- FindStat

Your data matches 562 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001876

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ 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

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

Description

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

Matching statistic: St001877

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension 2.

Matching statistic: St000655
(load all 50 compositions to match this statistic)

Mapping

Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%

Values

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

Description

The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].

Matching statistic: St000366
(load all 39 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

Description

The number of double descents of a permutation. A double descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) > \pi(i+1) > \pi(i+2)$ .

Matching statistic: St000371
(load all 31 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

Description

The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) > \pi(j) > \pi(k)$ . In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

Matching statistic: St000373
(load all 34 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

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: St000404
(load all 47 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

Description

The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].

Matching statistic: St000408
(load all 72 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

Description

The number of occurrences of the pattern 4231 in a permutation. It is a necessary condition that a permutation

$\pi$ avoids this pattern for the Schubert variety associated to

$\pi$ to be smooth [2].

Matching statistic: St000440
(load all 58 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

Description

The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].

Matching statistic: St000546
(load all 26 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%

Values

[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => 0
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => 0
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 0
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,2,6,3,4,7,5] => 0
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => 0
[6,2,2]
 => [3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,1,6,3,4,5,7] => 0
[5,3,1,1]
 => [5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => 0
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => 0
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => 0
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7,9] => 0
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8] => 0
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,7,6] => 0
[3,3,1,1,1,1]
 => [6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5] => 0
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => 0
[5,3,1,1,1]
 => [5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => 0
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,1,3,4,8,5,6,7] => ? = 0
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ? = 0
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => 0
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [4,5,1,2,3,7,6] => ? = 0
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 0
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => 0
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => ? = 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [2,6,1,3,4,5,7,8,9] => ? = 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? = 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,2,3,5,7,6] => 0
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,8,4,5,7] => ? = 0
[5,3,3,2]
 => [6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,6,7,5] => ? = 1
[5,3,3,1,1]
 => [6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,2,4,5,6,7] => ? = 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6,8,9,10] => ? = 0
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,4,2,1]
 => [5,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 0
[5,4,3,3,1]
 => [6,5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [3,4,1,2,6,7,5] => ? = 0
[6,4,4,2,1]
 => [3,3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? => ? = 0
[6,4,3,3,1]
 => [6,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => ? = 1
[6,4,3,2,2]
 => [3,3,3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? => ? = 1
[5,5,3,3,1]
 => [10,6,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,2]
 => [10,3,1,1,1,1]
 => [1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,5,3,2,1,1]
 => [10,3,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? => ? = 1
[5,4,4,3,1]
 => [5,3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,7,1,8,4,5,6] => ? = 0
[5,4,4,2,2]
 => [5,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? => ? = 1
[5,4,4,2,1,1]
 => [5,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? => ? = 1
[5,4,3,3,1,1]
 => [6,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => 0
[6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 0
[5,5,3,2,2,1]
 => ?
 => ?
 => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[5,4,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ? => ? = 1
[6,5,4,2,2]
 => ?
 => ?
 => ? => ? = 0
[6,5,3,2,2]
 => ?
 => ?
 => ? => ? = 1
[6,4,4,3,1]
 => ?
 => ?
 => ? => ? = 1
[6,5,3,3,1]
 => ?
 => ?
 => ? => ? = 0
[7,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,4,4,3,2,1]
 => ?
 => ?
 => ? => ? = 0
[6,6,5,3,3,2,1]
 => ?
 => ?
 => ? => ? = 1
[6,6,5,4,2,2,1]
 => ?
 => ?
 => ? => ? = 2
[6,6,5,4,3,1,1]
 => ?
 => ?
 => ? => ? = 1
[7,5,5,4,3,2]
 => ?
 => ?
 => ? => ? = 1
[7,6,4,4,3,2]
 => ?
 => ?
 => ? => ? = 2
[6,5,5,4,3,1]
 => ?
 => ?
 => ? => ? = 0

Description

The number of global descents of a permutation. The global descents are the integers in the set

$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$ In particular, if

$i\in C(\pi)$ then

$i$ is a descent. For the number of global ascents, see [[St000234]].

The following 552 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000011The number of touch points (or returns) of a Dyck path. St000382The first part of an integer composition. St000842The breadth of a permutation. St000920The logarithmic height of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000627The exponent of a binary word. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000297The number of leading ones in a binary word. St000632The jump number of the poset. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000264The girth of a graph, which is not a tree. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000660The number of rises of length at least 3 of a Dyck path. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000220The number of occurrences of the pattern 132 in a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000486The number of cycles of length at least 3 of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000929The constant term of the character polynomial of an integer partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000056The decomposition (or block) number of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000570The Edelman-Greene number of a permutation. St000652The maximal difference between successive positions of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001741The largest integer such that all patterns of this size are contained in the permutation. St001696The natural major index of a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000042The number of crossings of a perfect matching. St000234The number of global ascents of a permutation. St000352The Elizalde-Pak rank of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000054The first entry of the permutation. St000257The number of distinct parts of a partition that occur at least twice. St000733The row containing the largest entry of a standard tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000439The position of the first down step of a Dyck path. St000862The number of parts of the shifted shape of a permutation. St000013The height of a Dyck path. St000057The Shynar inversion number of a standard tableau. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000218The number of occurrences of the pattern 213 in a permutation. St000534The number of 2-rises of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001394The genus of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000374The number of exclusive right-to-left minima of a permutation. St000651The maximal size of a rise in a permutation. St000742The number of big ascents of a permutation after prepending zero. St000451The length of the longest pattern of the form k 1 2. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000527The width of the poset. St000022The number of fixed points of a permutation. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000153The number of adjacent cycles of a permutation. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000237The number of small exceedances. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000323The minimal crossing number of a graph. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000370The genus of a graph. St000389The number of runs of ones of odd length in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000648The number of 2-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001561The value of the elementary symmetric function evaluated at 1. St001584The area statistic between a Dyck path and its bounce path. St001593This is the number of standard Young tableaux of the given shifted shape. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001730The number of times the path corresponding to a binary word crosses the base line. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000007The number of saliances of the permutation. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000480The number of lower covers of a partition in dominance order. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000536The pathwidth of a graph. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000657The smallest part of an integer composition. St000659The number of rises of length at least 2 of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000769The major index of a composition regarded as a word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000847The number of standard Young tableaux whose descent set is the binary word. St000883The number of longest increasing subsequences of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000900The minimal number of repetitions of a part in an integer composition. St000913The number of ways to refine the partition into singletons. St000983The length of the longest alternating subword. St000996The number of exclusive left-to-right maxima of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001092The number of distinct even parts of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001437The flex of a binary word. St001568The smallest positive integer that does not appear twice in the partition. St001587Half of the largest even part of an integer partition. St001673The degree of asymmetry of an integer composition. St001732The number of peaks visible from the left. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001949The rigidity index of a graph. St000147The largest part of an integer partition. St000402Half the size of the symmetry class of a permutation. St000481The number of upper covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000905The number of different multiplicities of parts of an integer composition. St000995The largest even part of an integer partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001432The order dimension of the partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000010The length of the partition. St000159The number of distinct parts of the integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001261The Castelnuovo-Mumford regularity of a graph. St001471The magnitude of a Dyck path. St001484The number of singletons of an integer partition. St001670The connected partition number of a graph. St000346The number of coarsenings of a partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000511The number of invariant subsets when acting with a permutation of given cycle type. St001722The number of minimal chains with small intervals between a binary word and the top element. St000053The number of valleys of the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000124The cardinality of the preimage of the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000246The number of non-inversions of a permutation. St000386The number of factors DDU in a Dyck path. St000463The number of admissible inversions of a permutation. St000647The number of big descents of a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000884The number of isolated descents of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000617The number of global maxima of a Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St000058The order of a permutation. St000097The order of the largest clique of the graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000210Minimum over maximum difference of elements in cycles. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000701The protection number of a binary tree. St000990The first ascent of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001468The smallest fixpoint of a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000665The number of rafts of a permutation. St001330The hat guessing number of a graph. St000478Another weight of a partition according to Alladi. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001850The number of Hecke atoms of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000353The number of inner valleys of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001665The number of pure excedances of a permutation. St000098The chromatic number of a graph. St000201The number of leaf nodes in a binary tree. St000662The staircase size of the code of a permutation. St000834The number of right outer peaks of a permutation. St001725The harmonious chromatic number of a graph. St000401The size of the symmetry class of a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000232The number of crossings of a set partition. St000358The number of occurrences of the pattern 31-2. St000441The number of successions of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001510The number of self-evacuating linear extensions of a finite poset. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000253The crossing number of a set partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000710The number of big deficiencies of a permutation. St000729The minimal arc length of a set partition. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001075The minimal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000069The number of maximal elements of a poset. St000354The number of recoils of a permutation. St000764The number of strong records in an integer composition. St000836The number of descents of distance 2 of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001801Half the number of preimage-image pairs of different parity in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000461The rix statistic of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000035The number of left outer peaks of a permutation. St000383The last part of an integer composition. St000711The number of big exceedences of a permutation. St000886The number of permutations with the same antidiagonal sums. St000931The number of occurrences of the pattern UUU in a Dyck path. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001469The holeyness of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000381The largest part of an integer composition. St000619The number of cyclic descents of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000444The length of the maximal rise of a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000331The number of upper interactions of a Dyck path. St000443The number of long tunnels of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001524The degree of symmetry of a binary word. St001811The Castelnuovo-Mumford regularity of a permutation. St001947The number of ties in a parking function. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000120The number of left tunnels of a Dyck path. St000306The bounce count of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001423The number of distinct cubes in a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000891The number of distinct diagonal sums of a permutation matrix. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000144The pyramid weight of the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000322The skewness of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St001095The number of non-isomorphic posets with precisely one further covering relation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001434The number of negative sum pairs of a signed permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000031The number of cycles in the cycle decomposition of a permutation. St000640The rank of the largest boolean interval in a poset. St000741The Colin de Verdière graph invariant. St000910The number of maximal chains of minimal length in a poset. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001115The number of even descents of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000654The first descent of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000245The number of ascents of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000837The number of ascents of distance 2 of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001760The number of prefix or suffix reversals needed to sort a permutation. St000863The length of the first row of the shifted shape of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one.