Your data matches 32 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 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 8 = 7 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 8 = 7 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 9 = 8 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 7 = 6 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 8 = 7 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 10 = 9 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 9 = 8 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 10 = 9 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 9 = 8 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 10 = 9 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 11 = 10 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 9 = 8 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 10 = 9 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 11 = 10 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 12 = 11 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 8 = 7 + 1
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 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[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 = 5 + 1
[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 = 4 + 1
[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 = 4 + 1
[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 = 5 + 1
[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 = 3 + 1
[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 = 7 + 1
[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 = 6 + 1
[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 = 6 + 1
[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 = 7 + 1
[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 = 5 + 1
[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 = 6 + 1
[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 = 5 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 6 + 1
[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 = 5 + 1
[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 = 6 + 1
[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 = 7 + 1
[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 = 4 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 6 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 6 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 10 + 1
[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 = 11 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 7 + 1
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 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[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 = 7 + 1
[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 = 6 + 1
[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 = 6 + 1
[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 = 7 + 1
[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 = 5 + 1
[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 = 6 + 1
[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 = 5 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 6 + 1
[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 = 5 + 1
[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 = 6 + 1
[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 = 7 + 1
[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 = 4 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 6 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 6 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 10 + 1
[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 = 11 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 7 + 1
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,0,0]
 => [1,2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 2
[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
[1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[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
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
Description
The number of edges of a graph.
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 + 1
[1,0,1,0]
 => [2,1] => [1,2] => ([],2)
 => 4 = 3 + 1
[1,1,0,0]
 => [1,2] => [2,1] => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [2,3,1] => [1,3,2] => ([(1,2)],3)
 => 6 = 5 + 1
[1,0,1,1,0,0]
 => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1,3] => ([(1,2)],3)
 => 6 = 5 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 8 = 7 + 1
[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 = 6 + 1
[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 = 6 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
[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 = 5 + 1
[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 = 6 + 1
[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 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 8 = 7 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 9 = 8 + 1
[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 = 6 + 1
[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 = 5 + 1
[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 = 6 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 8 = 7 + 1
[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 = 4 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 8 + 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),(2,3),(2,4),(3,4)],5)
 => 10 = 9 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 6 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 6 + 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),(2,3),(2,4),(3,4)],5)
 => 10 = 9 + 1
[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 = 8 + 1
[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 = 10 + 1
[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 = 11 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 7 + 1
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: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,1,0,0]
 => 8 = 7 + 1
[1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => 8 = 7 + 1
[1,1,0,1,0,1,0,0]
 => 9 = 8 + 1
[1,1,0,1,1,0,0,0]
 => 7 = 6 + 1
[1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,0]
 => 7 = 6 + 1
[1,1,1,0,1,0,0,0]
 => 8 = 7 + 1
[1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 10 = 9 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 9 = 8 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 10 = 9 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 9 = 8 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 10 = 9 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 11 = 10 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 9 = 8 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 8 = 7 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 9 = 8 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 9 = 8 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 8 = 7 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 10 = 9 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 9 = 8 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 11 = 10 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 12 = 11 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 10 = 9 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 9 = 8 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 10 = 9 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 11 = 10 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 8 = 7 + 1
[]
 => ? = 0 + 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: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1,0]
 => [1] => ([],1)
 => 2 = 1 + 1
[1,0,1,0]
 => [2,1] => ([],2)
 => 4 = 3 + 1
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [2,3,1] => ([(1,2)],3)
 => 6 = 5 + 1
[1,0,1,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => [3,1,2] => ([(1,2)],3)
 => 6 = 5 + 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 7 = 6 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 8 = 7 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 8 = 7 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 9 = 8 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 7 = 6 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 6 = 5 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 7 = 6 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 8 = 7 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 10 = 9 + 1
[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 = 8 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 6 + 1
[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 = 8 + 1
[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 = 7 + 1
[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 = 7 + 1
[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 = 8 + 1
[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 = 6 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 10 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 12 = 11 + 1
[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 = 9 + 1
[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 = 8 + 1
[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 = 9 + 1
[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 = 10 + 1
[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 = 7 + 1
[]
 => [] => ([],0)
 => ? = 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: 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: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 2 = 1 + 1
[1,0,1,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 4 = 3 + 1
[1,1,0,0]
 => [1,2] => [2,1] => ([],2)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 6 = 5 + 1
[1,0,1,1,0,0]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 6 = 5 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => ([],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 7 = 6 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 8 = 7 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 8 = 7 + 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)
 => 9 = 8 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 7 = 6 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 6 = 5 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 7 = 6 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 8 = 7 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => 5 = 4 + 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)
 => 10 = 9 + 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)
 => 9 = 8 + 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)
 => 9 = 8 + 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)
 => 10 = 9 + 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)
 => 8 = 7 + 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)
 => 9 = 8 + 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)
 => 8 = 7 + 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)
 => 10 = 9 + 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)
 => 11 = 10 + 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)
 => 9 = 8 + 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)
 => 8 = 7 + 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)
 => 9 = 8 + 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)
 => 10 = 9 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => 7 = 6 + 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)
 => 9 = 8 + 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)
 => 8 = 7 + 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)
 => 8 = 7 + 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)
 => 9 = 8 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 7 = 6 + 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)
 => 10 = 9 + 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)
 => 9 = 8 + 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)
 => 11 = 10 + 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)
 => 12 = 11 + 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)
 => 10 = 9 + 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)
 => 9 = 8 + 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)
 => 10 = 9 + 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)
 => 11 = 10 + 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)
 => 8 = 7 + 1
[]
 => [] => [] => ([],0)
 => ? = 0 + 1
Description
The number of chains of 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: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [1] => [1] => ([],1)
 => ? = 1
[1,0,1,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 3
[1,1,0,0]
 => [1,2] => [2,1] => ([],2)
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 5
[1,0,1,1,0,0]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => 4
[1,1,0,0,1,0]
 => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => 4
[1,1,0,1,0,0]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 5
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => ([],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 7
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 6
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 7
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 5
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 6
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 5
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 7
[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
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 6
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 5
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 6
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 7
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => 4
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => 6
[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
[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
[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
[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
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 6
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[]
 => [] => [] => ([],0)
 => ? = 0
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: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [1] => [1] => ([],1)
 => ? = 1
[1,0,1,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 3
[1,1,0,0]
 => [1,2] => [2,1] => ([],2)
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 5
[1,0,1,1,0,0]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => 4
[1,1,0,0,1,0]
 => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => 4
[1,1,0,1,0,0]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 5
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => ([],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 7
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 6
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 7
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 5
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 6
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 5
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 7
[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
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 6
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 5
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 6
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 7
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => 4
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => 6
[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
[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
[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
[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
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 6
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[]
 => [] => [] => ([],0)
 => ? = 0
Description
The number of non-empty boolean intervals in a poset.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001616The number of neutral elements in a lattice. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001717The largest size of an interval in a poset. St000656The number of cuts of a poset. St000189The number of elements in the poset. St001894The depth of a signed permutation. St000019The cardinality of the support of a permutation.