Your data matches 206 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000011
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000306
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000306: 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]
 => [1,0,1,0]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [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]
 => 1
[1,0,1,1,0,0,1,0]
 => [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]
 => 1
[1,0,1,1,0,1,0,0]
 => [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]
 => 1
[1,0,1,1,1,0,0,0]
 => [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]
 => 1
[1,1,0,0,1,0,1,0]
 => [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]
 => 1
[1,1,0,0,1,1,0,0]
 => [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]
 => 1
[1,1,0,1,0,0,1,0]
 => [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]
 => 1
[1,1,0,1,0,1,0,0]
 => [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]
 => 1
[1,1,0,1,1,0,0,0]
 => [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]
 => 1
[1,1,1,0,0,0,1,0]
 => [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]
 => 1
[1,1,1,0,0,1,0,0]
 => [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]
 => 1
[1,1,1,0,1,0,0,0]
 => [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]
 => 1
[1,1,1,1,0,0,0,0]
 => [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]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [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]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [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]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [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]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [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]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [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]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [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]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [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]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [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]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [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]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [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]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [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]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [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]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [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]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [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]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [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]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [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]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [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]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [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]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [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]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [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]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [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]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [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]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [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]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [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]
 => 2
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St001484
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => []
 => 0 = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1 = 2 - 1
Description
The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000445
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000883
(load all 3 compositions to match this statistic)
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => [1] => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,5,4,3,6,2,7] => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => ? = 5
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,7,2,8] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,7,6,5,4,3,2,8] => ? = 6
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St000759
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => [1,0]
 => []
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,2,1]
 => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,1,1]
 => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,5,4,4,3,2,1]
 => ? = 6
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000439
(load all 3 compositions to match this statistic)
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 + 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 + 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 + 2
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 + 2
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 + 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 + 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000054
(load all 4 compositions to match this statistic)
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1] => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => ? = 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => ? = 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => ? = 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,4,2,5,6,7] => ? = 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => ? = 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [2,4,5,3,6,1,7] => ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,5,6,4,7,2,1] => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [4,5,3,6,2,7,1] => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,4,7,3,2,1] => ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? = 2
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St000214
(load all 4 compositions to match this statistic)
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,6,2,1] => [1,2,6,3,4,5,7] => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ? = 1 - 1
Description
The number of adjacencies of a permutation. An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
(load all 4 compositions to match this statistic)
Mapping
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1 - 1
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
The following 196 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001050The number of terminal closers of a set partition. St000247The number of singleton blocks of a set partition. St000648The number of 2-excedences of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000441The number of successions of a permutation. St000674The number of hills of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000248The number of anti-singletons of a set partition. St000502The number of successions of a set partitions. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000025The number of initial rises of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001061The number of indices that are both descents and recoils of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000649The number of 3-excedences of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000843The decomposition number of a perfect matching. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St000654The first descent of a permutation. St000335The difference of lower and upper interactions. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000367The number of simsun double descents of a permutation. St000444The length of the maximal rise of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000989The number of final rises of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000455The second largest eigenvalue of a graph if it is integral. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000069The number of maximal elements of a poset. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000731The number of double exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000068The number of minimal elements in a poset. St001720The minimal length of a chain of small intervals in a lattice. St000914The sum of the values of the Möbius function of a poset. St001621The number of atoms of a lattice. St000145The Dyson rank of a partition. St000314The number of left-to-right-maxima of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000035The number of left outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000884The number of isolated descents of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000133The "bounce" of a permutation. St000223The number of nestings in the permutation. St000358The number of occurrences of the pattern 31-2. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001846The number of elements which do not have a complement in the lattice. St001866The nesting alignments of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001569The maximal modular displacement of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001960The number of descents of a permutation minus one if its first entry is not one. St000390The number of runs of ones in a binary word. St000805The number of peaks of the associated bargraph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000983The length of the longest alternating subword. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001549The number of restricted non-inversions between exceedances. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001712The number of natural descents of a standard Young tableau. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000889The number of alternating sign matrices with the same antidiagonal sums. St000296The length of the symmetric border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.