Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Images
=>
Cc0012;cc-rep-0Cc0029;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>([(0,1),(0,2),(1,3),(2,3)],4)
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
[(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
[(1,4),(2,3)]=>[4,3,2,1]=>[2,3,4,1]=>([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,9),(5,9),(6,7),(6,10),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,12),(8,9),(9,10),(9,12),(10,11),(12,11)],13)
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,17),(6,17),(7,18),(8,18),(10,13),(11,12),(12,9),(13,9),(14,11),(15,10),(16,12),(16,13),(17,11),(17,16),(18,10),(18,16)],19)
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,9),(4,9),(5,10),(6,10),(7,11),(8,11),(9,14),(10,14),(10,15),(11,12),(12,13),(13,15),(14,16),(15,16)],17)
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,16),(3,16),(4,15),(5,14),(6,17),(7,17),(8,14),(8,15),(10,13),(11,9),(12,9),(13,11),(14,10),(15,10),(16,12),(17,11),(17,12)],18)
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,3,5,1,2,4,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,9),(5,9),(6,10),(7,10),(8,11),(9,12),(10,13),(11,12),(13,11)],14)
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,9),(7,9),(8,10),(9,11),(10,11),(12,10)],13)
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,2,4,5,3,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,13),(3,9),(4,9),(5,10),(6,11),(7,11),(8,10),(8,14),(9,16),(10,15),(11,12),(12,14),(13,16),(14,15),(15,13)],17)
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,2,1,4,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,11),(3,10),(4,9),(5,9),(6,12),(7,12),(8,10),(8,11),(9,16),(10,15),(11,15),(12,14),(13,16),(15,13),(16,14)],17)
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,10),(4,10),(5,9),(6,9),(7,11),(8,11),(9,15),(10,14),(11,12),(12,13),(13,14),(13,15),(14,16),(15,16)],17)
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[5,1,4,6,8,2,7,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,9),(7,9),(8,10),(9,11),(10,11),(12,10)],13)
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[2,8,1,3,6,5,7,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,13),(4,13),(5,13),(6,11),(7,12),(8,10),(9,12),(10,13),(11,10),(12,11)],14)
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,18),(3,17),(4,16),(5,9),(6,17),(6,18),(7,14),(7,19),(8,16),(8,19),(10,15),(11,13),(12,13),(13,9),(14,11),(15,14),(16,12),(17,10),(18,10),(19,11),(19,12)],20)
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,2,4,3,6,1,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,13),(3,13),(4,13),(5,9),(6,9),(7,10),(8,10),(9,12),(10,11),(11,13),(13,12)],14)
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,2,4,1,7,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,12),(4,12),(5,12),(6,12),(7,10),(8,11),(9,11),(10,12),(11,10)],13)
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,4,6,7,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,13),(3,9),(4,9),(5,10),(6,11),(7,11),(8,10),(8,14),(9,16),(10,15),(11,12),(12,14),(13,16),(14,15),(15,13)],17)
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,4,3,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,11),(3,10),(4,9),(5,9),(6,12),(7,12),(8,10),(8,11),(9,16),(10,15),(11,15),(12,14),(13,16),(15,13),(16,14)],17)
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,4,2,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,1,8,4,5,3,2,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,14),(3,14),(4,14),(5,9),(6,10),(7,10),(8,9),(8,11),(9,13),(10,11),(11,13),(12,14),(13,12)],15)
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,1,6,3,4,2,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,9),(4,12),(5,13),(6,14),(7,15),(8,9),(8,11),(9,16),(10,15),(11,16),(12,14),(13,12),(14,11),(15,13)],17)
[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,21),(2,21),(3,17),(4,16),(5,18),(6,16),(6,20),(7,17),(7,20),(8,18),(8,19),(10,11),(11,14),(12,10),(13,10),(14,9),(15,9),(16,12),(17,13),(18,15),(19,14),(19,15),(20,12),(20,13),(21,11),(21,19)],22)
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,3,5,8,2,4,1,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[2,3,4,1,8,5,7,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,12),(3,13),(4,10),(5,9),(6,11),(7,11),(8,9),(8,10),(9,15),(10,15),(11,12),(12,14),(13,16),(14,16),(15,13)],17)
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,10),(4,10),(5,12),(6,11),(7,13),(8,14),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,9),(4,9),(5,10),(6,10),(7,11),(8,11),(9,14),(10,14),(10,15),(11,12),(12,13),(13,15),(14,16),(15,16)],17)
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,16),(3,16),(4,15),(5,14),(6,17),(7,17),(8,14),(8,15),(10,13),(11,9),(12,9),(13,11),(14,10),(15,10),(16,12),(17,11),(17,12)],18)
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,6,7,8,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,12),(3,13),(4,10),(5,9),(6,11),(7,11),(8,9),(8,10),(9,15),(10,15),(11,12),(12,14),(13,16),(14,16),(15,13)],17)
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,14),(3,17),(4,16),(5,13),(6,12),(7,14),(7,15),(8,16),(8,17),(10,12),(11,13),(12,9),(13,9),(14,10),(15,10),(16,11),(17,11)],18)
[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,3,5,6,2,7,1,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12)
[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[3,5,6,7,8,4,2,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,17),(2,16),(3,15),(4,18),(5,14),(6,13),(6,18),(7,16),(7,19),(8,17),(8,19),(10,14),(11,10),(12,10),(13,9),(14,15),(15,13),(16,11),(17,12),(18,9),(19,11),(19,12)],20)
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,8,5,6,1,4,2,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,9),(4,9),(5,10),(6,10),(7,11),(8,11),(9,14),(10,14),(10,15),(11,12),(12,13),(13,15),(14,16),(15,16)],17)
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,6,1,7,2,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,6,5,7,3,4]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,9),(4,9),(5,10),(6,10),(7,11),(8,11),(9,15),(10,16),(11,12),(12,13),(12,16),(13,14),(14,15),(16,14)],17)
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,5,4,2,6,1,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,9),(6,10),(7,10),(8,9),(8,11),(9,12),(10,13),(11,12),(13,11)],14)
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,4,2,5,6,8,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[2,7,4,1,6,5,8,3]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
[(1,6),(2,4),(3,5),(7,8),(9,10)]=>[6,4,5,2,3,1,8,7,10,9]=>[3,5,2,6,4,1,8,7,10,9]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(2,17),(3,17),(4,17),(5,17),(6,11),(7,11),(8,12),(9,12),(10,13),(11,14),(12,14),(12,15),(13,15),(14,16),(15,16),(17,13)],18)
[(1,2),(3,8),(4,6),(5,7),(9,10)]=>[2,1,8,6,7,4,5,3,10,9]=>[2,1,5,7,4,8,6,3,10,9]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(2,17),(3,17),(4,17),(5,17),(6,12),(7,12),(8,11),(9,11),(10,13),(11,15),(12,14),(13,14),(13,15),(14,16),(15,16),(17,13)],18)
[(1,10),(2,8),(3,9),(4,6),(5,7)]=>[10,8,9,6,7,4,5,2,3,1]=>[3,5,2,7,4,9,6,10,8,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,12),(10,11),(12,11)],13)
[(1,2),(3,4),(5,10),(6,8),(7,9)]=>[2,1,4,3,10,8,9,6,7,5]=>[2,1,4,3,7,9,6,10,8,5]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(2,17),(3,17),(4,17),(5,17),(6,11),(7,11),(8,12),(9,12),(10,13),(11,14),(12,14),(12,15),(13,15),(14,16),(15,16),(17,13)],18)
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
Map
lattice of intervals
Description
The lattice of intervals of a permutation.
An interval of a permutation $\pi$ is a possibly empty interval of values that appear in consecutive positions of $\pi$. The lattice of intervals of $\pi$ has as elements the intervals of $\pi$, ordered by set inclusion.
An interval of a permutation $\pi$ is a possibly empty interval of values that appear in consecutive positions of $\pi$. The lattice of intervals of $\pi$ has as elements the intervals of $\pi$, ordered by set inclusion.
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