Your data matches 28 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000633
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
Description
The size of the automorphism group of a poset. A poset automorphism is a permutation of the elements of the poset preserving the order relation.
Matching statistic: St001399
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
Description
The distinguishing number of a poset. This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring. See also [[St000469]], which is the same concept for graphs.
Matching statistic: St001483
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[3,2,1]
 => [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
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 2
[4,3,2,1]
 => [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
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [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
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000317
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,3,4,6] => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,3,4,6] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,2,4,6] => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0 = 1 - 1
[4,2,2,1]
 => [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] => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,6] => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,6] => 1 = 2 - 1
[4,3,2,1]
 => [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] => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,6,4] => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,2,6,4] => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 0 = 1 - 1
Description
The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St000373
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0 = 1 - 1
[4,2,2,1]
 => [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] => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,6] => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,4,1,6] => 1 = 2 - 1
[4,3,2,1]
 => [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] => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3] => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 0 = 1 - 1
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
Matching statistic: St000648
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0 = 1 - 1
[4,2,2,1]
 => [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] => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,6] => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,4,1,6] => 1 = 2 - 1
[4,3,2,1]
 => [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] => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3] => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 0 = 1 - 1
Description
The number of 2-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000660
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000732
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,3,4,6] => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,3,4,6] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,2,4,6] => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0 = 1 - 1
[4,2,2,1]
 => [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] => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,6] => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,6] => 1 = 2 - 1
[4,3,2,1]
 => [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] => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,6,4] => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,2,6,4] => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 0 = 1 - 1
Description
The number of double deficiencies of a permutation. A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St000850
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000850: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The number of 1/2-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] exhibit various posets that have a $1/2$-balanced pair.
Matching statistic: St001552
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001552: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1 = 2 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0 = 1 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1 = 2 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0 = 1 - 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 0 = 1 - 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0 = 1 - 1
[4,2,2,1]
 => [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] => 0 = 1 - 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,6] => 0 = 1 - 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,4,1,6] => 1 = 2 - 1
[4,3,2,1]
 => [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] => 0 = 1 - 1
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => 0 = 1 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3] => 1 = 2 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => 1 = 2 - 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 0 = 1 - 1
[4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 1 = 2 - 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => 0 = 1 - 1
[4,3,2,2]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => 0 = 1 - 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 0 = 1 - 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 0 = 1 - 1
Description
The number of inversions between excedances and fixed points of a permutation. This is, $$\operatorname{iefp}(\sigma) = \#\{1 \leq i,j \leq n \mid i < j = \sigma(j) < \sigma(i) \}.$$
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001728The number of invisible descents of a permutation. St001964The interval resolution global dimension of a poset. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000454The largest eigenvalue of a graph if it is integral. St000068The number of minimal elements in a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.