- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001880: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 153 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

to poset

Description

Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.

Map

to ordered tree

Description

Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path

$D$ of semilength

$n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths

$D_1,\dots,D_k$ of respective semilengths

$n_1,\dots,n_k$ (so one has

$n = n_1 + \dots n_k$ ) each of which has no returns.
Denote by

$\tilde D_i$ the path of semilength

$n_i-1$ obtained from

$D_i$ by removing the initial up- and the final down-step.
This map then sends

$D$ to the tree

$T$ having a root note with ordered children

$T_1,\dots,T_k$ which are again ordered trees computed from

$D_1,\dots,D_k$ respectively.
The unique path of semilength

$1$ is sent to the tree consisting of a single node.