- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000527: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => ([],5) => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => ([],6) => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => ([],7) => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 138 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => ([],8) => 8

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

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

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

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

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$F_{1} = q^{2}$

$F_{2} = q^{2} + q^{3}$

$F_{3} = q^{2} + 3\ q^{3} + q^{4}$

$F_{4} = q^{2} + 7\ q^{3} + 5\ q^{4} + q^{5}$

$F_{5} = q^{2} + 15\ q^{3} + 18\ q^{4} + 7\ q^{5} + q^{6}$

Description

The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

Map

Hessenberg poset

Description

The Hessenberg poset of a Dyck path.
Let

$D$ be a Dyck path of semilength

$n$ , regarded as a subdiagonal path from

$(0,0)$ to

$(n,n)$ , and let

$\boldsymbol{m}_i$ be the

$x$ -coordinate of the

$i$ -th up step.
Then the Hessenberg poset (or natural unit interval order) corresponding to

$D$ has elements

$\{1,\dots,n\}$ with

$i < j$ if

$j < \boldsymbol{m}_i$ .

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.