- FindStat

Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001876

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1

Description

The number of 2-regular simple modules in the incidence algebra of the lattice.

Matching statistic: St001877

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,3,2] => [[6,5,3],[4,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => [3,2,3] => [[6,4,3],[3,2]]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2

Description

Number of indecomposable injective modules with projective dimension 2.

Matching statistic: St001964

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 0
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,2,3] => [[4,2,1,1],[1]]
 => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7)
 => ? = 0

Description

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Matching statistic: St000454

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St000455

Mapping

Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[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]
 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[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]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[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]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[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]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000022

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [6,3,1,7,5,4,2] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,3,1,6,5,4,2] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [2,7,6,1,5,4,3] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [2,7,6,1,5,4,3] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 1

Description

The number of fixed points of a permutation.

Matching statistic: St000373

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [6,3,1,7,5,4,2] => ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,3,1,6,5,4,2] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 2 = 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: St001394

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [6,3,1,7,5,4,2] => ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,3,1,6,5,4,2] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [2,7,6,1,5,4,3] => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 2 = 1 + 1

Description

The genus of a permutation. The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$ where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

Matching statistic: St000031

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 1 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [6,3,1,7,5,4,2] => ? = 1 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,3,1,6,5,4,2] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => ? = 1 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2

Description

The number of cycles in the cycle decomposition of a permutation.

Matching statistic: St000994

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [5,3,1,7,6,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [5,3,1,7,6,4,2] => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [2,7,1,6,5,4,3] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 1 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 1 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [6,3,1,7,5,4,2] => ? = 1 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,3,1,6,5,4,2] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [6,1,7,5,4,3,2] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [5,1,7,6,4,3,2] => ? = 1 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [2,7,6,1,5,4,3] => ? = 0 + 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 3 = 1 + 2

Description

The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation

$\pi$ is an index

$i$ such that

$\pi^{-1}(i) < i > \pi(i)$ . Analogously, a '''cycle valley''' is an index

$i$ such that

$\pi^{-1}(i) > i < \pi(i)$ . Clearly, every cycle of

$\pi$ contains as many peaks as valleys.

The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation.