Identifier
-
Mp00229:
Dyck paths
—Delest-Viennot⟶
Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤ
Values
[1,0] => [1,0] => [1] => ([],1) => 1
[1,0,1,0] => [1,1,0,0] => [2,1] => ([],2) => 2
[1,1,0,0] => [1,0,1,0] => [1,2] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => ([(1,2)],3) => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => ([(0,2),(1,2)],3) => 1
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,3,2] => ([(0,1),(0,2)],3) => 1
[1,1,0,1,0,0] => [1,1,1,0,0,0] => [3,1,2] => ([(1,2)],3) => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => ([(1,2),(2,3)],4) => 2
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 2
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 2
[1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => ([(1,2),(2,3)],4) => 2
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => 1
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => ([(0,3),(1,2)],4) => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 2
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 2
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 1
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => 2
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5) => 2
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 2
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => 2
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => 2
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => 1
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => 2
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6) => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6) => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6) => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6) => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6) => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6) => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6) => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,3,5] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6) => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6) => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6) => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6) => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6) => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,1,3,4] => ([(0,5),(1,4),(1,5),(4,2),(5,3)],6) => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6) => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,1,3,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6) => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6) => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,3,6,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6) => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6) => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6) => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 1
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Description
The length of the shortest maximal antichain in a poset.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let D be a Dyck path of semilength n. The parallelogram polyomino γ(D) is defined as follows: let ˜D=d0d1…d2n+1 be the Dyck path obtained by prepending an up step and appending a down step to D. Then, the upper path of γ(D) corresponds to the sequence of steps of ˜D with even indices, and the lower path of γ(D) corresponds to the sequence of steps of ˜D with odd indices.
The Delest-Viennot bijection β returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path (γ(−1)∘β)(D).
Let D be a Dyck path of semilength n. The parallelogram polyomino γ(D) is defined as follows: let ˜D=d0d1…d2n+1 be the Dyck path obtained by prepending an up step and appending a down step to D. Then, the upper path of γ(D) corresponds to the sequence of steps of ˜D with even indices, and the lower path of γ(D) corresponds to the sequence of steps of ˜D with odd indices.
The Delest-Viennot bijection β returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path (γ(−1)∘β)(D).
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength n in an n×n square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength n in an n×n square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
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