- FindStat

Identifier

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = 7\ q + 14\ q^{2} + 14\ q^{3} + 7\ q^{4}$

$F_{5} = 42\ q + 105\ q^{2} + 135\ q^{3} + 105\ q^{4} + 42\ q^{5}$

$F_{6} = 429\ q + 1287\ q^{2} + 2002\ q^{3} + 2002\ q^{4} + 1287\ q^{5} + 429\ q^{6}$

Description

The number of maximal elements of a poset.

Map

to Dyck path

Description

The Dyck path determined by the last diagonal of the monotone triangle of an alternating sign matrix.

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

$j < \boldsymbol{m}_i$ .