- FindStat

Your data matches 52 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001124

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000159

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St000318

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000340

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[.,[.,[.,.]]]
 => [3,2,1] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
 => [8,9,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
 => [7,8,9,6,5,4,3,2,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
 => [3,4,5,6,7,8,9,2,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
 => [9,10,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
 => [6,7,8,9,10,5,4,3,2,1] => [5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[[.,.],[[[[[.,.],.],.],.],.]]]]
 => [3,5,6,7,8,9,4,2,1] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
 => [3,4,5,6,7,8,9,10,2,1] => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
 => [8,7,9,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
 => [8,7,6,9,5,4,3,2,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
 => [8,7,6,5,9,4,3,2,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
 => [8,7,6,5,4,9,3,2,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
 => [8,7,6,5,4,3,9,2,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
 => [8,7,6,5,4,3,2,9,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => [8,7,6,5,4,3,2,1,9] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
 => [9,8,10,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]]
 => [6,8,7,9,5,4,3,2,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
 => [9,8,7,10,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]]
 => [5,8,7,6,9,4,3,2,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
 => [9,8,7,6,10,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]]
 => [4,8,7,6,5,9,3,2,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
 => [9,8,7,6,5,10,4,3,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]]
 => [3,8,7,6,5,4,9,2,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
 => [9,8,7,6,5,4,10,3,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]]
 => [2,8,7,6,5,4,3,9,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
 => [9,8,7,6,5,4,3,10,2,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,8,7,6,5,4,3,2,9] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
 => [9,8,7,6,5,4,3,2,10,1] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
 => [9,8,7,6,5,4,3,2,1,10] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
 => [1,3,5,7,9,10,8,6,4,2] => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
 => [1,6,7,5,8,4,9,3,10,2] => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]
 => [5,6,4,7,3,8,2,9,1,10] => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
 => [3,2,4,5,6,7,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
 => [2,1,3,4,5,6,7,8,10,9] => [8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
 => [2,4,3,5,6,7,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
 => [3,4,2,5,6,7,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
 => [3,2,4,5,6,7,8,9,10,1] => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,5,6,9,8,7] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[[[[[.,.],.],.],.],.],[[.,[.,.]],.]]
 => [1,2,3,4,5,8,7,9,6] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[[[[.,.],.],.],.],[[[.,[.,.]],.],.]]
 => [1,2,3,4,7,6,8,9,5] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[[[.,.],.],.],[[[[.,[.,.]],.],.],.]]
 => [1,2,3,6,5,7,8,9,4] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[[.,.],.],[[[[[.,[.,.]],.],.],.],.]]
 => [1,2,5,4,6,7,8,9,3] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[.,.],[[[[[[.,[.,.]],.],.],.],.],.]]
 => [1,4,3,5,6,7,8,9,2] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]
 => [7,8,6,5,4,3,2,1,9] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
 => [10,9,8,7,6,5,4,3,2,1,11] => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
 => [8,9,7,6,5,4,3,2,1,10] => [3,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000259

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

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

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St000260

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

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

Description

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

Matching statistic: St000302

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

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

Description

The determinant of the distance matrix of a connected graph.

Matching statistic: St000466

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

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

Description

The Gutman (or modified Schultz) index of a connected graph. This is

$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$ where

$d(u)$ is the degree of vertex

$u$ and

$d(u,v)$ is the distance between vertices

$u$ and

$v$ . For trees on

$n$ vertices, the modified Schultz index is related to the Wiener index via

$S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].

Matching statistic: St000467

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

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

Description

The hyper-Wiener index of a connected graph. This is

$\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.$

Matching statistic: St000771

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

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

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000455The second largest eigenvalue of a graph if it is integral. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001625The Möbius invariant of a lattice. St001545The second Elser number of a connected graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001488The number of corners of a skew partition. St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St001846The number of elements which do not have a complement in the lattice. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000068The number of minimal elements in a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001597The Frobenius rank of a skew partition.