,8] => ? => ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,2,5,6,8,7] => ? => ? = 3
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [3,1,4,6,7,2,5,8] => ? => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,1,4,6,2,5,7,8] => ? => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [3,1,4,7,8,2,5,6] => ? => ? = 2
[[.,[.,.]],[[[.,[[.,.],.]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6,8] => ? => ? = 2
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ? => ? = 2
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ? => ? = 1
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,6,3,7,5,8] => ? => ? = 2
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,4,2,5,6,3,8,7] => ? => ? = 3
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,4,2,5,6,3,7,8] => ? => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,3,6,8,7] => ? => ? = 3
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,2,3,6,8,5,7] => ? => ? = 2
[[[.,[.,.]],[.,.]],[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [4,1,5,2,6,7,3,8] => ? => ? = 3
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,2,5,3,6,8,4,7] => ? => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,6,4,8,7] => ? => ? = 3
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,6,3,7,8,5] => ? => ? = 3
[[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,6,2,3,7,5,8] => ? => ? = 2
[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [4,1,5,6,2,7,8,3] => ? => ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [4,1,5,6,2,7,3,8] => ? => ? = 3
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,5,6,2,3,7,4,8] => ? => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ? => ? = 2
[[[.,.],[.,[[[.,.],.],.]]],[.,.]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,4,5,2,3,6,8,7] => ? => ? = 2
[[[.,.],[[.,.],[.,[.,.]]]],[.,.]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,2,6,7,3,8,5] => ? => ? = 3
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
 => [4,1,5,7,2,3,8,6] => ? => ? = 3
[[[.,[.,.]],[[[.,.],.],.]],[.,.]]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
 => [4,1,5,2,3,6,8,7] => ? => ? = 3
[[[[.,.],[.,.]],[.,[.,.]]],[.,.]]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => [1,5,2,6,7,3,8,4] => ? => ? = 3
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => [1,5,6,2,7,3,8,4] => ? => ? = 3
[[[.,.],[.,[[.,[.,.]],[.,.]]]],.]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,4,5,7,2,8,3,6] => ? => ? = 2

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000389

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,3,1] => 01 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => 10 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,3,2] => 01 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3,1,2] => 10 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 001 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 101 => 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 100 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 001 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 010 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 101 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 001 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 010 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 100 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 100 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 0001 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 0010 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 0101 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 0100 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 1001 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1010 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 0100 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 1010 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 0001 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 0010 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0101 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => 0010 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0100 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => 1001 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => 1010 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 0001 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0010 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => 0101 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 1001 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 0010 => 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,8,5,7] => ? => ? = 2
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,6,5,7,8] => ? => ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,6,2,7,8,5] => ? => ? = 2
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,4,7,2,5,8,6] => ? => ? = 2
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,2,8,5,7] => ? => ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,4,2,7,5,6,8] => ? => ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,4,7,2,8,5,6] => ? => ? = 2
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,5,6,8,4,7] => ? => ? = 2
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,5,4,6,8,7] => ? => ? = 3
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,5,2,6,4,8,7] => ? => ? = 3
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => ? => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,8,7] => ? => ? = 3
[[.,.],[[[.,[.,[.,.]]],.],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,3,6,7,2,4,8,5] => ? => ? = 2
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,3,2,7,4,5,8,6] => ? => ? = 3
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,2,4,7,5,6,8] => ? => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ? => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5,8] => ? => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ? => ? = 2
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,6,8,2,4,5,7] => ? => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,3,2,7,8,4,5,6] => ? => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,2,6,8,7] => ? => ? = 3
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [3,1,4,5,7,2,6,8] => ? => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [3,1,4,5,2,8,6,7] => ? => ? = 3
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5,8] => ? => ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,2,5,7,6,8] => ? => ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,2,5,6,8,7] => ? => ? = 3
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [3,1,4,6,7,2,5,8] => ? => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,1,4,6,2,5,7,8] => ? => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [3,1,4,7,8,2,5,6] => ? => ? = 2
[[.,[.,.]],[[[.,[[.,.],.]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6,8] => ? => ? = 2
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ? => ? = 2
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ? => ? = 1
[[[.,.],.],[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,6,3,7,8,5] => ? => ? = 2
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,6,3,7,5,8] => ? => ? = 2
[[[.,.],.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,8,5,7] => ? => ? = 2
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,4,2,5,6,3,8,7] => ? => ? = 3
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,4,2,5,6,3,7,8] => ? => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,3,6,8,7] => ? => ? = 3
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,2,3,6,8,5,7] => ? => ? = 2
[[[.,[.,.]],[.,.]],[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [4,1,5,2,6,7,3,8] => ? => ? = 3
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,2,5,3,6,8,4,7] => ? => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,6,4,8,7] => ? => ? = 3
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,6,3,7,8,5] => ? => ? = 3
[[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,6,2,3,7,5,8] => ? => ? = 2
[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [4,1,5,6,2,7,8,3] => ? => ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [4,1,5,6,2,7,3,8] => ? => ? = 3
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,5,6,2,3,7,4,8] => ? => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ? => ? = 2
[[[.,.],[.,[[[.,.],.],.]]],[.,.]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,4,5,2,3,6,8,7] => ? => ? = 2
[[[.,.],[[.,.],[.,[.,.]]]],[.,.]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,2,6,7,3,8,5] => ? => ? = 3

Description

The number of runs of ones of odd length in a binary word.

Matching statistic: St000390
(load all 3 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,3,1] => 01 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => 10 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,3,2] => 01 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3,1,2] => 10 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 001 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 101 => 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 100 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 001 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 010 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 101 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 001 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 010 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 100 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 100 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 0001 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 0010 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 0101 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 0100 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 1001 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1010 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 0100 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 1010 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 0001 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 0010 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0101 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => 0010 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0100 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => 1001 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => 1010 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 0001 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0010 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => 0101 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 1001 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 0010 => 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,8,5,7] => ? => ? = 2
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,6,5,7,8] => ? => ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,6,2,7,8,5] => ? => ? = 2
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,4,7,2,5,8,6] => ? => ? = 2
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,2,8,5,7] => ? => ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,4,2,7,5,6,8] => ? => ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,4,7,2,8,5,6] => ? => ? = 2
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,5,6,8,4,7] => ? => ? = 2
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,5,4,6,8,7] => ? => ? = 3
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,5,2,6,4,8,7] => ? => ? = 3
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => ? => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,8,7] => ? => ? = 3
[[.,.],[[[.,[.,[.,.]]],.],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,3,6,7,2,4,8,5] => ? => ? = 2
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,3,2,7,4,5,8,6] => ? => ? = 3
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,2,4,7,5,6,8] => ? => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ? => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5,8] => ? => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ? => ? = 2
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,6,8,2,4,5,7] => ? => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,3,2,7,8,4,5,6] => ? => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,2,6,8,7] => ? => ? = 3
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [3,1,4,5,7,2,6,8] => ? => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [3,1,4,5,2,8,6,7] => ? => ? = 3
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5,8] => ? => ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,2,5,7,6,8] => ? => ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,2,5,6,8,7] => ? => ? = 3
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [3,1,4,6,7,2,5,8] => ? => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,1,4,6,2,5,7,8] => ? => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [3,1,4,7,8,2,5,6] => ? => ? = 2
[[.,[.,.]],[[[.,[[.,.],.]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6,8] => ? => ? = 2
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ? => ? = 2
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ? => ? = 1
[[[.,.],.],[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,6,3,7,8,5] => ? => ? = 2
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,6,3,7,5,8] => ? => ? = 2
[[[.,.],.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,8,5,7] => ? => ? = 2
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,4,2,5,6,3,8,7] => ? => ? = 3
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,4,2,5,6,3,7,8] => ? => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,3,6,8,7] => ? => ? = 3
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,2,3,6,8,5,7] => ? => ? = 2
[[[.,[.,.]],[.,.]],[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [4,1,5,2,6,7,3,8] => ? => ? = 3
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,2,5,3,6,8,4,7] => ? => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,6,4,8,7] => ? => ? = 3
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,6,3,7,8,5] => ? => ? = 3
[[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,6,2,3,7,5,8] => ? => ? = 2
[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [4,1,5,6,2,7,8,3] => ? => ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [4,1,5,6,2,7,3,8] => ? => ? = 3
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,5,6,2,3,7,4,8] => ? => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ? => ? = 2
[[[.,.],[.,[[[.,.],.],.]]],[.,.]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,4,5,2,3,6,8,7] => ? => ? = 2
[[[.,.],[[.,.],[.,[.,.]]]],[.,.]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,2,6,7,3,8,5] => ? => ? = 3

Description

The number of runs of ones in a binary word.

Matching statistic: St000142

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,8,5,7] => ?
 => ? = 2
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,6,5,7,8] => ?
 => ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,6,2,7,8,5] => ?
 => ? = 2
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,2,8,5,7] => ?
 => ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,4,2,7,5,6,8] => ?
 => ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,4,7,2,8,5,6] => ?
 => ? = 2
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,5,6,8,4,7] => ?
 => ? = 2
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 3
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => ?
 => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,8,7] => ?
 => ? = 3
[[.,.],[[[.,[.,[.,.]]],.],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,3,2,7,4,5,8,6] => ?
 => ? = 3
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,2,4,7,5,6,8] => ?
 => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ?
 => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5,8] => ?
 => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ?
 => ? = 2
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,6,8,2,4,5,7] => ?
 => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,3,2,7,8,4,5,6] => ?
 => ? = 2
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,2,6,8,7] => ?
 => ? = 3
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [3,1,4,5,7,2,6,8] => ?
 => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [3,1,4,5,2,8,6,7] => ?
 => ? = 3
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5,8] => ?
 => ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,2,5,7,6,8] => ?
 => ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,2,5,6,8,7] => ?
 => ? = 3
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [3,1,4,6,7,2,5,8] => ?
 => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,1,4,6,2,5,7,8] => ?
 => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [3,1,4,7,8,2,5,6] => ?
 => ? = 2
[[.,[.,.]],[[[.,[[.,.],.]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6,8] => ?
 => ? = 2
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ?
 => ? = 2
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ?
 => ? = 1
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,6,3,7,5,8] => ?
 => ? = 2
[[[.,.],.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[[[.,.],.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,4,2,5,6,3,8,7] => ?
 => ? = 3
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,4,2,5,6,3,7,8] => ?
 => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,3,6,8,7] => ?
 => ? = 3
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,2,3,6,8,5,7] => ?
 => ? = 2
[[[.,[.,.]],[.,.]],[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [4,1,5,2,6,7,3,8] => ?
 => ? = 3
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,2,5,3,6,8,4,7] => ?
 => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,6,4,8,7] => ?
 => ? = 3
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,6,3,7,8,5] => ?
 => ? = 3
[[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,6,2,3,7,5,8] => ?
 => ? = 2
[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [4,1,5,6,2,7,8,3] => ?
 => ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [4,1,5,6,2,7,3,8] => ?
 => ? = 3
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,5,6,2,3,7,4,8] => ?
 => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ?
 => ? = 2

Description

The number of even parts of a partition.

Matching statistic: St000566

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,8,5,7] => ?
 => ? = 2
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,6,5,7,8] => ?
 => ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,6,2,7,8,5] => ?
 => ? = 2
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,2,8,5,7] => ?
 => ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,4,2,7,5,6,8] => ?
 => ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,4,7,2,8,5,6] => ?
 => ? = 2
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,5,6,8,4,7] => ?
 => ? = 2
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 3
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => ?
 => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,8,7] => ?
 => ? = 3
[[.,.],[[[.,[.,[.,.]]],.],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,3,2,7,4,5,8,6] => ?
 => ? = 3
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,2,4,7,5,6,8] => ?
 => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ?
 => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5,8] => ?
 => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ?
 => ? = 2
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,6,8,2,4,5,7] => ?
 => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,3,2,7,8,4,5,6] => ?
 => ? = 2
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,2,6,8,7] => ?
 => ? = 3
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [3,1,4,5,7,2,6,8] => ?
 => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [3,1,4,5,2,8,6,7] => ?
 => ? = 3
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5,8] => ?
 => ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,2,5,7,6,8] => ?
 => ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,2,5,6,8,7] => ?
 => ? = 3
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [3,1,4,6,7,2,5,8] => ?
 => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,1,4,6,2,5,7,8] => ?
 => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [3,1,4,7,8,2,5,6] => ?
 => ? = 2
[[.,[.,.]],[[[.,[[.,.],.]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6,8] => ?
 => ? = 2
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ?
 => ? = 2
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ?
 => ? = 1
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,6,3,7,5,8] => ?
 => ? = 2
[[[.,.],.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[[[.,.],.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,4,2,5,6,3,8,7] => ?
 => ? = 3
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,4,2,5,6,3,7,8] => ?
 => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,3,6,8,7] => ?
 => ? = 3
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,2,3,6,8,5,7] => ?
 => ? = 2
[[[.,[.,.]],[.,.]],[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [4,1,5,2,6,7,3,8] => ?
 => ? = 3
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,2,5,3,6,8,4,7] => ?
 => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,6,4,8,7] => ?
 => ? = 3
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,6,3,7,8,5] => ?
 => ? = 3
[[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,6,2,3,7,5,8] => ?
 => ? = 2
[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [4,1,5,6,2,7,8,3] => ?
 => ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [4,1,5,6,2,7,3,8] => ?
 => ? = 3
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,5,6,2,3,7,4,8] => ?
 => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ?
 => ? = 2

Description

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if

$\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is

$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$

Matching statistic: St001251

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,8,5,7] => ?
 => ? = 2
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,6,5,7,8] => ?
 => ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,6,2,7,8,5] => ?
 => ? = 2
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,2,8,5,7] => ?
 => ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,4,2,7,5,6,8] => ?
 => ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,3,4,7,2,8,5,6] => ?
 => ? = 2
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,5,6,8,4,7] => ?
 => ? = 2
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 3
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => ?
 => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,8,7] => ?
 => ? = 3
[[.,.],[[[.,[.,[.,.]]],.],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,3,2,7,4,5,8,6] => ?
 => ? = 3
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,3,2,4,7,5,6,8] => ?
 => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ?
 => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,6,7,2,4,5,8] => ?
 => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ?
 => ? = 2
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
 => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,6,8,2,4,5,7] => ?
 => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,3,2,7,8,4,5,6] => ?
 => ? = 2
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,5,2,6,8,7] => ?
 => ? = 3
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [3,1,4,5,7,2,6,8] => ?
 => ? = 2
[[.,[.,.]],[.,[[[.,.],[.,.]],.]]]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [3,1,4,5,2,8,6,7] => ?
 => ? = 3
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5,8] => ?
 => ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,2,5,7,6,8] => ?
 => ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,2,5,6,8,7] => ?
 => ? = 3
[[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [3,1,4,6,7,2,5,8] => ?
 => ? = 2
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,1,4,6,2,5,7,8] => ?
 => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [3,1,4,7,8,2,5,6] => ?
 => ? = 2
[[.,[.,.]],[[[.,[[.,.],.]],.],.]]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6,8] => ?
 => ? = 2
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ?
 => ? = 2
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ?
 => ? = 1
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,6,3,7,5,8] => ?
 => ? = 2
[[[.,.],.],[[[.,.],[.,.]],[.,.]]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[[[.,.],.],[[.,[[.,.],[.,.]]],.]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,4,2,5,6,3,8,7] => ?
 => ? = 3
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
 => [1,4,2,5,6,3,7,8] => ?
 => ? = 2
[[[.,[.,.]],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,3,6,8,7] => ?
 => ? = 3
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,2,3,6,8,5,7] => ?
 => ? = 2
[[[.,[.,.]],[.,.]],[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [4,1,5,2,6,7,3,8] => ?
 => ? = 3
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,2,5,3,6,8,4,7] => ?
 => ? = 2
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,6,4,8,7] => ?
 => ? = 3
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,6,3,7,8,5] => ?
 => ? = 3
[[[.,.],[[.,[.,.]],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,6,2,3,7,5,8] => ?
 => ? = 2
[[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [4,1,5,6,2,7,8,3] => ?
 => ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [4,1,5,6,2,7,3,8] => ?
 => ? = 3
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,5,6,2,3,7,4,8] => ?
 => ? = 2
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6,8] => ?
 => ? = 2

Description

The number of parts of a partition that are not congruent 1 modulo 3.

The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001252Half the sum of the even parts of a partition. St001657The number of twos in an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000157The number of descents of a standard tableau. St000251The number of nonsingleton blocks of a set partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000834The number of right outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000884The number of isolated descents of a permutation. St001737The number of descents of type 2 in a permutation. St000665The number of rafts of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St000662The staircase size of the code of a permutation. St000354The number of recoils of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St000245The number of ascents of a permutation. St001427The number of descents of a signed permutation. St000647The number of big descents of a permutation. St000021The number of descents of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000353The number of inner valleys of a permutation. St000023The number of inner peaks of a permutation. St000711The number of big exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001896The number of right descents of a signed permutations. St001597The Frobenius rank of a skew partition. St001946The number of descents in a parking function. St000920The logarithmic height of a Dyck path.