- FindStat

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000779

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The tier of a permutation. This is the number of elements

$i$ such that

$[i+1,k,i]$ is an occurrence of the pattern

$[2,3,1]$ . For example,

$[3,5,6,1,2,4]$ has tier

$2$ , with witnesses

$[3,5,2]$ (or

$[3,6,2]$ ) and

$[5,6,4]$ . According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].

Matching statistic: St000701
(load all 5 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The protection number of a binary tree. This is the minimal distance from the root to a leaf.

Matching statistic: St000920

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St000260

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 50%

Values

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

Description

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

Matching statistic: St001181
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001503

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001503: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.

Matching statistic: St001200
(load all 3 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001570

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%

Values

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

Description

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.

Matching statistic: St000264

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 0 + 3
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 0 + 3
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? = 0 + 3
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? = 0 + 3
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? = 0 + 3
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ? = 1 + 3
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 0 + 3
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 0 + 3
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 0 + 3
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 1 + 3
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 0 + 3
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 1 + 3
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 1 + 3
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 1 + 3
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 0 + 3
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 0 + 3
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 0 + 3
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 1 + 3
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 0 + 3
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 1 + 3
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 1 + 3
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 1 + 3
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 1 + 3
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 0 + 3
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 1 + 3
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,[.,.]],[[.,.],[.,.]]]
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,.],.],[[.,.],[.,.]]]
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,.],[.,.]],[.,[.,.]]]
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,.],[.,.]],[[.,.],.]]
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 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: St001060

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%

Values

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

Description

The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.