- FindStat

Identifier

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2384 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

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.

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 $j < \boldsymbol{m}_i$.