- FindStat

Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001875

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules with projective dimension at most 1.

Matching statistic: St001622

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .