searching the database
Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001880
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 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)
=> 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[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]
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> 2
[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]
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 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)
=> 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 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]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 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]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> 2
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> 2
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 3
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> 3
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> 2
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> 1
[6,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 3
[6,4,3,2,1,1]
=> [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]
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> 2
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[6,5,3,2,2,1]
=> [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]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[6,4,3,2,2,1]
=> [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]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[6,5,4,2,1,1]
=> [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]
=> ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001010
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001010: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 57%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001010: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 57%
Values
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[4,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[5,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4 - 1
[6,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 5 - 1
[4,2,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[6,2,1]
=> [4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[5,2,1,1]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[6,2,1,1]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[5,4,1]
=> [3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[5,2,1,1,1]
=> [3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 5 - 1
[6,2,1,1,1]
=> [3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[5,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[5,3,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[6,5,1]
=> [3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[6,3,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[6,2,1,1,1,1]
=> [3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5,4,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 6 - 1
[5,3,2,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 5 - 1
[6,3,2,1,1]
=> [4,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 1
[6,3,1,1,1,1]
=> [3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[5,4,3,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 6 - 1
[5,4,2,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 5 - 1
[5,3,2,2,1]
=> [3,3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[6,5,2,1]
=> [4,3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[6,4,3,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 7 - 1
[6,4,1,1,1,1]
=> [4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[6,3,2,1,1,1]
=> [4,4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[6,5,2,1,1]
=> [3,3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[6,4,2,1,1,1]
=> [4,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0]
=> ?
=> ? = 3 - 1
[6,3,2,2,1,1]
=> [5,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[5,4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 6 - 1
[6,5,4,1]
=> [3,3,3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ? = 3 - 1
[6,5,2,1,1,1]
=> [3,2,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ?
=> ? = 2 - 1
[6,4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 7 - 1
[6,4,2,2,1,1]
=> [4,2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ?
=> ? = 3 - 1
[6,3,2,2,2,1]
=> [3,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[6,5,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 7 - 1
[6,5,3,1,1,1]
=> [3,2,2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 3 - 1
[6,4,3,2,1,1]
=> [5,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 6 - 1
[6,4,2,2,2,1]
=> [3,3,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ? = 2 - 1
[6,5,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 7 - 1
[6,5,3,2,2,1]
=> ?
=> ?
=> ?
=> ? = 7 - 1
[6,4,3,2,2,1]
=> ?
=> ?
=> ?
=> ? = 7 - 1
[6,5,4,2,1,1]
=> ?
=> ?
=> ?
=> ? = 6 - 1
[6,5,3,2,1,1]
=> ?
=> ?
=> ?
=> ? = 6 - 1
[6,5,4,3,1]
=> ?
=> ?
=> ?
=> ? = 7 - 1
[6,5,4,2,1]
=> ?
=> ?
=> ?
=> ? = 7 - 1
Description
Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000528
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 71%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 71%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ([(0,4),(0,6),(2,10),(3,9),(4,7),(5,2),(5,8),(6,3),(6,7),(7,5),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 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]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 5
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,9),(2,10),(3,12),(4,7),(5,2),(5,8),(6,3),(6,8),(7,9),(8,10),(8,12),(10,11),(11,1),(11,7),(12,4),(12,11)],13)
=> ? = 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 7
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 2
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,5),(4,10),(5,2),(5,11),(6,4),(6,7),(7,10),(8,9),(10,11),(11,1),(11,8)],12)
=> ? = 2
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 3
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,6),(1,8),(2,9),(3,7),(4,2),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,5),(11,9)],12)
=> ? = 3
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ? = 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,8),(4,7),(5,2),(5,8),(6,1),(6,7),(7,10),(8,9),(10,3),(10,5)],11)
=> ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,9),(2,10),(3,8),(4,7),(5,2),(5,11),(6,5),(6,7),(7,11),(8,9),(10,1),(10,8),(11,3),(11,10)],12)
=> ? = 2
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ([(0,4),(0,6),(1,7),(2,9),(3,8),(4,11),(5,3),(5,10),(6,1),(6,11),(7,5),(7,9),(9,10),(10,8),(11,2),(11,7)],12)
=> ? = 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,8),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,10),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1
[6,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ? = 3
[6,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 2
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 7
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 7
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 6
[6,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,5),(4,10),(5,2),(5,11),(6,4),(6,7),(7,10),(8,9),(10,11),(11,1),(11,8)],12)
=> ? = 6
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 7
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 7
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St001807
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 57%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 57%
Values
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 4
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => 5
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,1,6,7,2,3,4] => ? = 2
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => 5
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,7,3,5] => ? = 4
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [6,1,7,8,2,3,4,5] => ? = 2
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [5,1,7,2,8,3,4,6] => ? = 2
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0]
=> [6,1,8,2,9,3,4,5,7] => ? = 2
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,1,6,7,2,4,5] => ? = 2
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0,0]
=> [5,1,8,2,3,9,4,6,7] => ? = 1
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [3,4,1,5,7,2,6] => ? = 5
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 1
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [4,5,1,6,8,2,3,7] => ? = 6
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 2
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,1,7,8,2,4,5,6] => ? = 2
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 3
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 1
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,1,4,6,8,2,5,7] => ? = 6
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? => ? = 5
[6,3,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 3
[6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 1
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,8,4,7] => ? = 6
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? => ? = 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? => ? = 6
[6,5,2,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 3
[6,4,3,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,4,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 2
[6,3,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 2
[6,5,2,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 3
[6,4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 3
[6,3,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 2
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> ? => ? = 6
[6,5,4,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ?
=> ? => ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 2
[6,4,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,4,2,2,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 1
[6,5,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 3
[6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 2
[6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,5,3,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,5,4,2,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 6
[6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? => ? = 6
[6,5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? => ? = 7
[6,5,4,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? => ? = 7
Description
The lower middle entry of a permutation.
This is the entry $\sigma(\frac{n+1}{2})$ when $n$ is odd, and $\sigma(\frac{n}{2})$ when $n$ is even, where $n$ is the size of the permutation $\sigma$.
Matching statistic: St001232
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 57%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 57%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 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]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 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]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 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]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 7 - 1
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [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]
=> ? = 3 - 1
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 - 1
[6,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 7 - 1
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> 6 = 7 - 1
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6 = 7 - 1
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[6,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 1
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 7 - 1
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000337
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [8,3,4,5,7,1,2,6] => ? = 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,8,1,2,7] => ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? = 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 4
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [8,9,4,5,6,1,2,3,7] => ? = 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [9,6,4,5,1,8,2,3,7] => ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,3,4,8,1,2,5,6] => ? = 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> [9,5,4,1,6,8,2,3,7] => ? = 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8,3,1,5,7,2,4,6] => ? = 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [8,3,4,5,9,1,2,6,7] => ? = 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,6,5,1,2,3,4,7] => ? = 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [9,4,1,5,6,8,2,3,7] => ? = 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8,7,4,1,2,3,5,6] => ? = 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => ? = 5
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,5,1,6,2,3,4,7] => ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,1,5,6,8,2,4,7] => ? = 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [7,4,1,8,2,3,5,6] => ? = 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => ? = 6
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [9,8,4,5,1,2,3,6,7] => ? = 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [9,3,8,6,1,2,4,5,7] => ? = 7
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [4,3,1,9,6,8,2,5,7] => ? = 2
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,1,5,6,2,3,4,7] => ? = 2
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [8,5,4,1,9,2,3,6,7] => ? = 3
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [8,4,1,9,6,2,3,5,7] => ? = 3
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,9,1,5,2,8,3,4,7] => ? = 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 6
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [9,3,4,8,1,2,5,6,7] => ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [8,4,1,5,9,2,3,6,7] => ? = 2
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,9,8,1,2,3,4,5,7] => ? = 7
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [9,4,1,6,2,8,3,5,7] => ? = 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [5,9,1,2,6,8,3,4,7] => ? = 1
[6,5,3,2,1]
=> [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]
=> [9,8,5,1,2,3,4,6,7] => ? = 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [8,3,1,5,9,2,4,6,7] => ? = 3
[6,4,3,2,1,1]
=> [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]
=> [9,8,1,6,2,3,4,5,7] => ? = 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [9,4,1,2,6,8,3,5,7] => ? = 2
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 7
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [5,8,1,2,9,3,4,6,7] => ? = 7
[6,4,3,2,2,1]
=> [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]
=> [9,8,1,2,6,3,4,5,7] => ? = 7
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,4,1,8,2,3,5,6,7] => ? = 6
[6,5,3,2,1,1]
=> [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]
=> [9,8,1,5,2,3,4,6,7] => ? = 6
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [9,3,8,1,2,4,5,6,7] => ? = 7
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 7
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000725
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [8,3,4,5,7,1,2,6] => ? = 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,8,1,2,7] => ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? = 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 4
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [8,9,4,5,6,1,2,3,7] => ? = 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [9,6,4,5,1,8,2,3,7] => ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,3,4,8,1,2,5,6] => ? = 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> [9,5,4,1,6,8,2,3,7] => ? = 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8,3,1,5,7,2,4,6] => ? = 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [8,3,4,5,9,1,2,6,7] => ? = 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,6,5,1,2,3,4,7] => ? = 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [9,4,1,5,6,8,2,3,7] => ? = 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8,7,4,1,2,3,5,6] => ? = 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => ? = 5
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,5,1,6,2,3,4,7] => ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,1,5,6,8,2,4,7] => ? = 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [7,4,1,8,2,3,5,6] => ? = 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => ? = 6
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [9,8,4,5,1,2,3,6,7] => ? = 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [9,3,8,6,1,2,4,5,7] => ? = 7
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [4,3,1,9,6,8,2,5,7] => ? = 2
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,1,5,6,2,3,4,7] => ? = 2
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [8,5,4,1,9,2,3,6,7] => ? = 3
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [8,4,1,9,6,2,3,5,7] => ? = 3
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,9,1,5,2,8,3,4,7] => ? = 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 6
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [9,3,4,8,1,2,5,6,7] => ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [8,4,1,5,9,2,3,6,7] => ? = 2
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,9,8,1,2,3,4,5,7] => ? = 7
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [9,4,1,6,2,8,3,5,7] => ? = 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [5,9,1,2,6,8,3,4,7] => ? = 1
[6,5,3,2,1]
=> [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]
=> [9,8,5,1,2,3,4,6,7] => ? = 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [8,3,1,5,9,2,4,6,7] => ? = 3
[6,4,3,2,1,1]
=> [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]
=> [9,8,1,6,2,3,4,5,7] => ? = 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [9,4,1,2,6,8,3,5,7] => ? = 2
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 7
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [5,8,1,2,9,3,4,6,7] => ? = 7
[6,4,3,2,2,1]
=> [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]
=> [9,8,1,2,6,3,4,5,7] => ? = 7
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,4,1,8,2,3,5,6,7] => ? = 6
[6,5,3,2,1,1]
=> [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]
=> [9,8,1,5,2,3,4,6,7] => ? = 6
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [9,3,8,1,2,4,5,6,7] => ? = 7
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 7
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000906
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ([(0,4),(0,6),(2,10),(3,9),(4,7),(5,2),(5,8),(6,3),(6,7),(7,5),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 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]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 5
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,9),(2,10),(3,12),(4,7),(5,2),(5,8),(6,3),(6,8),(7,9),(8,10),(8,12),(10,11),(11,1),(11,7),(12,4),(12,11)],13)
=> ? = 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 7
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 2
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,5),(4,10),(5,2),(5,11),(6,4),(6,7),(7,10),(8,9),(10,11),(11,1),(11,8)],12)
=> ? = 2
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 3
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,6),(1,8),(2,9),(3,7),(4,2),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,5),(11,9)],12)
=> ? = 3
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ? = 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,8),(4,7),(5,2),(5,8),(6,1),(6,7),(7,10),(8,9),(10,3),(10,5)],11)
=> ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,9),(2,10),(3,8),(4,7),(5,2),(5,11),(6,5),(6,7),(7,11),(8,9),(10,1),(10,8),(11,3),(11,10)],12)
=> ? = 2
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ([(0,4),(0,6),(1,7),(2,9),(3,8),(4,11),(5,3),(5,10),(6,1),(6,11),(7,5),(7,9),(9,10),(10,8),(11,2),(11,7)],12)
=> ? = 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,8),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,10),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1
[6,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ? = 3
[6,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 2
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 7
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 7
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 6
[6,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,5),(4,10),(5,2),(5,11),(6,4),(6,7),(7,10),(8,9),(10,11),(11,1),(11,8)],12)
=> ? = 6
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 7
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 7
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St001636
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ([(0,4),(0,6),(2,10),(3,9),(4,7),(5,2),(5,8),(6,3),(6,7),(7,5),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 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]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 5
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ([(0,5),(0,6),(1,9),(2,10),(3,12),(4,7),(5,2),(5,8),(6,3),(6,8),(7,9),(8,10),(8,12),(10,11),(11,1),(11,7),(12,4),(12,11)],13)
=> ? = 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,7),(3,7),(4,8),(5,1),(5,9),(6,5),(6,8),(8,9),(9,10),(10,2),(10,3)],11)
=> ? = 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 7
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 2
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,5),(4,10),(5,2),(5,11),(6,4),(6,7),(7,10),(8,9),(10,11),(11,1),(11,8)],12)
=> ? = 2
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 3
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,6),(1,8),(2,9),(3,7),(4,2),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,5),(11,9)],12)
=> ? = 3
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ? = 2
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,8),(4,7),(5,2),(5,8),(6,1),(6,7),(7,10),(8,9),(10,3),(10,5)],11)
=> ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,9),(2,10),(3,8),(4,7),(5,2),(5,11),(6,5),(6,7),(7,11),(8,9),(10,1),(10,8),(11,3),(11,10)],12)
=> ? = 2
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ([(0,4),(0,6),(1,7),(2,9),(3,8),(4,11),(5,3),(5,10),(6,1),(6,11),(7,5),(7,9),(9,10),(10,8),(11,2),(11,7)],12)
=> ? = 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,8),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,10),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1
[6,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 7
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ? = 3
[6,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 6
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
=> ? = 2
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 7
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 7
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
=> ? = 6
[6,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,5),(4,10),(5,2),(5,11),(6,4),(6,7),(7,10),(8,9),(10,11),(11,1),(11,8)],12)
=> ? = 6
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,9),(2,8),(3,2),(3,7),(4,1),(4,7),(5,6),(6,3),(6,4),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 7
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 7
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
Matching statistic: St000019
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 4 = 3 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 5 = 4 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [8,3,4,5,7,1,2,6] => ? = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 5 = 4 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,8,1,2,7] => ? = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? = 2 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 4 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [8,9,4,5,6,1,2,3,7] => ? = 2 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 2 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [9,6,4,5,1,8,2,3,7] => ? = 2 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,3,4,8,1,2,5,6] => ? = 2 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 6 = 5 + 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> [9,5,4,1,6,8,2,3,7] => ? = 1 + 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? = 6 + 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8,3,1,5,7,2,4,6] => ? = 2 + 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [8,3,4,5,9,1,2,6,7] => ? = 2 + 1
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,6,5,1,2,3,4,7] => ? = 3 + 1
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [9,4,1,5,6,8,2,3,7] => ? = 1 + 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [8,7,4,1,2,3,5,6] => ? = 6 + 1
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => ? = 5 + 1
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,5,1,6,2,3,4,7] => ? = 3 + 1
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,1,5,6,8,2,4,7] => ? = 1 + 1
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 6 + 1
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [7,4,1,8,2,3,5,6] => ? = 5 + 1
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => ? = 6 + 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [9,8,4,5,1,2,3,6,7] => ? = 3 + 1
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [9,3,8,6,1,2,4,5,7] => ? = 7 + 1
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [4,3,1,9,6,8,2,5,7] => ? = 2 + 1
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [9,8,1,5,6,2,3,4,7] => ? = 2 + 1
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [8,5,4,1,9,2,3,6,7] => ? = 3 + 1
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [8,4,1,9,6,2,3,5,7] => ? = 3 + 1
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,9,1,5,2,8,3,4,7] => ? = 2 + 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 7 = 6 + 1
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [9,3,4,8,1,2,5,6,7] => ? = 3 + 1
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [8,4,1,5,9,2,3,6,7] => ? = 2 + 1
[6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,9,8,1,2,3,4,5,7] => ? = 7 + 1
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [9,4,1,6,2,8,3,5,7] => ? = 3 + 1
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [5,9,1,2,6,8,3,4,7] => ? = 1 + 1
[6,5,3,2,1]
=> [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]
=> [9,8,5,1,2,3,4,6,7] => ? = 7 + 1
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [8,3,1,5,9,2,4,6,7] => ? = 3 + 1
[6,4,3,2,1,1]
=> [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]
=> [9,8,1,6,2,3,4,5,7] => ? = 6 + 1
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [9,4,1,2,6,8,3,5,7] => ? = 2 + 1
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 8 = 7 + 1
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [5,8,1,2,9,3,4,6,7] => ? = 7 + 1
[6,4,3,2,2,1]
=> [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]
=> [9,8,1,2,6,3,4,5,7] => ? = 7 + 1
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,4,1,8,2,3,5,6,7] => ? = 6 + 1
[6,5,3,2,1,1]
=> [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]
=> [9,8,1,5,2,3,4,6,7] => ? = 6 + 1
[6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [9,3,8,1,2,4,5,6,7] => ? = 7 + 1
[6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 7 + 1
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000080The rank of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001090The number of pop-stack-sorts needed to sort a permutation. St001782The order of rowmotion on the set of order ideals of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!