- FindStat

Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001107
(load all 17 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

Matching statistic: St000974
(load all 3 compositions to match this statistic)

Mapping

St000974: Ordered trees ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the trunk of an ordered tree. This is the length of the path from the root to the first vertex which has not exactly one child.

Matching statistic: St000674

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.

Matching statistic: St000993

Mapping

Mp00246: Ordered trees —rotate⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%

Values

[[],[]]
 => [[],[]]
 => [1,0,1,0]
 => [1]
 => ? = 0 + 1
[[[]]]
 => [[[]]]
 => [1,1,0,0]
 => []
 => ? = 2 + 1
[[],[],[]]
 => [[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1 = 0 + 1
[[],[[]]]
 => [[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => 1 = 0 + 1
[[[]],[]]
 => [[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 0 + 1
[[[],[]]]
 => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 2 = 1 + 1
[[[[]]]]
 => [[[[]]]]
 => [1,1,1,0,0,0]
 => []
 => ? = 3 + 1
[[],[],[],[]]
 => [[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 1 = 0 + 1
[[],[],[[]]]
 => [[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1 = 0 + 1
[[],[[]],[]]
 => [[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1 = 0 + 1
[[],[[],[]]]
 => [[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1 = 0 + 1
[[],[[[]]]]
 => [[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 = 0 + 1
[[[]],[],[]]
 => [[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1 = 0 + 1
[[[]],[[]]]
 => [[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1 = 0 + 1
[[[],[]],[]]
 => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1 = 0 + 1
[[[[]]],[]]
 => [[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0 + 1
[[[],[],[]]]
 => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2 = 1 + 1
[[[],[[]]]]
 => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 2 = 1 + 1
[[[[],[]]]]
 => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3 = 2 + 1
[[[[[]]]]]
 => [[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 4 + 1
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 1 = 0 + 1
[[[]],[[[]]]]
 => [[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => [[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1 = 0 + 1
[[[[]]],[[]]]
 => [[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => [[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[[[],[],[],[]]]
 => [[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 2 = 1 + 1
[[[],[],[[]]]]
 => [[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 2 = 1 + 1
[[[],[[]],[]]]
 => [[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 2 = 1 + 1
[[[],[[],[]]]]
 => [[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 2 = 1 + 1
[[[],[[[]]]]]
 => [[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2 = 1 + 1
[[[[]],[],[]]]
 => [[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 2 = 1 + 1
[[[[]],[[]]]]
 => [[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2 = 1 + 1
[[[[],[]],[]]]
 => [[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 2 = 1 + 1
[[[[[[]]]]]]
 => [[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 5 + 1
[[],[],[],[],[],[]]
 => [[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 0 + 1
[[],[],[],[],[[]]]
 => [[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => ? = 0 + 1
[[],[],[],[[]],[]]
 => [[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => ? = 0 + 1
[[],[],[],[[],[]]]
 => [[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => ? = 0 + 1
[[],[],[[]],[],[]]
 => [[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => ? = 0 + 1
[[],[],[[]],[[]]]
 => [[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => ? = 0 + 1
[[],[],[[],[]],[]]
 => [[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => ? = 0 + 1
[[],[[]],[],[],[]]
 => [[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 0 + 1
[[],[[]],[],[[]]]
 => [[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => ? = 0 + 1
[[],[[]],[[]],[]]
 => [[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => ? = 0 + 1
[[],[[],[]],[],[]]
 => [[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 0 + 1
[[[],[],[],[]],[]]
 => [[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => ? = 0 + 1
[[[[[[]]]]],[]]
 => [[[[[[],[]]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[[[],[],[],[],[]]]
 => [[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => ? = 1 + 1
[[[],[],[],[[]]]]
 => [[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => ? = 1 + 1
[[[],[],[[]],[]]]
 => [[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => ? = 1 + 1
[[[],[[]],[],[]]]
 => [[[]],[],[],[[]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => ? = 1 + 1
[[[[[[[]]]]]]]
 => [[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 6 + 1
[[],[],[],[],[],[],[]]
 => [[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => ? = 0 + 1
[[],[],[],[],[],[[]]]
 => [[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => ? = 0 + 1
[[],[],[],[],[[]],[]]
 => [[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => ? = 0 + 1
[[],[],[],[],[[],[]]]
 => [[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => ? = 0 + 1
[[],[],[],[],[[[]]]]
 => [[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => ? = 0 + 1
[[],[],[],[[]],[],[]]
 => [[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => ? = 0 + 1
[[],[],[],[[]],[[]]]
 => [[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => ? = 0 + 1
[[],[],[],[[],[]],[]]
 => [[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => ? = 0 + 1
[[],[],[],[[[]]],[]]
 => [[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => ? = 0 + 1
[[],[],[],[[],[],[]]]
 => [[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => ? = 0 + 1
[[],[],[],[[],[[]]]]
 => [[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => ? = 0 + 1
[[],[],[],[[[]],[]]]
 => [[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => ? = 0 + 1
[[],[],[],[[[],[]]]]
 => [[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,2,1]
 => ? = 0 + 1
[[],[],[],[[[[]]]]]
 => [[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => ? = 0 + 1
[[],[],[[]],[],[],[]]
 => [[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => ? = 0 + 1
[[],[],[[]],[],[[]]]
 => [[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => ? = 0 + 1
[[],[],[[]],[[]],[]]
 => [[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1,1]
 => ? = 0 + 1
[[],[],[[]],[[],[]]]
 => [[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1,1]
 => ? = 0 + 1
[[],[],[[]],[[[]]]]
 => [[],[[]],[[[]]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,1,1]
 => ? = 0 + 1
[[],[],[[],[]],[],[]]
 => [[],[[],[]],[],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => ? = 0 + 1
[[],[],[[[]]],[],[]]
 => [[],[[[]]],[],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1,1,1]
 => ? = 0 + 1
[[],[],[[],[]],[[]]]
 => [[],[[],[]],[[]],[]]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1,1]
 => ? = 0 + 1
[[],[],[[[]]],[[]]]
 => [[],[[[]]],[[]],[]]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [6,4,4,1,1,1]
 => ? = 0 + 1
[[],[],[[],[],[]],[]]
 => [[],[[],[],[]],[],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,1]
 => ? = 0 + 1

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000264

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 19%●distinct values known / distinct values provided: 12%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St000907
(load all 2 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%

Values

[[],[]]
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 3 = 2 + 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 1 = 0 + 1
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1 = 0 + 1
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1 = 0 + 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 1 = 0 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 1 = 0 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 1 = 0 + 1
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1 = 0 + 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1 = 0 + 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 1 = 0 + 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 1 = 0 + 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => 2 = 1 + 1
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[],[[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[],[[[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[]],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[],[]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[],[[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[],[[[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[],[[[[]]]]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[]],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[[]],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[[]],[[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[[]],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[[],[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[]]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[],[]],[[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[]]],[[]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[],[[],[],[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[],[[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[]],[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[],[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[[[[]]]],[]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
 => ? = 0 + 1
[[],[],[[],[],[[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[[],[],[[],[[]],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[[],[],[[],[[],[]]]]
 => ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[],[[[]]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[]],[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[[],[],[[[]],[[]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[],[]],[]]]
 => ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[[]]],[]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[],[[[],[],[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[[[],[[]]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[[[[]],[]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
 => ? = 0 + 1
[[],[],[[[[],[]]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
 => ? = 0 + 1
[[],[],[[[[[]]]]]]
 => ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
 => ? = 0 + 1
[[],[[]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[],[[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[[]],[[]]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[[],[[]],[[],[]],[]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 0 + 1

Description

The number of maximal antichains of minimal length in a poset.

Matching statistic: St000461

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%

Values

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

Description

The rix statistic of a permutation. This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then $rix(w) := 0$ if $i = 1 < k$, $rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and $rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.

Matching statistic: St001091

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.

Matching statistic: St000160

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%

Values

[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1 = 0 + 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => 3 = 2 + 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2 = 1 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 4 = 3 + 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 3 = 2 + 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5 = 4 + 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 1 + 1
[[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[],[[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[],[[[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[]],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[],[]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[],[[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[],[[[[]]]]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[]],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[]],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[]],[[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[]],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[]]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[]],[[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[]]],[[]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[],[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[]],[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[],[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[[]]]],[]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[],[[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[[]],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[[],[]]]]
 => ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[],[[[]]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[]],[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[]],[[]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[],[]],[]]]
 => ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[[]]],[]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[],[],[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[],[[]]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[[]],[]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[[],[]]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[],[[[[[]]]]]]
 => ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[],[[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[[]],[[]]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
[[],[[]],[[],[]],[]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1

Description

The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].

Matching statistic: St000260
(load all 3 compositions to match this statistic)

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 16%●distinct values known / distinct values provided: 12%

Values

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

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000475The number of parts equal to 1 in a partition. St001933The largest multiplicity of a part in an integer partition. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000312The number of leaves in a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001479The number of bridges of a graph.