- FindStat

Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000395
(load all 8 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 6
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 8
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 7
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 7
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 7
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 8
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 9
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 7
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 7
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 10
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 9
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 9
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 9
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 10
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 11
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 9
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 9
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 9
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 8
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 8
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 9
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 7
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 10
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 9
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 11
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 12
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 9
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 11
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 8

Description

The sum of the heights of the peaks of a Dyck path.

Matching statistic: St000394
(load all 3 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 6
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 8
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 7
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 7
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 7
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 6
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 8
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 9
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 7
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 6
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 7
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 10
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => 9
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => 9
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => 9
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => 10
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => 11
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => 9
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => 8
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => 9
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => 9
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => 8
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => 8
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => 9
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => 7
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => 10
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => 9
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => 11
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 12
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => 9
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => 11
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 8

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St001034
(load all 4 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,1,0,0]
 => 2
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 6
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 8
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 7
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 7
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 6
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 7
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 6
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 8
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 9
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 7
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 7
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 10
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 9
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 9
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 9
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 10
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 11
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 9
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 8
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 9
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 9
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 8
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 8
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 9
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 7
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 10
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 9
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 11
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 12
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 9
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 11
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 8

Description

The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].

Matching statistic: St000300

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of independent sets of vertices of a graph. An independent set of vertices of a graph

$G$ is a subset

$U \subset V(G)$ such that no two vertices in

$U$ are adjacent. This is also the number of vertex covers of

$G$ as the map

$U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers. The size of the largest independent set, also called independence number of

$G$ , is [[St000093]]

Matching statistic: St001213
(load all 8 compositions to match this statistic)

Mapping

St001213: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 99%●distinct values known / distinct values provided: 94%

Values

[1,0]
 => 2
[1,0,1,0]
 => 4
[1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => 6
[1,0,1,1,0,0]
 => 5
[1,1,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => 6
[1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => 8
[1,0,1,0,1,1,0,0]
 => 7
[1,0,1,1,0,0,1,0]
 => 7
[1,0,1,1,0,1,0,0]
 => 8
[1,0,1,1,1,0,0,0]
 => 6
[1,1,0,0,1,0,1,0]
 => 7
[1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,0,0,1,0]
 => 8
[1,1,0,1,0,1,0,0]
 => 9
[1,1,0,1,1,0,0,0]
 => 7
[1,1,1,0,0,0,1,0]
 => 6
[1,1,1,0,0,1,0,0]
 => 7
[1,1,1,0,1,0,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => 10
[1,0,1,0,1,0,1,1,0,0]
 => 9
[1,0,1,0,1,1,0,0,1,0]
 => 9
[1,0,1,0,1,1,0,1,0,0]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => 9
[1,0,1,1,0,0,1,1,0,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => 10
[1,0,1,1,0,1,0,1,0,0]
 => 11
[1,0,1,1,0,1,1,0,0,0]
 => 9
[1,0,1,1,1,0,0,0,1,0]
 => 8
[1,0,1,1,1,0,0,1,0,0]
 => 9
[1,0,1,1,1,0,1,0,0,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => 9
[1,1,0,0,1,0,1,1,0,0]
 => 8
[1,1,0,0,1,1,0,0,1,0]
 => 8
[1,1,0,0,1,1,0,1,0,0]
 => 9
[1,1,0,0,1,1,1,0,0,0]
 => 7
[1,1,0,1,0,0,1,0,1,0]
 => 10
[1,1,0,1,0,0,1,1,0,0]
 => 9
[1,1,0,1,0,1,0,0,1,0]
 => 11
[1,1,0,1,0,1,0,1,0,0]
 => 12
[1,1,0,1,0,1,1,0,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => 9
[1,1,0,1,1,0,0,1,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => 11
[1,1,0,1,1,1,0,0,0,0]
 => 8
[]
 => ? = 1

Description

The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.

Matching statistic: St000070
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 99%●distinct values known / distinct values provided: 94%

Values

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

Description

The number of antichains in a poset. An antichain in a poset

$P$ is a subset of elements of

$P$ which are pairwise incomparable. An order ideal is a subset

$I$ of

$P$ such that

$a\in I$ and

$b \leq_P a$ implies

$b \in I$ . Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.

Matching statistic: St000639
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 88% ●values known / values provided: 99%●distinct values known / distinct values provided: 88%

Values

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

Description

The number of relations in a poset. This is the number of intervals

$x,y$ with

$x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.

Matching statistic: St000641
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 88% ●values known / values provided: 99%●distinct values known / distinct values provided: 88%

Values

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

Description

The number of non-empty boolean intervals in a poset.

Matching statistic: St000180
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000180: Posets ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 94%

Values

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

Description

The number of chains of a poset.

Matching statistic: St000081

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of edges of a graph.

The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001616The number of neutral elements in a lattice. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001428The number of B-inversions of a signed permutation. St000018The number of inversions of a permutation. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000030The sum of the descent differences of a permutations. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.