Identifier
-
Mp00150:
Perfect matchings
—to Dyck path⟶
Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[(1,5),(2,3),(4,6)] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 3
[(1,6),(2,3),(4,5)] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 3
[(1,3),(2,5),(4,6)] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 3
[(1,3),(2,6),(4,5)] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 3
[(1,2),(3,4),(5,6),(7,8)] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,8),(2,6),(3,4),(5,7)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,7),(2,6),(3,4),(5,8)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,6),(2,7),(3,4),(5,8)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,6),(2,8),(3,4),(5,7)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,7),(2,8),(3,4),(5,6)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,8),(2,7),(3,4),(5,6)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,4),(2,8),(3,6),(5,7)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,4),(2,7),(3,6),(5,8)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,4),(2,6),(3,7),(5,8)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,6),(2,4),(3,7),(5,8)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,7),(2,4),(3,6),(5,8)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,8),(2,4),(3,6),(5,7)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,8),(2,4),(3,7),(5,6)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,7),(2,4),(3,8),(5,6)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,6),(2,4),(3,8),(5,7)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,4),(2,6),(3,8),(5,7)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,4),(2,7),(3,8),(5,6)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,4),(2,8),(3,7),(5,6)] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[(1,7),(2,3),(4,5),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,8),(2,3),(4,5),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,9),(2,3),(4,5),(6,7),(8,10)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,3),(4,5),(6,7),(8,9)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,3),(2,7),(4,5),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,2),(3,7),(4,5),(6,8),(9,10)] => [1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[(1,2),(3,8),(4,5),(6,7),(9,10)] => [1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[(1,3),(2,8),(4,5),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,3),(2,9),(4,5),(6,7),(8,10)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,2),(3,9),(4,5),(6,7),(8,10)] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[(1,2),(3,10),(4,5),(6,7),(8,9)] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[(1,3),(2,10),(4,5),(6,7),(8,9)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,2),(3,5),(4,7),(6,8),(9,10)] => [1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[(1,3),(2,5),(4,7),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,5),(2,3),(4,7),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,5),(2,3),(4,8),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,3),(2,5),(4,8),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[(1,2),(3,5),(4,8),(6,7),(9,10)] => [1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[(1,10),(2,9),(3,7),(4,5),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,10),(3,7),(4,5),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,10),(3,7),(4,5),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,10),(3,8),(4,5),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,9),(3,8),(4,5),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,9),(3,7),(4,5),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,8),(3,7),(4,5),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,8),(3,7),(4,5),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,7),(3,8),(4,5),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,7),(3,8),(4,5),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,7),(3,9),(4,5),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,8),(3,9),(4,5),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,2),(3,5),(4,9),(6,7),(8,10)] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[(1,3),(2,5),(4,9),(6,7),(8,10)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,5),(2,3),(4,9),(6,7),(8,10)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,5),(2,3),(4,10),(6,7),(8,9)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,3),(2,5),(4,10),(6,7),(8,9)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,2),(3,5),(4,10),(6,7),(8,9)] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[(1,7),(2,8),(3,10),(4,5),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,7),(3,10),(4,5),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,7),(3,10),(4,5),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,7),(3,9),(4,5),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,8),(3,9),(4,5),(6,7)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,8),(3,10),(4,5),(6,7)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,9),(3,10),(4,5),(6,7)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,9),(3,10),(4,5),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,10),(3,9),(4,5),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,10),(3,9),(4,5),(6,7)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,10),(3,8),(4,5),(6,7)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,9),(3,8),(4,5),(6,7)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,10),(3,9),(4,7),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,9),(3,10),(4,7),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,8),(3,10),(4,7),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,7),(3,10),(4,8),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,5),(3,9),(4,7),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,5),(3,10),(4,7),(6,8)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,5),(3,10),(4,7),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,5),(3,10),(4,8),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,5),(3,9),(4,8),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,5),(3,9),(4,7),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,5),(3,8),(4,7),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,5),(3,8),(4,7),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,7),(3,9),(4,8),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,8),(3,9),(4,7),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,9),(3,8),(4,7),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,10),(3,8),(4,7),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,10),(3,7),(4,8),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,9),(3,7),(4,8),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,8),(3,7),(4,9),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,7),(3,8),(4,9),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,10),(2,5),(3,7),(4,8),(6,9)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,5),(3,7),(4,8),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,5),(3,7),(4,9),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,7),(2,5),(3,8),(4,9),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,5),(2,3),(4,7),(6,9),(8,10)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,3),(2,5),(4,7),(6,9),(8,10)] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,2),(3,5),(4,7),(6,9),(8,10)] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[(1,7),(2,8),(3,5),(4,9),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,8),(2,7),(3,5),(4,9),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[(1,9),(2,7),(3,5),(4,8),(6,10)] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
>>> Load all 193 entries. <<<
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
Barnabei-Castronuovo involution
Description
The Barnabei-Castronuovo Schützenberger involution on Dyck paths.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let $D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is $\dots, 1, D_2, 1, D_1$.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let $D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is $\dots, 1, D_2, 1, D_1$.
Map
to Dyck path
Description
The Dyck path corresponding to the opener-closer sequence of the perfect matching.
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 $j < \boldsymbol{m}_i$.
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 $j < \boldsymbol{m}_i$.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!