- 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: 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: 95% ●values known / values provided: 96%●distinct values known / distinct values provided: 95%

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
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? = 14
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 21
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 19
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 11
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 19
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 12
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 13
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 20
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 14
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 15
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 17
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 20

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: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%

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
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,4,1,5,6,7,2] => [2,7,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 15
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 18
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,7,1,2,3,5,6] => [6,5,3,2,1,7,4] => ([(0,1),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 16
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,1,7,2,3,4,6] => [6,4,3,2,7,1,5] => ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 16
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [5,7,1,2,3,4,6] => [6,4,3,2,1,7,5] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 17
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 18
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,1,2,8] => [8,2,1,7,6,5,4,3] => ([(0,1),(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,4,5,6,7,8,2,3] => [3,2,8,7,6,5,4,1] => ([(0,1),(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3,8] => [8,3,2,1,7,6,5,4] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,8,2,3,4] => [4,3,2,8,7,6,5,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5,8] => [8,5,4,3,2,1,7,6] => ([(0,1),(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [6,8,1,2,3,4,5,7] => [7,5,4,3,2,1,8,6] => ([(0,1),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 20
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,7,8,2,3,4,5,6] => [6,5,4,3,2,8,7,1] => ([(0,1),(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,8,1,2,3,9] => [9,3,2,1,8,7,6,5,4] => ?
 => ? = 25
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,8,9,2,3,4] => [4,3,2,9,8,7,6,5,1] => ?
 => ? = 25
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7,9] => [9,7,6,5,4,3,2,1,8] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 17
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,9,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,9,1] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 17
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 18
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,1] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 18
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,9,1,6] => [6,1,9,8,7,5,4,3,2] => ([(0,1),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 20

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: 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: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%

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
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,1,3,7] => ([(0,6),(1,4),(1,6),(2,5),(3,2),(4,3),(6,5)],7)
 => ? = 15
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => ? = 10
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ? = 10
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,4,1,5,6,7,2] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
 => ? = 15
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [3,4,5,1,6,2,7] => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
 => ? = 15
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 18
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [3,1,2,4,7,5,6] => ([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
 => ? = 12
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [1,4,5,6,2,7,3] => ([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
 => ? = 15
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,4,2,3,7,5,6] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
 => ? = 12
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [4,6,1,2,3,5,7] => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
 => ? = 15
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,7,1,2,3,5,6] => ([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
 => ? = 16
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,5,2,7,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(4,3),(4,6)],7)
 => ? = 14
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,5,7,2,3,4,6] => ([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
 => ? = 15
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [5,1,6,2,3,4,7] => ([(0,6),(1,4),(1,6),(2,5),(3,2),(4,3),(6,5)],7)
 => ? = 15
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,1,7,2,3,4,6] => ([(0,5),(0,6),(1,4),(1,6),(2,5),(3,2),(4,3)],7)
 => ? = 16
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 18
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 16
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,1,2] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 21
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,1,2,8] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 19
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8] => ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
 => ? = 11
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3,8] => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 21
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [4,1,2,3,5,6,7,8] => ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
 => ? = 12
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [5,1,2,3,4,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
 => ? = 13
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5,8] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 19
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [6,8,1,2,3,4,5,7] => ([(0,6),(1,3),(1,7),(2,7),(4,5),(5,2),(6,4)],8)
 => ? = 20
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [6,1,2,3,4,5,7,8] => ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
 => ? = 14
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6,8] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 15
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
 => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,8,6,7] => ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
 => ? = 11
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,2,3,4,8,5,6,7] => ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
 => ? = 12
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,2,3,8,4,5,6,7] => ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
 => ? = 13
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
 => ? = 14
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 16
[]
 => [] => ([],0)
 => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,8,1,2,3,9] => ([(0,7),(1,6),(2,8),(3,8),(4,5),(5,3),(6,4),(7,2)],9)
 => ? = 25
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7,9] => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ? = 17
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 18
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,1] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 18
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,9,1,6] => ([(0,8),(1,6),(3,7),(4,5),(5,2),(6,3),(7,4),(7,8)],9)
 => ? = 20

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: 73% ●values known / values provided: 83%●distinct values known / distinct values provided: 73%

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
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,1,3,7] => [7,3,1,6,5,4,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,5,6,7,2,4] => [4,2,7,6,5,3,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,4,1,5,6,7,2] => [2,7,6,5,1,4,3] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
 => ? = 15 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [3,4,5,1,6,2,7] => [7,2,6,1,5,4,3] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
 => ? = 18 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,1,2,7] => [7,2,1,6,5,4,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 16 - 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [1,4,5,6,2,7,3] => [3,7,2,6,5,4,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 16 - 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 17 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,2,3,4] => [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 17 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
 => ? = 16 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 10 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,8,2] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ? = 15 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ? = 21 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,1,2,8] => [8,2,1,7,6,5,4,3] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7)],8)
 => ? = 19 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8] => [8,7,6,5,4,2,1,3] => ([(5,7),(6,7)],8)
 => ? = 11 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,4,5,6,7,8,2,3] => [3,2,8,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7)],8)
 => ? = 19 - 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3,8] => [8,3,2,1,7,6,5,4] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 21 - 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [4,1,2,3,5,6,7,8] => [8,7,6,5,3,2,1,4] => ([(4,7),(5,7),(6,7)],8)
 => ? = 12 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,8,2,3,4] => [4,3,2,8,7,6,5,1] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 21 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 25 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [5,1,2,3,4,6,7,8] => [8,7,6,4,3,2,1,5] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 13 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5,8] => [8,5,4,3,2,1,7,6] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 19 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [6,8,1,2,3,4,5,7] => [7,5,4,3,2,1,8,6] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 20 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [6,1,2,3,4,5,7,8] => [8,7,5,4,3,2,1,6] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 14 - 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,7,8,2,3,4,5,6] => [6,5,4,3,2,8,7,1] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 19 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6,8] => [8,6,5,4,3,2,1,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 15 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => [7,8,6,5,4,3,2,1] => ([(6,7)],8)
 => ? = 10 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1] => ([(5,7),(6,7)],8)
 => ? = 11 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,2,3,4,8,5,6,7] => [7,6,5,8,4,3,2,1] => ([(4,7),(5,7),(6,7)],8)
 => ? = 12 - 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,2,3,8,4,5,6,7] => [7,6,5,4,8,3,2,1] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 13 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => [7,6,5,4,3,8,2,1] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 14 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,2,3,4,5,6,7] => [7,6,5,4,3,2,8,1] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 15 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 16 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ([],8)
 => ? = 9 - 1
[]
 => [] => [] => ([],0)
 => ? = 1 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,8,1,2,3,9] => [9,3,2,1,8,7,6,5,4] => ?
 => ? = 25 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,8,9,2,3,4] => [4,3,2,9,8,7,6,5,1] => ?
 => ? = 25 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7,9] => [9,7,6,5,4,3,2,1,8] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 17 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,9,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,9,1] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 17 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 18 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ([],9)
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,1] => [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
 => ? = 18 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,9,1,6] => [6,1,9,8,7,5,4,3,2] => ([(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8)],9)
 => ? = 20 - 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: 73% ●values known / values provided: 83%●distinct values known / distinct values provided: 73%

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
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,1,3,7] => [7,3,1,6,5,4,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,5,6,7,2,4] => [4,2,7,6,5,3,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,4,1,5,6,7,2] => [2,7,6,5,1,4,3] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
 => ? = 15 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [3,4,5,1,6,2,7] => [7,2,6,1,5,4,3] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
 => ? = 18 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,1,2,7] => [7,2,1,6,5,4,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 16 - 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [1,4,5,6,2,7,3] => [3,7,2,6,5,4,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 15 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 16 - 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 17 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,2,3,4] => [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 17 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
 => ? = 16 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 10 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,8,2] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ? = 15 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
 => ? = 21 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,1,2,8] => [8,2,1,7,6,5,4,3] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7)],8)
 => ? = 19 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8] => [8,7,6,5,4,2,1,3] => ([(5,7),(6,7)],8)
 => ? = 11 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,4,5,6,7,8,2,3] => [3,2,8,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7)],8)
 => ? = 19 - 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3,8] => [8,3,2,1,7,6,5,4] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 21 - 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [4,1,2,3,5,6,7,8] => [8,7,6,5,3,2,1,4] => ([(4,7),(5,7),(6,7)],8)
 => ? = 12 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,8,2,3,4] => [4,3,2,8,7,6,5,1] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 21 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 25 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [5,1,2,3,4,6,7,8] => [8,7,6,4,3,2,1,5] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 13 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5,8] => [8,5,4,3,2,1,7,6] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 19 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [6,8,1,2,3,4,5,7] => [7,5,4,3,2,1,8,6] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 20 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [6,1,2,3,4,5,7,8] => [8,7,5,4,3,2,1,6] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 14 - 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,7,8,2,3,4,5,6] => [6,5,4,3,2,8,7,1] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 19 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6,8] => [8,6,5,4,3,2,1,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 15 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => [7,8,6,5,4,3,2,1] => ([(6,7)],8)
 => ? = 10 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1] => ([(5,7),(6,7)],8)
 => ? = 11 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,2,3,4,8,5,6,7] => [7,6,5,8,4,3,2,1] => ([(4,7),(5,7),(6,7)],8)
 => ? = 12 - 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,2,3,8,4,5,6,7] => [7,6,5,4,8,3,2,1] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 13 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => [7,6,5,4,3,8,2,1] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 14 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,2,3,4,5,6,7] => [7,6,5,4,3,2,8,1] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 15 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 16 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ([],8)
 => ? = 9 - 1
[]
 => [] => [] => ([],0)
 => ? = 1 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,8,1,2,3,9] => [9,3,2,1,8,7,6,5,4] => ?
 => ? = 25 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,8,9,2,3,4] => [4,3,2,9,8,7,6,5,1] => ?
 => ? = 25 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7,9] => [9,7,6,5,4,3,2,1,8] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 17 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,9,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,9,1] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 17 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 18 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ([],9)
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,1] => [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
 => ? = 18 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,9,1,6] => [6,1,9,8,7,5,4,3,2] => ([(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8)],9)
 => ? = 20 - 1

Description

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

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

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 80%●distinct values known / distinct values provided: 77%

Values

[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 4
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 5
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 5
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 6
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 8
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 7
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [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]
 => [1,1,1,0,1,0,0,1,0,0]
 => 8
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 7
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 6
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 8
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 9
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 7
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 7
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 10
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 9
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 9
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 8
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 9
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 8
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 10
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 11
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 9
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 8
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 9
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 10
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 9
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 8
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 8
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 9
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 7
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 10
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 9
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 11
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 12
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 9
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 10
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 11
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 14
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 15
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 14
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 15
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 15
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 18
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 16
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 12
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 17
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 15
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 14
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 16
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 11
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 15
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 16
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 16
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 17
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 12
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 18
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 13
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 14
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 16
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 15
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 21
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 19
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 11
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 19
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 21
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 12
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 21
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 25
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 13
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 19
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 20
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 14
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 19
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 15
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 11
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 12
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 13
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 14
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 15
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 16
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 25
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 25
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 17

Description

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

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

Mapping

St001213: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 72%●distinct values known / distinct values provided: 68%

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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 14
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 15
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 10
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 13
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 15
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 14
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 15
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 15
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 18
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 16
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 12
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 12
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 15
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 16
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 12
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 17
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 15
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 14
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 16
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 11
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 17
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 14
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 15
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? = 15
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 16
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? = 16
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 17
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 12
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 15
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 16
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 18
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 13
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 9
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 10
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 11
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 12
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 13
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 14
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 16
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 15
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 21
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 19
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 11
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 19
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 21

Description

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

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: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 77%

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: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 55%

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.