Identifier
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Images
=>
Cc0014;cc-rep-2Cc0029;cc-rep-3
[1]=>[1]=>([],1)=>([(0,1)],2) [1,2]=>[1,2]=>([(0,1)],2)=>([(0,2),(2,1)],3) [2,1]=>[2,1]=>([(0,1)],2)=>([(0,2),(2,1)],3) [1,2,3]=>[1,2,3]=>([(0,2),(2,1)],3)=>([(0,3),(2,1),(3,2)],4) [1,3,2]=>[1,3,2]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) [2,1,3]=>[2,1,3]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) [2,3,1]=>[3,2,1]=>([(0,2),(2,1)],3)=>([(0,3),(2,1),(3,2)],4) [3,1,2]=>[3,1,2]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) [3,2,1]=>[2,3,1]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) [1,2,3,4]=>[1,2,3,4]=>([(0,3),(2,1),(3,2)],4)=>([(0,4),(2,3),(3,1),(4,2)],5) [1,2,4,3]=>[1,2,4,3]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [1,3,2,4]=>[1,3,2,4]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [1,3,4,2]=>[1,4,3,2]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [1,4,2,3]=>[1,4,2,3]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [1,4,3,2]=>[1,3,4,2]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [2,1,3,4]=>[2,1,3,4]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [2,1,4,3]=>[2,1,4,3]=>([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)=>([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9) [2,3,1,4]=>[3,2,1,4]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [2,3,4,1]=>[4,2,3,1]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [2,4,1,3]=>[4,2,1,3]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [2,4,3,1]=>[3,2,4,1]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [3,1,2,4]=>[3,1,2,4]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [3,1,4,2]=>[3,4,1,2]=>([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)=>([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9) [3,2,1,4]=>[2,3,1,4]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [3,2,4,1]=>[4,3,2,1]=>([(0,3),(2,1),(3,2)],4)=>([(0,4),(2,3),(3,1),(4,2)],5) [3,4,1,2]=>[4,1,3,2]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [3,4,2,1]=>[2,4,3,1]=>([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)=>([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) [4,1,2,3]=>[4,1,2,3]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [4,1,3,2]=>[4,3,1,2]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [4,2,3,1]=>[3,4,2,1]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [4,3,2,1]=>[2,3,4,1]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) [1,2,3,4,5]=>[1,2,3,4,5]=>([(0,4),(2,3),(3,1),(4,2)],5)=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6) [1,2,3,5,4]=>[1,2,3,5,4]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [1,2,4,5,3]=>[1,2,5,4,3]=>([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)=>([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20) [1,4,3,5,2]=>[1,5,4,3,2]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [2,1,3,4,5]=>[2,1,3,4,5]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [2,1,4,5,3]=>[2,1,5,4,3]=>([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)=>([(0,9),(1,13),(1,17),(3,16),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,17),(11,13),(11,14),(12,16),(13,18),(14,18),(15,1),(15,10),(15,11),(16,2),(17,4),(17,18),(18,3),(18,12)],19) [2,3,1,4,5]=>[3,2,1,4,5]=>([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)=>([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20) [2,3,1,5,4]=>[3,2,1,5,4]=>([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)=>([(0,9),(1,13),(1,17),(3,16),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,17),(11,13),(11,14),(12,16),(13,18),(14,18),(15,1),(15,10),(15,11),(16,2),(17,4),(17,18),(18,3),(18,12)],19) [3,2,4,1,5]=>[4,3,2,1,5]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [3,4,2,5,1]=>[5,4,3,2,1]=>([(0,4),(2,3),(3,1),(4,2)],5)=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6) [3,5,1,4,2]=>[5,4,3,1,2]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [3,5,2,4,1]=>[4,5,3,2,1]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [4,2,5,1,3]=>[5,4,1,2,3]=>([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)=>([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20) [4,3,1,5,2]=>[3,4,5,2,1]=>([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)=>([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20) [5,1,2,3,4]=>[5,1,2,3,4]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [5,2,4,1,3]=>[4,5,1,2,3]=>([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)=>([(0,9),(1,13),(1,17),(3,16),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,17),(11,13),(11,14),(12,16),(13,18),(14,18),(15,1),(15,10),(15,11),(16,2),(17,4),(17,18),(18,3),(18,12)],19) [5,2,4,3,1]=>[3,4,5,1,2]=>([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)=>([(0,9),(1,13),(1,17),(3,16),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,17),(11,13),(11,14),(12,16),(13,18),(14,18),(15,1),(15,10),(15,11),(16,2),(17,4),(17,18),(18,3),(18,12)],19) [5,4,3,2,1]=>[2,3,4,5,1]=>([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)=>([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15) [1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) [4,3,5,2,6,1]=>[6,5,4,3,2,1]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],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)=>([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) [4,5,3,6,2,7,1]=>[7,6,5,4,3,2,1]=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)=>([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
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
  • 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
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
order ideals
Description
The lattice of order ideals of a poset.
An order ideal $\mathcal I$ in a poset $P$ is a downward closed set, i.e., $a \in \mathcal I$ and $b \leq a$ implies $b \in \mathcal I$. This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.