Identifier
-
Mp00151:
Permutations
—to cycle type⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤ
Values
[1,2,3] => {{1},{2},{3}} => [1,2,3] => ([(0,2),(2,1)],3) => 2
[1,2,3,4] => {{1},{2},{3},{4}} => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[1,3,2,4] => {{1},{2,3},{4}} => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[1,3,2,4,5] => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[1,3,4,2,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
[1,4,2,3,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
[1,4,3,2,5] => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
[1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}} => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[1,2,4,3,5,6] => {{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[1,2,4,5,3,6] => {{1},{2},{3,4,5},{6}} => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
[1,2,5,3,4,6] => {{1},{2},{3,4,5},{6}} => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
[1,2,5,4,3,6] => {{1},{2},{3,5},{4},{6}} => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 10
[1,3,2,4,5,6] => {{1},{2,3},{4},{5},{6}} => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
[1,3,4,2,5,6] => {{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
[1,3,4,5,2,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[1,3,5,2,4,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[1,3,5,4,2,6] => {{1},{2,3,5},{4},{6}} => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
[1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
[1,4,2,5,3,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[1,4,3,2,5,6] => {{1},{2,4},{3},{5},{6}} => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 10
[1,4,3,5,2,6] => {{1},{2,4,5},{3},{6}} => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
[1,4,5,2,3,6] => {{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 8
[1,4,5,3,2,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[1,5,2,3,4,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[1,5,2,4,3,6] => {{1},{2,3,5},{4},{6}} => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
[1,5,3,2,4,6] => {{1},{2,4,5},{3},{6}} => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
[1,5,3,4,2,6] => {{1},{2,5},{3},{4},{6}} => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
[1,5,4,2,3,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
[1,5,4,3,2,6] => {{1},{2,5},{3,4},{6}} => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
[1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}} => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,2,3,4,6,5,7] => {{1},{2},{3},{4},{5,6},{7}} => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 7
[1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}} => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 7
[1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}} => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 8
[1,2,3,6,4,5,7] => {{1},{2},{3},{4,5,6},{7}} => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 8
[1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}} => [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 11
[1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}} => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 7
[1,2,4,5,3,6,7] => {{1},{2},{3,4,5},{6},{7}} => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 8
[1,2,4,5,6,3,7] => {{1},{2},{3,4,5,6},{7}} => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 9
[1,2,4,6,3,5,7] => {{1},{2},{3,4,5,6},{7}} => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 9
[1,2,4,6,5,3,7] => {{1},{2},{3,4,6},{5},{7}} => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 11
[1,2,5,3,4,6,7] => {{1},{2},{3,4,5},{6},{7}} => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 8
[1,2,5,3,6,4,7] => {{1},{2},{3,4,5,6},{7}} => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 9
[1,2,5,4,3,6,7] => {{1},{2},{3,5},{4},{6},{7}} => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 11
[1,2,5,4,6,3,7] => {{1},{2},{3,5,6},{4},{7}} => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 11
[1,2,5,6,3,4,7] => {{1},{2},{3,5},{4,6},{7}} => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 9
[1,2,5,6,4,3,7] => {{1},{2},{3,4,5,6},{7}} => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 9
[1,2,6,3,4,5,7] => {{1},{2},{3,4,5,6},{7}} => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 9
[1,2,6,3,5,4,7] => {{1},{2},{3,4,6},{5},{7}} => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 11
[1,2,6,4,3,5,7] => {{1},{2},{3,5,6},{4},{7}} => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 11
[1,2,6,4,5,3,7] => {{1},{2},{3,6},{4},{5},{7}} => [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 13
[1,2,6,5,3,4,7] => {{1},{2},{3,4,5,6},{7}} => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 9
[1,2,6,5,4,3,7] => {{1},{2},{3,6},{4,5},{7}} => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 17
[1,3,2,4,5,6,7] => {{1},{2,3},{4},{5},{6},{7}} => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 7
[1,3,2,4,6,5,7] => {{1},{2,3},{4},{5,6},{7}} => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 8
[1,3,4,2,5,6,7] => {{1},{2,3,4},{5},{6},{7}} => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 8
[1,3,4,5,2,6,7] => {{1},{2,3,4,5},{6},{7}} => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 9
[1,3,4,5,6,2,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,3,4,6,2,5,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,3,4,6,5,2,7] => {{1},{2,3,4,6},{5},{7}} => [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 12
[1,3,5,2,4,6,7] => {{1},{2,3,4,5},{6},{7}} => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 9
[1,3,5,2,6,4,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,3,5,4,2,6,7] => {{1},{2,3,5},{4},{6},{7}} => [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 11
[1,3,5,4,6,2,7] => {{1},{2,3,5,6},{4},{7}} => [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 11
[1,3,5,6,2,4,7] => {{1},{2,3,5},{4,6},{7}} => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 9
[1,3,5,6,4,2,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,3,6,2,4,5,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,3,6,2,5,4,7] => {{1},{2,3,4,6},{5},{7}} => [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 12
[1,3,6,4,2,5,7] => {{1},{2,3,5,6},{4},{7}} => [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 11
[1,3,6,4,5,2,7] => {{1},{2,3,6},{4},{5},{7}} => [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 13
[1,3,6,5,2,4,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,3,6,5,4,2,7] => {{1},{2,3,6},{4,5},{7}} => [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 16
[1,4,2,3,5,6,7] => {{1},{2,3,4},{5},{6},{7}} => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 8
[1,4,2,5,3,6,7] => {{1},{2,3,4,5},{6},{7}} => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 9
[1,4,2,5,6,3,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,4,2,6,3,5,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,4,2,6,5,3,7] => {{1},{2,3,4,6},{5},{7}} => [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 12
[1,4,3,2,5,6,7] => {{1},{2,4},{3},{5},{6},{7}} => [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 11
[1,4,3,5,2,6,7] => {{1},{2,4,5},{3},{6},{7}} => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 11
[1,4,3,5,6,2,7] => {{1},{2,4,5,6},{3},{7}} => [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 12
[1,4,3,6,2,5,7] => {{1},{2,4,5,6},{3},{7}} => [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 12
[1,4,5,2,3,6,7] => {{1},{2,4},{3,5},{6},{7}} => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 9
[1,4,5,2,6,3,7] => {{1},{2,4},{3,5,6},{7}} => [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 9
[1,4,5,3,2,6,7] => {{1},{2,3,4,5},{6},{7}} => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 9
[1,4,5,3,6,2,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,4,5,6,2,3,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,4,5,6,3,2,7] => {{1},{2,4,6},{3,5},{7}} => [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 15
[1,4,6,2,3,5,7] => {{1},{2,4},{3,5,6},{7}} => [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 9
[1,4,6,2,5,3,7] => {{1},{2,4},{3,6},{5},{7}} => [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 10
[1,4,6,3,2,5,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,4,6,3,5,2,7] => {{1},{2,3,4,6},{5},{7}} => [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 12
[1,4,6,5,2,3,7] => {{1},{2,4,5},{3,6},{7}} => [1,4,6,5,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 12
[1,4,6,5,3,2,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,5,2,3,4,6,7] => {{1},{2,3,4,5},{6},{7}} => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 9
[1,5,2,3,6,4,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
[1,5,2,4,3,6,7] => {{1},{2,3,5},{4},{6},{7}} => [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 11
[1,5,2,4,6,3,7] => {{1},{2,3,5,6},{4},{7}} => [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 11
[1,5,2,6,3,4,7] => {{1},{2,3,5},{4,6},{7}} => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 9
[1,5,2,6,4,3,7] => {{1},{2,3,4,5,6},{7}} => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 10
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Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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