- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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

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

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: St000701

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1,0]
 => [.,.]
 => 1
[.,[.,.]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => 2
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
 => ? = 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
 => ? = 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
 => ? = 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
 => ? = 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => ? = 2
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
 => ? = 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
 => ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => ? = 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [[[.,.],[[.,[.,[.,[.,.]]]],.]],.]
 => ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
 => ? = 2
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [[[.,.],.],[[.,[.,[.,[.,.]]]],.]]
 => ? = 2
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
 => ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
 => ? = 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
 => ? = 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => ? = 2
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => ? = 2
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
 => [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]]
 => ? = 2
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => ? = 2
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 2
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
 => ? = 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
 => ? = 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[.,[[.,[.,[.,.]]],.]],.],.],.]
 => ? = 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[.,.],.],[.,[[.,[.,[.,.]]],.]]]
 => ? = 2
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
 => ? = 1
[.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[[.,[.,[.,.]]],.],.]],.],.]
 => ? = 1
[.,[.,[[[.,.],.],[.,[[.,.],.]]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [[.,[.,[[.,.],[.,[.,[.,.]]]]]],.]
 => ? = 1
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
 => ? = 1
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [[.,[[[[.,[.,[.,.]]],.],.],.]],.]
 => ? = 1
[.,[.,[[[.,.],[.,.]],[[.,.],.]]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [[.,[[[.,.],[.,[.,[.,.]]]],.]],.]
 => ? = 1
[.,[.,[[[[.,.],.],.],[[.,.],.]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [[.,[[[.,.],.],[.,[.,[.,.]]]]],.]
 => ? = 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => ? = 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [[[.,.],[[[.,[.,[.,.]]],.],.]],.]
 => ? = 1
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [[[.,.],.],[[[.,[.,[.,.]]],.],.]]
 => ? = 2
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [[[[.,.],[[.,[.,[.,.]]],.]],.],.]
 => ? = 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [[[.,[.,.]],[[.,[.,[.,.]]],.]],.]
 => ? = 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
 => ? = 1
[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
 => ? = 2
[.,[.,[[.,[[[.,.],[.,.]],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
 => [[.,[[.,.],.]],[[.,[.,[.,.]]],.]]
 => ? = 2
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
 => ? = 2
[.,[.,[[.,[[[[.,.],.],.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [[[.,.],[[.,[.,[.,.]]],.]],[.,.]]
 => ? = 2
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
 => ? = 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
 => ? = 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,[.,[.,.]]]],.],.]
 => ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
 => ? = 1
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],[.,[.,[.,.]]]],.]
 => ? = 1
[.,[.,[[[[[.,.],.],.],[.,.]],.]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
 => ? = 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
 => [[.,[.,[[.,.],.]]],[.,[.,[.,.]]]]
 => ? = 2
[.,[.,[[[[[.,.],.],[.,.]],.],.]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
 => [[.,[[.,.],[.,[.,[.,.]]]]],[.,.]]
 => ? = 2
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => ? = 2
[.,[.,[[[[[.,.],[.,.]],.],.],.]]]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
 => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => ? = 2

Description

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

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: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%

Values

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

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: 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: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

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

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: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[.,.]
 => [1] => [1,0]
 => [1,1,0,0]
 => 1
[.,[.,.]]
 => [2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[[.,.],.]
 => [1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[[.,.],[.,.]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [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
[.,[.,[.,[[.,.],.]]]]
 => [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
[.,[.,[[.,.],[.,.]]]]
 => [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
[.,[.,[[.,[.,.]],.]]]
 => [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
[.,[.,[[[.,.],.],.]]]
 => [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
[.,[[.,.],[.,[.,.]]]]
 => [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
[.,[[.,.],[[.,.],.]]]
 => [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
[.,[[.,[.,.]],[.,.]]]
 => [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
[.,[[[.,.],.],[.,.]]]
 => [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
[.,[[.,[.,[.,.]]],.]]
 => [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
[.,[[.,[[.,.],.]],.]]
 => [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
[.,[[[.,.],[.,.]],.]]
 => [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
[.,[[[.,[.,.]],.],.]]
 => [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
[.,[[[[.,.],.],.],.]]
 => [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
[[.,.],[.,[.,[.,.]]]]
 => [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
[[.,.],[.,[[.,.],.]]]
 => [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
[[.,.],[[.,.],[.,.]]]
 => [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
[[.,.],[[.,[.,.]],.]]
 => [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
[[.,.],[[[.,.],.],.]]
 => [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
[[.,[.,.]],[.,[.,.]]]
 => [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
[[.,[.,.]],[[.,.],.]]
 => [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
[[[.,.],.],[.,[.,.]]]
 => [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
[[[.,.],.],[[.,.],.]]
 => [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
[[.,[.,[.,.]]],[.,.]]
 => [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
[[.,[[.,.],.]],[.,.]]
 => [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
[[[.,.],[.,.]],[.,.]]
 => [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
[[[.,[.,.]],.],[.,.]]
 => [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
[[[[.,.],.],.],[.,.]]
 => [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
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 2
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 2
[.,[.,[[[.,.],[.,.]],.]]]
 => [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]
 => ? = 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]
 => ? = 2
[.,[.,[[[[.,.],.],.],.]]]
 => [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]
 => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 2
[.,[[.,[.,.]],[.,[.,.]]]]
 => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 2
[.,[[.,[[.,.],[.,.]]],.]]
 => [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]
 => ? = 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]
 => ? = 2
[.,[[.,[[[.,.],.],.]],.]]
 => [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]
 => ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
 => [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]
 => ? = 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]
 => ? = 2
[.,[[[.,[.,.]],[.,.]],.]]
 => [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]
 => ? = 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]
 => ? = 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]
 => ? = 2
[.,[[[.,[[.,.],.]],.],.]]
 => [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]
 => ? = 2
[.,[[[[.,.],[.,.]],.],.]]
 => [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]
 => ? = 2
[.,[[[[.,[.,.]],.],.],.]]
 => [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]
 => ? = 2
[.,[[[[[.,.],.],.],.],.]]
 => [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]
 => ? = 2
[[.,.],[.,[.,[.,[.,.]]]]]
 => [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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 2
[[.,.],[[.,.],[.,[.,.]]]]
 => [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]
 => ? = 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]
 => ? = 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]
 => ? = 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: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%

Values

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

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: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

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

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: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

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

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: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

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

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.