Identifier
Mp00235:
Permutations
—descent views to invisible inversion bottoms⟶
Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00209: Permutations —pattern poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Images
=>
Cc0014;cc-rep-2Cc0029;cc-rep-3
[1]=>[1]=>([],1)=>([],1)
[1,2]=>[1,2]=>([(0,1)],2)=>([(0,1)],2)
[2,1]=>[2,1]=>([(0,1)],2)=>([(0,1)],2)
[1,2,3]=>[1,2,3]=>([(0,2),(2,1)],3)=>([(0,2),(2,1)],3)
[1,3,2]=>[1,3,2]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([],1)
[2,1,3]=>[2,1,3]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([],1)
[2,3,1]=>[3,2,1]=>([(0,2),(2,1)],3)=>([(0,2),(2,1)],3)
[3,1,2]=>[3,1,2]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([],1)
[3,2,1]=>[2,3,1]=>([(0,1),(0,2),(1,3),(2,3)],4)=>([],1)
[1,2,3,4]=>[1,2,3,4]=>([(0,3),(2,1),(3,2)],4)=>([(0,3),(2,1),(3,2)],4)
[1,2,4,3]=>[1,2,4,3]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[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)=>([],1)
[1,3,4,2]=>[1,4,3,2]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[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)=>([],1)
[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)=>([],1)
[2,1,3,4]=>[2,1,3,4]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[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)=>([(0,1)],2)
[2,3,1,4]=>[3,2,1,4]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[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)=>([],1)
[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)=>([],1)
[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)=>([],1)
[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)=>([],1)
[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)=>([(0,1)],2)
[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)=>([],1)
[3,2,4,1]=>[4,3,2,1]=>([(0,3),(2,1),(3,2)],4)=>([(0,3),(2,1),(3,2)],4)
[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)=>([],1)
[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)=>([],1)
[4,1,2,3]=>[4,1,2,3]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[4,1,3,2]=>[4,3,1,2]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[4,2,3,1]=>[3,4,2,1]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[4,3,2,1]=>[2,3,4,1]=>([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)=>([(0,2),(2,1)],3)
[1,2,3,4,5]=>[1,2,3,4,5]=>([(0,4),(2,3),(3,1),(4,2)],5)=>([(0,4),(2,3),(3,1),(4,2)],5)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[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)=>([],1)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[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)=>([],1)
[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)=>([],1)
[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)=>([],1)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[3,4,2,5,1]=>[5,4,3,2,1]=>([(0,4),(2,3),(3,1),(4,2)],5)=>([(0,4),(2,3),(3,1),(4,2)],5)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[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)=>([],1)
[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)=>([],1)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[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)=>([],1)
[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)=>([],1)
[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)=>([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[1,2,3,5,6,4]=>[1,2,3,6,5,4]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[1,2,5,4,6,3]=>[1,2,6,5,4,3]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[1,4,5,3,6,2]=>[1,6,5,4,3,2]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[2,1,5,4,6,3]=>[2,1,6,5,4,3]=>([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[2,3,1,4,5,6]=>[3,2,1,4,5,6]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[2,3,1,5,6,4]=>[3,2,1,6,5,4]=>([(0,1),(0,2),(1,4),(1,10),(2,3),(2,10),(3,5),(3,8),(4,5),(4,9),(5,11),(7,6),(8,7),(8,11),(9,7),(9,11),(10,8),(10,9),(11,6)],12)=>([(0,1)],2)
[3,2,4,1,5,6]=>[4,3,2,1,5,6]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[3,2,4,1,6,5]=>[4,3,2,1,6,5]=>([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[3,4,2,5,1,6]=>[5,4,3,2,1,6]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[4,1,5,2,6,3]=>[4,5,6,1,2,3]=>([(0,1),(0,2),(1,4),(1,10),(2,3),(2,10),(3,5),(3,8),(4,5),(4,9),(5,11),(7,6),(8,7),(8,11),(9,7),(9,11),(10,8),(10,9),(11,6)],12)=>([(0,1)],2)
[4,3,5,2,6,1]=>[6,5,4,3,2,1]=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)=>([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
[4,3,6,1,5,2]=>[6,5,4,3,1,2]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[4,3,6,2,5,1]=>[5,6,4,3,2,1]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[5,1,3,6,2,4]=>[5,6,1,2,3,4]=>([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[5,2,6,1,4,3]=>[6,5,4,1,2,3]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[5,2,6,3,4,1]=>[4,5,6,3,2,1]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[5,4,3,1,6,2]=>[3,4,5,6,1,2]=>([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[6,1,2,3,4,5]=>[6,1,2,3,4,5]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[6,1,3,5,2,4]=>[6,5,1,2,3,4]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[6,2,5,4,3,1]=>[3,4,5,6,2,1]=>([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)=>([(0,3),(2,1),(3,2)],4)
[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)=>([(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)
[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
[1,2,3,4,5,7,6]=>[1,2,3,4,5,7,6]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[1,2,3,4,6,7,5]=>[1,2,3,4,7,6,5]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[1,2,3,6,5,7,4]=>[1,2,3,7,6,5,4]=>([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)=>([],1)
[1,2,5,6,4,7,3]=>[1,2,7,6,5,4,3]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[1,5,4,6,3,7,2]=>[1,7,6,5,4,3,2]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[2,1,5,6,4,7,3]=>[2,1,7,6,5,4,3]=>([(0,5),(0,6),(1,11),(2,3),(2,13),(3,4),(3,10),(4,9),(4,14),(5,1),(5,12),(6,2),(6,12),(7,14),(9,8),(10,9),(10,14),(11,7),(12,11),(12,13),(13,7),(13,10),(14,8)],15)=>([(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)
[2,3,1,4,5,6,7]=>[3,2,1,4,5,6,7]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[2,3,1,6,5,7,4]=>[3,2,1,7,6,5,4]=>([(0,5),(0,6),(1,14),(2,4),(2,13),(3,1),(3,11),(4,9),(4,14),(5,2),(5,12),(6,3),(6,12),(8,10),(8,15),(9,10),(9,15),(10,7),(11,8),(11,14),(12,11),(12,13),(13,8),(13,9),(14,15),(15,7)],16)=>([],1)
[3,2,4,1,5,6,7]=>[4,3,2,1,5,6,7]=>([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)=>([],1)
[3,2,4,1,6,7,5]=>[4,3,2,1,7,6,5]=>([(0,5),(0,6),(1,14),(2,4),(2,13),(3,1),(3,11),(4,9),(4,14),(5,2),(5,12),(6,3),(6,12),(8,10),(8,15),(9,10),(9,15),(10,7),(11,8),(11,14),(12,11),(12,13),(13,8),(13,9),(14,15),(15,7)],16)=>([],1)
[3,4,2,5,1,6,7]=>[5,4,3,2,1,6,7]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[3,4,2,5,1,7,6]=>[5,4,3,2,1,7,6]=>([(0,5),(0,6),(1,11),(2,3),(2,13),(3,4),(3,10),(4,9),(4,14),(5,1),(5,12),(6,2),(6,12),(7,14),(9,8),(10,9),(10,14),(11,7),(12,11),(12,13),(13,7),(13,10),(14,8)],15)=>([(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)
[4,3,5,2,6,1,7]=>[6,5,4,3,2,1,7]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[4,5,3,6,2,7,1]=>[7,6,5,4,3,2,1]=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)=>([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
[4,5,3,7,1,6,2]=>[7,6,5,4,3,1,2]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[4,5,3,7,2,6,1]=>[6,7,5,4,3,2,1]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[4,6,2,7,1,5,3]=>[7,6,5,4,1,2,3]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[4,6,2,7,3,5,1]=>[5,6,7,4,3,2,1]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[6,2,4,7,1,3,5]=>[7,6,1,2,3,4,5]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[6,2,5,3,7,1,4]=>[7,6,5,1,2,3,4]=>([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)=>([],1)
[6,2,5,4,1,7,3]=>[4,5,6,7,3,2,1]=>([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)=>([],1)
[6,5,4,3,1,7,2]=>[3,4,5,6,7,2,1]=>([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)=>([(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)
[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
[7,2,4,6,1,3,5]=>[6,7,1,2,3,4,5]=>([(0,5),(0,6),(1,11),(2,3),(2,13),(3,4),(3,10),(4,9),(4,14),(5,1),(5,12),(6,2),(6,12),(7,14),(9,8),(10,9),(10,14),(11,7),(12,11),(12,13),(13,7),(13,10),(14,8)],15)=>([(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)
[7,2,6,5,4,3,1]=>[3,4,5,6,7,1,2]=>([(0,5),(0,6),(1,11),(2,3),(2,13),(3,4),(3,10),(4,9),(4,14),(5,1),(5,12),(6,2),(6,12),(7,14),(9,8),(10,9),(10,14),(11,7),(12,11),(12,13),(13,7),(13,10),(14,8)],15)=>([(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)
[7,3,5,4,1,6,2]=>[4,5,6,7,1,2,3]=>([(0,5),(0,6),(1,14),(2,4),(2,13),(3,1),(3,11),(4,9),(4,14),(5,2),(5,12),(6,3),(6,12),(8,10),(8,15),(9,10),(9,15),(10,7),(11,8),(11,14),(12,11),(12,13),(13,8),(13,9),(14,15),(15,7)],16)=>([],1)
[7,3,6,2,5,1,4]=>[5,6,7,1,2,3,4]=>([(0,5),(0,6),(1,14),(2,4),(2,13),(3,1),(3,11),(4,9),(4,14),(5,2),(5,12),(6,3),(6,12),(8,10),(8,15),(9,10),(9,15),(10,7),(11,8),(11,14),(12,11),(12,13),(13,8),(13,9),(14,15),(15,7)],16)=>([],1)
[7,6,5,4,3,2,1]=>[2,3,4,5,6,7,1]=>([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)=>([(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)
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
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.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
antichains of maximal size
Description
The lattice of antichains of maximal size in a poset.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
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