- FindStat

Your data matches 269 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001198

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The 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

$eA$ .

Matching statistic: St001206

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The 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

$eA$ .

Matching statistic: St000125

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree. [[oeis:A005773]] counts binary trees avoiding this pattern.

Matching statistic: St001906
(load all 4 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. Let

$\pi$ be a permutation. Its total displacement [[St000830]] is

$D(\pi) = \sum_i |\pi(i) - i|$ , and its absolute length [[St000216]] is the minimal number

$T(\pi)$ of transpositions whose product is

$\pi$ . Finally, let

$I(\pi)$ be the number of inversions [[St000018]] of

$\pi$ . This statistic equals

$\left(D(\pi)-T(\pi)-I(\pi)\right)/2$ . Diaconis and Graham [1] proved that this statistic is always nonnegative.

Matching statistic: St000455

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 50%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001578
(load all 7 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001578: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 50%

Values

[1,2] => [2,1] => ([(0,1)],2)
 => ([],1)
 => 0 = 2 - 2
[2,1] => [1,2] => ([],2)
 => ([],0)
 => ? = 2 - 2
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 2 - 2
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => ([],1)
 => 0 = 2 - 2
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => ([],1)
 => 0 = 2 - 2
[3,2,1] => [1,2,3] => ([],3)
 => ([],0)
 => ? = 2 - 2
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 2 - 2
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 2 - 2
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 2 - 2
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 2 - 2
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 2 - 2
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
 => ([],1)
 => 0 = 2 - 2
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 2 - 2
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 2 - 2
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 2 - 2
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
 => ([],1)
 => 0 = 2 - 2
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
 => ([],1)
 => 0 = 2 - 2
[4,3,2,1] => [1,2,3,4] => ([],4)
 => ([],0)
 => ? = 2 - 2
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 2 - 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 2
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 2 - 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,5,4,2,3] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3 - 2
[2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 2
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 2
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 0 = 2 - 2
[2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 2
[2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 2 - 2
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,4,5,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 2 - 2
[3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 2
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 - 2
[3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 2 - 2
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 0 = 2 - 2
[3,2,4,5,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[3,2,5,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 2 - 2
[3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[3,4,2,1,5] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[5,4,3,2,1] => [1,2,3,4,5] => ([],5)
 => ([],0)
 => ? = 2 - 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,4,3,5,6] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3 - 2
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,4,6,5,3] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,5,4,6,3] => [3,6,4,5,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,5,6,4,3] => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 - 2
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
[1,2,6,4,5,3] => [3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 - 2
[1,2,6,5,3,4] => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 - 2
[1,2,6,5,4,3] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 2 - 2

Description

The minimal number of edges to add or remove to make a graph a line graph.

Matching statistic: St001330
(load all 6 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%

Values

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

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001630
(load all 7 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%

Values

[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => ? = 2
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => ? = 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => ? = 2
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2

Description

The global dimension of the incidence algebra of the lattice over the rational numbers.

Matching statistic: St001877
(load all 10 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%

Values

[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => ? = 2 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => ? = 2 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => ? = 2 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3 - 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3 - 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 - 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 1
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1

Description

Number of indecomposable injective modules with projective dimension 2.

Matching statistic: St000929
(load all 2 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%

Values

[1,2] => [2,1] => ([],2)
 => [2]
 => 0 = 2 - 2
[2,1] => [1,2] => ([(0,1)],2)
 => [1]
 => ? = 2 - 2
[1,2,3] => [3,2,1] => ([],3)
 => [3,3]
 => 0 = 2 - 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 0 = 2 - 2
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 0 = 2 - 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => ? = 2 - 2
[1,2,3,4] => [4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => ? = 2 - 2
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => [4,4,4]
 => 0 = 2 - 2
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => [8]
 => 0 = 2 - 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => [8]
 => 0 = 2 - 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => [4]
 => 0 = 2 - 2
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => [4,4,4]
 => 0 = 2 - 2
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => [4,2]
 => 0 = 2 - 2
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => [8]
 => 0 = 2 - 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 0 = 2 - 2
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 0 = 2 - 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 0 = 2 - 2
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => [4]
 => 0 = 2 - 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 0 = 2 - 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [2]
 => 0 = 2 - 2
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 0 = 2 - 2
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0 = 2 - 2
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0 = 2 - 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 0 = 2 - 2
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => 0 = 2 - 2
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => ? = 2 - 2
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2 - 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2 - 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => ? = 2 - 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => ? = 2 - 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 2 - 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2 - 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => ? = 2 - 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 3 - 2
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 2 - 2
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 2 - 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => ? = 2 - 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 2 - 2
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 2 - 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 2 - 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 2 - 2
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => 0 = 2 - 2
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 2 - 2
[1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => 0 = 2 - 2
[1,5,4,2,3] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => 0 = 2 - 2
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => [5]
 => 0 = 2 - 2
[2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 3 - 2
[2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => ? = 2 - 2
[2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => ? = 2 - 2
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 2 - 2
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 2 - 2
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => 0 = 2 - 2
[2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => ? = 2 - 2
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 2 - 2
[2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 2 - 2
[2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => ? = 2 - 2
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 2 - 2
[2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 2 - 2
[2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => 0 = 2 - 2
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 2 - 2
[2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => 0 = 2 - 2
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 2 - 2
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => 0 = 2 - 2
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => 0 = 2 - 2
[2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => 0 = 2 - 2
[2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => 0 = 2 - 2
[3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => ? = 2 - 2
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 2 - 2
[3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 2 - 2
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 2 - 2
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => 0 = 2 - 2
[3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 2 - 2
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => 0 = 2 - 2
[3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => 0 = 2 - 2
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => 0 = 2 - 2
[3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => 0 = 2 - 2
[3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 2 - 2
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 2 - 2
[3,4,2,1,5] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => 0 = 2 - 2
[3,4,2,5,1] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => 0 = 2 - 2
[3,4,5,1,2] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => 0 = 2 - 2
[3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => 0 = 2 - 2
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 2 - 2
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => 0 = 2 - 2
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => 0 = 2 - 2
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 0 = 2 - 2
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 2 - 2
[4,1,3,5,2] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => 0 = 2 - 2
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 2 - 2
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => 0 = 2 - 2
[4,2,1,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => ? = 2 - 2
[4,2,1,5,3] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => 0 = 2 - 2
[4,2,3,1,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => 0 = 2 - 2
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 0 = 2 - 2
[5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => ? = 2 - 2
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => ? = 2 - 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 2 - 2
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 2 - 2
[1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(3,4),(3,5)],6)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2 - 2
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ? = 2 - 2

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

The following 259 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000068The number of minimal elements in a poset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001490The number of connected components of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000031The number of cycles in the cycle decomposition of a permutation. St000475The number of parts equal to 1 in a partition. St001720The minimal length of a chain of small intervals in a lattice. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001568The smallest positive integer that does not appear twice in the partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001472The permanent of the Coxeter matrix of the poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000488The number of cycles of a permutation of length at most 2. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000260The radius of a connected graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. 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. St001625The Möbius invariant of a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000635The number of strictly order preserving maps of a poset into itself. St001889The size of the connectivity set of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000256The number of parts from which one can substract 2 and still get an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001616The number of neutral elements in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001845The number of join irreducibles minus the rank of a lattice. St000878The number of ones minus the number of zeros of a binary word. St000627The exponent of a binary word. St000629The defect of a binary word. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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

$eA$ . St001429The number of negative entries in a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000098The chromatic number of a graph. St001624The breadth of a lattice. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000258The burning number of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001430The number of positive entries in a signed permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001691The number of kings in a graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000297The number of leading ones in a binary word. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000084The number of subtrees. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000328The maximum number of child nodes in a tree. St000876The number of factors in the Catalan decomposition of a binary word. St000905The number of different multiplicities of parts of an integer composition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001884The number of borders of a binary word. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000893The number of distinct diagonal sums of an alternating sign matrix. St000942The number of critical left to right maxima of the parking functions. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001256Number of simple reflexive modules that are 2-stable reflexive. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001410The minimal entry of a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000074The number of special entries. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000366The number of double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000508Eigenvalues of the random-to-random operator acting on a simple module. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001381The fertility of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001862The number of crossings of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000907The number of maximal antichains of minimal length in a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001765The number of connected components of the friends and strangers graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000782The indicator function of whether a given perfect matching is an L & P matching.