Identifier
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0005;cc-rep-0Cc0012;cc-rep-1
[1,0]=>[(1,2)]=>[2,1]=>[2,1]=>0 [1,0,1,0]=>[(1,2),(3,4)]=>[2,1,4,3]=>[4,2,1,3]=>000 [1,1,0,0]=>[(1,4),(2,3)]=>[3,4,2,1]=>[3,4,2,1]=>000 [1,0,1,0,1,0]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[6,4,2,1,3,5]=>00000 [1,0,1,1,0,0]=>[(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[5,6,2,1,4,3]=>00000 [1,1,0,0,1,0]=>[(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[6,3,4,2,1,5]=>00000 [1,1,0,1,0,0]=>[(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[5,3,6,2,4,1]=>00000 [1,1,1,0,0,0]=>[(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[4,5,6,3,2,1]=>00000 [1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[8,6,4,2,1,3,5,7]=>0000000 [1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[7,8,4,2,1,3,6,5]=>0000000 [1,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[8,5,6,2,1,4,3,7]=>0000000 [1,0,1,1,0,1,0,0]=>[(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[7,5,8,2,1,4,6,3]=>0000000 [1,0,1,1,1,0,0,0]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[6,7,8,2,1,5,4,3]=>0000000 [1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[8,6,3,4,2,1,5,7]=>0000000 [1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[7,8,3,4,2,1,6,5]=>0000000 [1,1,0,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[8,5,3,6,2,4,1,7]=>0000000 [1,1,0,1,0,1,0,0]=>[(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[7,5,3,8,2,4,6,1]=>0000000 [1,1,0,1,1,0,0,0]=>[(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[6,7,3,8,2,5,4,1]=>0000000 [1,1,1,0,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[8,4,5,6,3,2,1,7]=>0000000 [1,1,1,0,0,1,0,0]=>[(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[7,4,5,8,3,2,6,1]=>0000000 [1,1,1,0,1,0,0,0]=>[(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[6,4,7,8,3,5,2,1]=>0000000 [1,1,1,1,0,0,0,0]=>[(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[5,6,7,8,4,3,2,1]=>0000000 [1,0,1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[10,8,6,4,2,1,3,5,7,9]=>000000000 [1,0,1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,6),(7,10),(8,9)]=>[2,1,4,3,6,5,9,10,8,7]=>[9,10,6,4,2,1,3,5,8,7]=>000000000 [1,0,1,0,1,1,0,0,1,0]=>[(1,2),(3,4),(5,8),(6,7),(9,10)]=>[2,1,4,3,7,8,6,5,10,9]=>[10,7,8,4,2,1,3,6,5,9]=>000000000 [1,0,1,0,1,1,0,1,0,0]=>[(1,2),(3,4),(5,10),(6,7),(8,9)]=>[2,1,4,3,7,9,6,10,8,5]=>[9,7,10,4,2,1,3,6,8,5]=>000000000 [1,0,1,0,1,1,1,0,0,0]=>[(1,2),(3,4),(5,10),(6,9),(7,8)]=>[2,1,4,3,8,9,10,7,6,5]=>[8,9,10,4,2,1,3,7,6,5]=>000000000 [1,0,1,1,0,0,1,0,1,0]=>[(1,2),(3,6),(4,5),(7,8),(9,10)]=>[2,1,5,6,4,3,8,7,10,9]=>[10,8,5,6,2,1,4,3,7,9]=>000000000 [1,0,1,1,0,0,1,1,0,0]=>[(1,2),(3,6),(4,5),(7,10),(8,9)]=>[2,1,5,6,4,3,9,10,8,7]=>[9,10,5,6,2,1,4,3,8,7]=>000000000 [1,0,1,1,0,1,0,0,1,0]=>[(1,2),(3,8),(4,5),(6,7),(9,10)]=>[2,1,5,7,4,8,6,3,10,9]=>[10,7,5,8,2,1,4,6,3,9]=>000000000 [1,0,1,1,0,1,0,1,0,0]=>[(1,2),(3,10),(4,5),(6,7),(8,9)]=>[2,1,5,7,4,9,6,10,8,3]=>[9,7,5,10,2,1,4,6,8,3]=>000000000 [1,0,1,1,0,1,1,0,0,0]=>[(1,2),(3,10),(4,5),(6,9),(7,8)]=>[2,1,5,8,4,9,10,7,6,3]=>[8,9,5,10,2,1,4,7,6,3]=>000000000 [1,0,1,1,1,0,0,0,1,0]=>[(1,2),(3,8),(4,7),(5,6),(9,10)]=>[2,1,6,7,8,5,4,3,10,9]=>[10,6,7,8,2,1,5,4,3,9]=>000000000 [1,0,1,1,1,0,0,1,0,0]=>[(1,2),(3,10),(4,7),(5,6),(8,9)]=>[2,1,6,7,9,5,4,10,8,3]=>[9,6,7,10,2,1,5,4,8,3]=>000000000 [1,0,1,1,1,0,1,0,0,0]=>[(1,2),(3,10),(4,9),(5,6),(7,8)]=>[2,1,6,8,9,5,10,7,4,3]=>[8,6,9,10,2,1,5,7,4,3]=>000000000 [1,0,1,1,1,1,0,0,0,0]=>[(1,2),(3,10),(4,9),(5,8),(6,7)]=>[2,1,7,8,9,10,6,5,4,3]=>[7,8,9,10,2,1,6,5,4,3]=>000000000 [1,1,0,0,1,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8),(9,10)]=>[3,4,2,1,6,5,8,7,10,9]=>[10,8,6,3,4,2,1,5,7,9]=>000000000 [1,1,0,0,1,0,1,1,0,0]=>[(1,4),(2,3),(5,6),(7,10),(8,9)]=>[3,4,2,1,6,5,9,10,8,7]=>[9,10,6,3,4,2,1,5,8,7]=>000000000 [1,1,0,0,1,1,0,0,1,0]=>[(1,4),(2,3),(5,8),(6,7),(9,10)]=>[3,4,2,1,7,8,6,5,10,9]=>[10,7,8,3,4,2,1,6,5,9]=>000000000 [1,1,0,0,1,1,0,1,0,0]=>[(1,4),(2,3),(5,10),(6,7),(8,9)]=>[3,4,2,1,7,9,6,10,8,5]=>[9,7,10,3,4,2,1,6,8,5]=>000000000 [1,1,0,0,1,1,1,0,0,0]=>[(1,4),(2,3),(5,10),(6,9),(7,8)]=>[3,4,2,1,8,9,10,7,6,5]=>[8,9,10,3,4,2,1,7,6,5]=>000000000 [1,1,0,1,0,0,1,0,1,0]=>[(1,6),(2,3),(4,5),(7,8),(9,10)]=>[3,5,2,6,4,1,8,7,10,9]=>[10,8,5,3,6,2,4,1,7,9]=>000000000 [1,1,0,1,0,0,1,1,0,0]=>[(1,6),(2,3),(4,5),(7,10),(8,9)]=>[3,5,2,6,4,1,9,10,8,7]=>[9,10,5,3,6,2,4,1,8,7]=>000000000 [1,1,0,1,0,1,0,0,1,0]=>[(1,8),(2,3),(4,5),(6,7),(9,10)]=>[3,5,2,7,4,8,6,1,10,9]=>[10,7,5,3,8,2,4,6,1,9]=>000000000 [1,1,0,1,0,1,0,1,0,0]=>[(1,10),(2,3),(4,5),(6,7),(8,9)]=>[3,5,2,7,4,9,6,10,8,1]=>[9,7,5,3,10,2,4,6,8,1]=>000000000 [1,1,0,1,0,1,1,0,0,0]=>[(1,10),(2,3),(4,5),(6,9),(7,8)]=>[3,5,2,8,4,9,10,7,6,1]=>[8,9,5,3,10,2,4,7,6,1]=>000000000 [1,1,0,1,1,0,0,0,1,0]=>[(1,8),(2,3),(4,7),(5,6),(9,10)]=>[3,6,2,7,8,5,4,1,10,9]=>[10,6,7,3,8,2,5,4,1,9]=>000000000 [1,1,0,1,1,0,0,1,0,0]=>[(1,10),(2,3),(4,7),(5,6),(8,9)]=>[3,6,2,7,9,5,4,10,8,1]=>[9,6,7,3,10,2,5,4,8,1]=>000000000 [1,1,0,1,1,0,1,0,0,0]=>[(1,10),(2,3),(4,9),(5,6),(7,8)]=>[3,6,2,8,9,5,10,7,4,1]=>[8,6,9,3,10,2,5,7,4,1]=>000000000 [1,1,0,1,1,1,0,0,0,0]=>[(1,10),(2,3),(4,9),(5,8),(6,7)]=>[3,7,2,8,9,10,6,5,4,1]=>[7,8,9,3,10,2,6,5,4,1]=>000000000 [1,1,1,0,0,0,1,0,1,0]=>[(1,6),(2,5),(3,4),(7,8),(9,10)]=>[4,5,6,3,2,1,8,7,10,9]=>[10,8,4,5,6,3,2,1,7,9]=>000000000 [1,1,1,0,0,0,1,1,0,0]=>[(1,6),(2,5),(3,4),(7,10),(8,9)]=>[4,5,6,3,2,1,9,10,8,7]=>[9,10,4,5,6,3,2,1,8,7]=>000000000 [1,1,1,0,0,1,0,0,1,0]=>[(1,8),(2,5),(3,4),(6,7),(9,10)]=>[4,5,7,3,2,8,6,1,10,9]=>[10,7,4,5,8,3,2,6,1,9]=>000000000 [1,1,1,0,0,1,0,1,0,0]=>[(1,10),(2,5),(3,4),(6,7),(8,9)]=>[4,5,7,3,2,9,6,10,8,1]=>[9,7,4,5,10,3,2,6,8,1]=>000000000 [1,1,1,0,0,1,1,0,0,0]=>[(1,10),(2,5),(3,4),(6,9),(7,8)]=>[4,5,8,3,2,9,10,7,6,1]=>[8,9,4,5,10,3,2,7,6,1]=>000000000 [1,1,1,0,1,0,0,0,1,0]=>[(1,8),(2,7),(3,4),(5,6),(9,10)]=>[4,6,7,3,8,5,2,1,10,9]=>[10,6,4,7,8,3,5,2,1,9]=>000000000 [1,1,1,0,1,0,0,1,0,0]=>[(1,10),(2,7),(3,4),(5,6),(8,9)]=>[4,6,7,3,9,5,2,10,8,1]=>[9,6,4,7,10,3,5,2,8,1]=>000000000 [1,1,1,0,1,0,1,0,0,0]=>[(1,10),(2,9),(3,4),(5,6),(7,8)]=>[4,6,8,3,9,5,10,7,2,1]=>[8,6,4,9,10,3,5,7,2,1]=>000000000 [1,1,1,0,1,1,0,0,0,0]=>[(1,10),(2,9),(3,4),(5,8),(6,7)]=>[4,7,8,3,9,10,6,5,2,1]=>[7,8,4,9,10,3,6,5,2,1]=>000000000 [1,1,1,1,0,0,0,0,1,0]=>[(1,8),(2,7),(3,6),(4,5),(9,10)]=>[5,6,7,8,4,3,2,1,10,9]=>[10,5,6,7,8,4,3,2,1,9]=>000000000 [1,1,1,1,0,0,0,1,0,0]=>[(1,10),(2,7),(3,6),(4,5),(8,9)]=>[5,6,7,9,4,3,2,10,8,1]=>[9,5,6,7,10,4,3,2,8,1]=>000000000 [1,1,1,1,0,0,1,0,0,0]=>[(1,10),(2,9),(3,6),(4,5),(7,8)]=>[5,6,8,9,4,3,10,7,2,1]=>[8,5,6,9,10,4,3,7,2,1]=>000000000 [1,1,1,1,0,1,0,0,0,0]=>[(1,10),(2,9),(3,8),(4,5),(6,7)]=>[5,7,8,9,4,10,6,3,2,1]=>[7,5,8,9,10,4,6,3,2,1]=>000000000 [1,1,1,1,1,0,0,0,0,0]=>[(1,10),(2,9),(3,8),(4,7),(5,6)]=>[6,7,8,9,10,5,4,3,2,1]=>[6,7,8,9,10,5,4,3,2,1]=>000000000 [1,0,1,0,1,0,1,0,1,0,1,0]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[12,10,8,6,4,2,1,3,5,7,9,11]=>00000000000 [1,0,1,0,1,0,1,0,1,1,0,0]=>[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]=>[2,1,4,3,6,5,8,7,11,12,10,9]=>[11,12,8,6,4,2,1,3,5,7,10,9]=>00000000000 [1,0,1,0,1,0,1,1,0,0,1,0]=>[(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]=>[2,1,4,3,6,5,9,10,8,7,12,11]=>[12,9,10,6,4,2,1,3,5,8,7,11]=>00000000000 [1,0,1,0,1,0,1,1,0,1,0,0]=>[(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]=>[2,1,4,3,6,5,9,11,8,12,10,7]=>[11,9,12,6,4,2,1,3,5,8,10,7]=>00000000000 [1,0,1,0,1,0,1,1,1,0,0,0]=>[(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]=>[2,1,4,3,6,5,10,11,12,9,8,7]=>[10,11,12,6,4,2,1,3,5,9,8,7]=>00000000000 [1,0,1,0,1,1,0,0,1,0,1,0]=>[(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]=>[2,1,4,3,7,8,6,5,10,9,12,11]=>[12,10,7,8,4,2,1,3,6,5,9,11]=>00000000000 [1,0,1,0,1,1,0,0,1,1,0,0]=>[(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]=>[2,1,4,3,7,8,6,5,11,12,10,9]=>[11,12,7,8,4,2,1,3,6,5,10,9]=>00000000000 [1,0,1,0,1,1,0,1,0,0,1,0]=>[(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]=>[2,1,4,3,7,9,6,10,8,5,12,11]=>[12,9,7,10,4,2,1,3,6,8,5,11]=>00000000000 [1,0,1,0,1,1,0,1,0,1,0,0]=>[(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]=>[2,1,4,3,7,9,6,11,8,12,10,5]=>[11,9,7,12,4,2,1,3,6,8,10,5]=>00000000000 [1,0,1,0,1,1,0,1,1,0,0,0]=>[(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]=>[2,1,4,3,7,10,6,11,12,9,8,5]=>[10,11,7,12,4,2,1,3,6,9,8,5]=>00000000000 [1,0,1,0,1,1,1,0,0,0,1,0]=>[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]=>[2,1,4,3,8,9,10,7,6,5,12,11]=>[12,8,9,10,4,2,1,3,7,6,5,11]=>00000000000 [1,0,1,0,1,1,1,0,0,1,0,0]=>[(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]=>[2,1,4,3,8,9,11,7,6,12,10,5]=>[11,8,9,12,4,2,1,3,7,6,10,5]=>00000000000 [1,0,1,0,1,1,1,0,1,0,0,0]=>[(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]=>[2,1,4,3,8,10,11,7,12,9,6,5]=>[10,8,11,12,4,2,1,3,7,9,6,5]=>00000000000 [1,0,1,0,1,1,1,1,0,0,0,0]=>[(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]=>[2,1,4,3,9,10,11,12,8,7,6,5]=>[9,10,11,12,4,2,1,3,8,7,6,5]=>00000000000 [1,0,1,1,0,0,1,0,1,0,1,0]=>[(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]=>[2,1,5,6,4,3,8,7,10,9,12,11]=>[12,10,8,5,6,2,1,4,3,7,9,11]=>00000000000 [1,0,1,1,0,0,1,0,1,1,0,0]=>[(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]=>[2,1,5,6,4,3,8,7,11,12,10,9]=>[11,12,8,5,6,2,1,4,3,7,10,9]=>00000000000 [1,0,1,1,0,0,1,1,0,0,1,0]=>[(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]=>[2,1,5,6,4,3,9,10,8,7,12,11]=>[12,9,10,5,6,2,1,4,3,8,7,11]=>00000000000 [1,0,1,1,0,0,1,1,0,1,0,0]=>[(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]=>[2,1,5,6,4,3,9,11,8,12,10,7]=>[11,9,12,5,6,2,1,4,3,8,10,7]=>00000000000 [1,0,1,1,0,0,1,1,1,0,0,0]=>[(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]=>[2,1,5,6,4,3,10,11,12,9,8,7]=>[10,11,12,5,6,2,1,4,3,9,8,7]=>00000000000 [1,0,1,1,0,1,0,0,1,0,1,0]=>[(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]=>[2,1,5,7,4,8,6,3,10,9,12,11]=>[12,10,7,5,8,2,1,4,6,3,9,11]=>00000000000 [1,0,1,1,0,1,0,0,1,1,0,0]=>[(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]=>[2,1,5,7,4,8,6,3,11,12,10,9]=>[11,12,7,5,8,2,1,4,6,3,10,9]=>00000000000 [1,0,1,1,0,1,0,1,0,0,1,0]=>[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]=>[2,1,5,7,4,9,6,10,8,3,12,11]=>[12,9,7,5,10,2,1,4,6,8,3,11]=>00000000000 [1,0,1,1,0,1,0,1,0,1,0,0]=>[(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]=>[2,1,5,7,4,9,6,11,8,12,10,3]=>[11,9,7,5,12,2,1,4,6,8,10,3]=>00000000000 [1,0,1,1,0,1,0,1,1,0,0,0]=>[(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]=>[2,1,5,7,4,10,6,11,12,9,8,3]=>[10,11,7,5,12,2,1,4,6,9,8,3]=>00000000000 [1,0,1,1,0,1,1,0,0,0,1,0]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]=>[2,1,5,8,4,9,10,7,6,3,12,11]=>[12,8,9,5,10,2,1,4,7,6,3,11]=>00000000000 [1,0,1,1,0,1,1,0,0,1,0,0]=>[(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]=>[2,1,5,8,4,9,11,7,6,12,10,3]=>[11,8,9,5,12,2,1,4,7,6,10,3]=>00000000000 [1,0,1,1,0,1,1,0,1,0,0,0]=>[(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]=>[2,1,5,8,4,10,11,7,12,9,6,3]=>[10,8,11,5,12,2,1,4,7,9,6,3]=>00000000000 [1,0,1,1,0,1,1,1,0,0,0,0]=>[(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]=>[2,1,5,9,4,10,11,12,8,7,6,3]=>[9,10,11,5,12,2,1,4,8,7,6,3]=>00000000000 [1,0,1,1,1,0,0,0,1,0,1,0]=>[(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]=>[2,1,6,7,8,5,4,3,10,9,12,11]=>[12,10,6,7,8,2,1,5,4,3,9,11]=>00000000000 [1,0,1,1,1,0,0,0,1,1,0,0]=>[(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]=>[2,1,6,7,8,5,4,3,11,12,10,9]=>[11,12,6,7,8,2,1,5,4,3,10,9]=>00000000000 [1,0,1,1,1,0,0,1,0,0,1,0]=>[(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]=>[2,1,6,7,9,5,4,10,8,3,12,11]=>[12,9,6,7,10,2,1,5,4,8,3,11]=>00000000000 [1,0,1,1,1,0,0,1,0,1,0,0]=>[(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]=>[2,1,6,7,9,5,4,11,8,12,10,3]=>[11,9,6,7,12,2,1,5,4,8,10,3]=>00000000000 [1,0,1,1,1,0,0,1,1,0,0,0]=>[(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]=>[2,1,6,7,10,5,4,11,12,9,8,3]=>[10,11,6,7,12,2,1,5,4,9,8,3]=>00000000000 [1,0,1,1,1,0,1,0,0,0,1,0]=>[(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]=>[2,1,6,8,9,5,10,7,4,3,12,11]=>[12,8,6,9,10,2,1,5,7,4,3,11]=>00000000000 [1,0,1,1,1,0,1,0,0,1,0,0]=>[(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]=>[2,1,6,8,9,5,11,7,4,12,10,3]=>[11,8,6,9,12,2,1,5,7,4,10,3]=>00000000000 [1,0,1,1,1,0,1,0,1,0,0,0]=>[(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]=>[2,1,6,8,10,5,11,7,12,9,4,3]=>[10,8,6,11,12,2,1,5,7,9,4,3]=>00000000000 [1,0,1,1,1,0,1,1,0,0,0,0]=>[(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]=>[2,1,6,9,10,5,11,12,8,7,4,3]=>[9,10,6,11,12,2,1,5,8,7,4,3]=>00000000000 [1,0,1,1,1,1,0,0,0,0,1,0]=>[(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]=>[2,1,7,8,9,10,6,5,4,3,12,11]=>[12,7,8,9,10,2,1,6,5,4,3,11]=>00000000000 [1,0,1,1,1,1,0,0,0,1,0,0]=>[(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]=>[2,1,7,8,9,11,6,5,4,12,10,3]=>[11,7,8,9,12,2,1,6,5,4,10,3]=>00000000000 [1,0,1,1,1,1,0,0,1,0,0,0]=>[(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]=>[2,1,7,8,10,11,6,5,12,9,4,3]=>[10,7,8,11,12,2,1,6,5,9,4,3]=>00000000000 [1,0,1,1,1,1,0,1,0,0,0,0]=>[(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]=>[2,1,7,9,10,11,6,12,8,5,4,3]=>[9,7,10,11,12,2,1,6,8,5,4,3]=>00000000000 [1,0,1,1,1,1,1,0,0,0,0,0]=>[(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]=>[2,1,8,9,10,11,12,7,6,5,4,3]=>[8,9,10,11,12,2,1,7,6,5,4,3]=>00000000000 [1,1,0,0,1,0,1,0,1,0,1,0]=>[(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]=>[3,4,2,1,6,5,8,7,10,9,12,11]=>[12,10,8,6,3,4,2,1,5,7,9,11]=>00000000000 [1,1,0,0,1,0,1,0,1,1,0,0]=>[(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]=>[3,4,2,1,6,5,8,7,11,12,10,9]=>[11,12,8,6,3,4,2,1,5,7,10,9]=>00000000000 [1,1,0,0,1,0,1,1,0,0,1,0]=>[(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]=>[3,4,2,1,6,5,9,10,8,7,12,11]=>[12,9,10,6,3,4,2,1,5,8,7,11]=>00000000000 [1,1,0,0,1,0,1,1,0,1,0,0]=>[(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]=>[3,4,2,1,6,5,9,11,8,12,10,7]=>[11,9,12,6,3,4,2,1,5,8,10,7]=>00000000000 [1,1,0,0,1,0,1,1,1,0,0,0]=>[(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]=>[3,4,2,1,6,5,10,11,12,9,8,7]=>[10,11,12,6,3,4,2,1,5,9,8,7]=>00000000000 [1,1,0,0,1,1,0,0,1,0,1,0]=>[(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]=>[3,4,2,1,7,8,6,5,10,9,12,11]=>[12,10,7,8,3,4,2,1,6,5,9,11]=>00000000000 [1,1,0,0,1,1,0,0,1,1,0,0]=>[(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]=>[3,4,2,1,7,8,6,5,11,12,10,9]=>[11,12,7,8,3,4,2,1,6,5,10,9]=>00000000000 [1,1,0,0,1,1,0,1,0,0,1,0]=>[(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]=>[3,4,2,1,7,9,6,10,8,5,12,11]=>[12,9,7,10,3,4,2,1,6,8,5,11]=>00000000000 [1,1,0,0,1,1,0,1,0,1,0,0]=>[(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]=>[3,4,2,1,7,9,6,11,8,12,10,5]=>[11,9,7,12,3,4,2,1,6,8,10,5]=>00000000000 [1,1,0,0,1,1,0,1,1,0,0,0]=>[(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]=>[3,4,2,1,7,10,6,11,12,9,8,5]=>[10,11,7,12,3,4,2,1,6,9,8,5]=>00000000000 [1,1,0,0,1,1,1,0,0,0,1,0]=>[(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]=>[3,4,2,1,8,9,10,7,6,5,12,11]=>[12,8,9,10,3,4,2,1,7,6,5,11]=>00000000000 [1,1,0,0,1,1,1,0,0,1,0,0]=>[(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]=>[3,4,2,1,8,9,11,7,6,12,10,5]=>[11,8,9,12,3,4,2,1,7,6,10,5]=>00000000000 [1,1,0,0,1,1,1,0,1,0,0,0]=>[(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]=>[3,4,2,1,8,10,11,7,12,9,6,5]=>[10,8,11,12,3,4,2,1,7,9,6,5]=>00000000000 [1,1,0,0,1,1,1,1,0,0,0,0]=>[(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]=>[3,4,2,1,9,10,11,12,8,7,6,5]=>[9,10,11,12,3,4,2,1,8,7,6,5]=>00000000000 [1,1,0,1,0,0,1,0,1,0,1,0]=>[(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]=>[3,5,2,6,4,1,8,7,10,9,12,11]=>[12,10,8,5,3,6,2,4,1,7,9,11]=>00000000000 [1,1,0,1,0,0,1,0,1,1,0,0]=>[(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]=>[3,5,2,6,4,1,8,7,11,12,10,9]=>[11,12,8,5,3,6,2,4,1,7,10,9]=>00000000000 [1,1,0,1,0,0,1,1,0,0,1,0]=>[(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]=>[3,5,2,6,4,1,9,10,8,7,12,11]=>[12,9,10,5,3,6,2,4,1,8,7,11]=>00000000000 [1,1,0,1,0,0,1,1,0,1,0,0]=>[(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]=>[3,5,2,6,4,1,9,11,8,12,10,7]=>[11,9,12,5,3,6,2,4,1,8,10,7]=>00000000000 [1,1,0,1,0,0,1,1,1,0,0,0]=>[(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]=>[3,5,2,6,4,1,10,11,12,9,8,7]=>[10,11,12,5,3,6,2,4,1,9,8,7]=>00000000000 [1,1,0,1,0,1,0,0,1,0,1,0]=>[(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]=>[3,5,2,7,4,8,6,1,10,9,12,11]=>[12,10,7,5,3,8,2,4,6,1,9,11]=>00000000000 [1,1,0,1,0,1,0,0,1,1,0,0]=>[(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]=>[3,5,2,7,4,8,6,1,11,12,10,9]=>[11,12,7,5,3,8,2,4,6,1,10,9]=>00000000000 [1,1,0,1,0,1,0,1,0,0,1,0]=>[(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]=>[3,5,2,7,4,9,6,10,8,1,12,11]=>[12,9,7,5,3,10,2,4,6,8,1,11]=>00000000000 [1,1,0,1,0,1,0,1,0,1,0,0]=>[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]=>[3,5,2,7,4,9,6,11,8,12,10,1]=>[11,9,7,5,3,12,2,4,6,8,10,1]=>00000000000 [1,1,0,1,0,1,0,1,1,0,0,0]=>[(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]=>[3,5,2,7,4,10,6,11,12,9,8,1]=>[10,11,7,5,3,12,2,4,6,9,8,1]=>00000000000 [1,1,0,1,0,1,1,0,0,0,1,0]=>[(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]=>[3,5,2,8,4,9,10,7,6,1,12,11]=>[12,8,9,5,3,10,2,4,7,6,1,11]=>00000000000 [1,1,0,1,0,1,1,0,0,1,0,0]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]=>[3,5,2,8,4,9,11,7,6,12,10,1]=>[11,8,9,5,3,12,2,4,7,6,10,1]=>00000000000 [1,1,0,1,0,1,1,0,1,0,0,0]=>[(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]=>[3,5,2,8,4,10,11,7,12,9,6,1]=>[10,8,11,5,3,12,2,4,7,9,6,1]=>00000000000 [1,1,0,1,0,1,1,1,0,0,0,0]=>[(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]=>[3,5,2,9,4,10,11,12,8,7,6,1]=>[9,10,11,5,3,12,2,4,8,7,6,1]=>00000000000 [1,1,0,1,1,0,0,0,1,0,1,0]=>[(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]=>[3,6,2,7,8,5,4,1,10,9,12,11]=>[12,10,6,7,3,8,2,5,4,1,9,11]=>00000000000 [1,1,0,1,1,0,0,0,1,1,0,0]=>[(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]=>[3,6,2,7,8,5,4,1,11,12,10,9]=>[11,12,6,7,3,8,2,5,4,1,10,9]=>00000000000 [1,1,0,1,1,0,0,1,0,0,1,0]=>[(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]=>[3,6,2,7,9,5,4,10,8,1,12,11]=>[12,9,6,7,3,10,2,5,4,8,1,11]=>00000000000 [1,1,0,1,1,0,0,1,0,1,0,0]=>[(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]=>[3,6,2,7,9,5,4,11,8,12,10,1]=>[11,9,6,7,3,12,2,5,4,8,10,1]=>00000000000 [1,1,0,1,1,0,0,1,1,0,0,0]=>[(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]=>[3,6,2,7,10,5,4,11,12,9,8,1]=>[10,11,6,7,3,12,2,5,4,9,8,1]=>00000000000 [1,1,0,1,1,0,1,0,0,0,1,0]=>[(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]=>[3,6,2,8,9,5,10,7,4,1,12,11]=>[12,8,6,9,3,10,2,5,7,4,1,11]=>00000000000 [1,1,0,1,1,0,1,0,0,1,0,0]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)]=>[3,6,2,8,9,5,11,7,4,12,10,1]=>[11,8,6,9,3,12,2,5,7,4,10,1]=>00000000000 [1,1,0,1,1,0,1,0,1,0,0,0]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)]=>[3,6,2,8,10,5,11,7,12,9,4,1]=>[10,8,6,11,3,12,2,5,7,9,4,1]=>00000000000 [1,1,0,1,1,0,1,1,0,0,0,0]=>[(1,12),(2,3),(4,11),(5,6),(7,10),(8,9)]=>[3,6,2,9,10,5,11,12,8,7,4,1]=>[9,10,6,11,3,12,2,5,8,7,4,1]=>00000000000 [1,1,0,1,1,1,0,0,0,0,1,0]=>[(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]=>[3,7,2,8,9,10,6,5,4,1,12,11]=>[12,7,8,9,3,10,2,6,5,4,1,11]=>00000000000 [1,1,0,1,1,1,0,0,0,1,0,0]=>[(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]=>[3,7,2,8,9,11,6,5,4,12,10,1]=>[11,7,8,9,3,12,2,6,5,4,10,1]=>00000000000 [1,1,0,1,1,1,0,0,1,0,0,0]=>[(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)]=>[3,7,2,8,10,11,6,5,12,9,4,1]=>[10,7,8,11,3,12,2,6,5,9,4,1]=>00000000000 [1,1,0,1,1,1,0,1,0,0,0,0]=>[(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]=>[3,7,2,9,10,11,6,12,8,5,4,1]=>[9,7,10,11,3,12,2,6,8,5,4,1]=>00000000000 [1,1,0,1,1,1,1,0,0,0,0,0]=>[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]=>[3,8,2,9,10,11,12,7,6,5,4,1]=>[8,9,10,11,3,12,2,7,6,5,4,1]=>00000000000 [1,1,1,0,0,0,1,0,1,0,1,0]=>[(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]=>[4,5,6,3,2,1,8,7,10,9,12,11]=>[12,10,8,4,5,6,3,2,1,7,9,11]=>00000000000 [1,1,1,0,0,0,1,0,1,1,0,0]=>[(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]=>[4,5,6,3,2,1,8,7,11,12,10,9]=>[11,12,8,4,5,6,3,2,1,7,10,9]=>00000000000 [1,1,1,0,0,0,1,1,0,0,1,0]=>[(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]=>[4,5,6,3,2,1,9,10,8,7,12,11]=>[12,9,10,4,5,6,3,2,1,8,7,11]=>00000000000 [1,1,1,0,0,0,1,1,0,1,0,0]=>[(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]=>[4,5,6,3,2,1,9,11,8,12,10,7]=>[11,9,12,4,5,6,3,2,1,8,10,7]=>00000000000 [1,1,1,0,0,0,1,1,1,0,0,0]=>[(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]=>[4,5,6,3,2,1,10,11,12,9,8,7]=>[10,11,12,4,5,6,3,2,1,9,8,7]=>00000000000 [1,1,1,0,0,1,0,0,1,0,1,0]=>[(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)]=>[4,5,7,3,2,8,6,1,10,9,12,11]=>[12,10,7,4,5,8,3,2,6,1,9,11]=>00000000000 [1,1,1,0,0,1,0,0,1,1,0,0]=>[(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]=>[4,5,7,3,2,8,6,1,11,12,10,9]=>[11,12,7,4,5,8,3,2,6,1,10,9]=>00000000000 [1,1,1,0,0,1,0,1,0,0,1,0]=>[(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]=>[4,5,7,3,2,9,6,10,8,1,12,11]=>[12,9,7,4,5,10,3,2,6,8,1,11]=>00000000000 [1,1,1,0,0,1,0,1,0,1,0,0]=>[(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]=>[4,5,7,3,2,9,6,11,8,12,10,1]=>[11,9,7,4,5,12,3,2,6,8,10,1]=>00000000000 [1,1,1,0,0,1,0,1,1,0,0,0]=>[(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]=>[4,5,7,3,2,10,6,11,12,9,8,1]=>[10,11,7,4,5,12,3,2,6,9,8,1]=>00000000000 [1,1,1,0,0,1,1,0,0,0,1,0]=>[(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]=>[4,5,8,3,2,9,10,7,6,1,12,11]=>[12,8,9,4,5,10,3,2,7,6,1,11]=>00000000000 [1,1,1,0,0,1,1,0,0,1,0,0]=>[(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]=>[4,5,8,3,2,9,11,7,6,12,10,1]=>[11,8,9,4,5,12,3,2,7,6,10,1]=>00000000000 [1,1,1,0,0,1,1,0,1,0,0,0]=>[(1,12),(2,5),(3,4),(6,11),(7,8),(9,10)]=>[4,5,8,3,2,10,11,7,12,9,6,1]=>[10,8,11,4,5,12,3,2,7,9,6,1]=>00000000000 [1,1,1,0,0,1,1,1,0,0,0,0]=>[(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]=>[4,5,9,3,2,10,11,12,8,7,6,1]=>[9,10,11,4,5,12,3,2,8,7,6,1]=>00000000000 [1,1,1,0,1,0,0,0,1,0,1,0]=>[(1,8),(2,7),(3,4),(5,6),(9,10),(11,12)]=>[4,6,7,3,8,5,2,1,10,9,12,11]=>[12,10,6,4,7,8,3,5,2,1,9,11]=>00000000000 [1,1,1,0,1,0,0,0,1,1,0,0]=>[(1,8),(2,7),(3,4),(5,6),(9,12),(10,11)]=>[4,6,7,3,8,5,2,1,11,12,10,9]=>[11,12,6,4,7,8,3,5,2,1,10,9]=>00000000000 [1,1,1,0,1,0,0,1,0,0,1,0]=>[(1,10),(2,7),(3,4),(5,6),(8,9),(11,12)]=>[4,6,7,3,9,5,2,10,8,1,12,11]=>[12,9,6,4,7,10,3,5,2,8,1,11]=>00000000000 [1,1,1,0,1,0,0,1,0,1,0,0]=>[(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]=>[4,6,7,3,9,5,2,11,8,12,10,1]=>[11,9,6,4,7,12,3,5,2,8,10,1]=>00000000000 [1,1,1,0,1,0,0,1,1,0,0,0]=>[(1,12),(2,7),(3,4),(5,6),(8,11),(9,10)]=>[4,6,7,3,10,5,2,11,12,9,8,1]=>[10,11,6,4,7,12,3,5,2,9,8,1]=>00000000000 [1,1,1,0,1,0,1,0,0,0,1,0]=>[(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]=>[4,6,8,3,9,5,10,7,2,1,12,11]=>[12,8,6,4,9,10,3,5,7,2,1,11]=>00000000000 [1,1,1,0,1,0,1,0,0,1,0,0]=>[(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]=>[4,6,8,3,9,5,11,7,2,12,10,1]=>[11,8,6,4,9,12,3,5,7,2,10,1]=>00000000000 [1,1,1,0,1,0,1,0,1,0,0,0]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]=>[4,6,8,3,10,5,11,7,12,9,2,1]=>[10,8,6,4,11,12,3,5,7,9,2,1]=>00000000000 [1,1,1,0,1,0,1,1,0,0,0,0]=>[(1,12),(2,11),(3,4),(5,6),(7,10),(8,9)]=>[4,6,9,3,10,5,11,12,8,7,2,1]=>[9,10,6,4,11,12,3,5,8,7,2,1]=>00000000000 [1,1,1,0,1,1,0,0,0,0,1,0]=>[(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]=>[4,7,8,3,9,10,6,5,2,1,12,11]=>[12,7,8,4,9,10,3,6,5,2,1,11]=>00000000000 [1,1,1,0,1,1,0,0,0,1,0,0]=>[(1,12),(2,9),(3,4),(5,8),(6,7),(10,11)]=>[4,7,8,3,9,11,6,5,2,12,10,1]=>[11,7,8,4,9,12,3,6,5,2,10,1]=>00000000000 [1,1,1,0,1,1,0,0,1,0,0,0]=>[(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)]=>[4,7,8,3,10,11,6,5,12,9,2,1]=>[10,7,8,4,11,12,3,6,5,9,2,1]=>00000000000 [1,1,1,0,1,1,0,1,0,0,0,0]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]=>[4,7,9,3,10,11,6,12,8,5,2,1]=>[9,7,10,4,11,12,3,6,8,5,2,1]=>00000000000 [1,1,1,0,1,1,1,0,0,0,0,0]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]=>[4,8,9,3,10,11,12,7,6,5,2,1]=>[8,9,10,4,11,12,3,7,6,5,2,1]=>00000000000 [1,1,1,1,0,0,0,0,1,0,1,0]=>[(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]=>[5,6,7,8,4,3,2,1,10,9,12,11]=>[12,10,5,6,7,8,4,3,2,1,9,11]=>00000000000 [1,1,1,1,0,0,0,0,1,1,0,0]=>[(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]=>[5,6,7,8,4,3,2,1,11,12,10,9]=>[11,12,5,6,7,8,4,3,2,1,10,9]=>00000000000 [1,1,1,1,0,0,0,1,0,0,1,0]=>[(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]=>[5,6,7,9,4,3,2,10,8,1,12,11]=>[12,9,5,6,7,10,4,3,2,8,1,11]=>00000000000 [1,1,1,1,0,0,0,1,0,1,0,0]=>[(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]=>[5,6,7,9,4,3,2,11,8,12,10,1]=>[11,9,5,6,7,12,4,3,2,8,10,1]=>00000000000 [1,1,1,1,0,0,0,1,1,0,0,0]=>[(1,12),(2,7),(3,6),(4,5),(8,11),(9,10)]=>[5,6,7,10,4,3,2,11,12,9,8,1]=>[10,11,5,6,7,12,4,3,2,9,8,1]=>00000000000 [1,1,1,1,0,0,1,0,0,0,1,0]=>[(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]=>[5,6,8,9,4,3,10,7,2,1,12,11]=>[12,8,5,6,9,10,4,3,7,2,1,11]=>00000000000 [1,1,1,1,0,0,1,0,0,1,0,0]=>[(1,12),(2,9),(3,6),(4,5),(7,8),(10,11)]=>[5,6,8,9,4,3,11,7,2,12,10,1]=>[11,8,5,6,9,12,4,3,7,2,10,1]=>00000000000 [1,1,1,1,0,0,1,0,1,0,0,0]=>[(1,12),(2,11),(3,6),(4,5),(7,8),(9,10)]=>[5,6,8,10,4,3,11,7,12,9,2,1]=>[10,8,5,6,11,12,4,3,7,9,2,1]=>00000000000 [1,1,1,1,0,0,1,1,0,0,0,0]=>[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9)]=>[5,6,9,10,4,3,11,12,8,7,2,1]=>[9,10,5,6,11,12,4,3,8,7,2,1]=>00000000000 [1,1,1,1,0,1,0,0,0,0,1,0]=>[(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]=>[5,7,8,9,4,10,6,3,2,1,12,11]=>[12,7,5,8,9,10,4,6,3,2,1,11]=>00000000000 [1,1,1,1,0,1,0,0,0,1,0,0]=>[(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]=>[5,7,8,9,4,11,6,3,2,12,10,1]=>[11,7,5,8,9,12,4,6,3,2,10,1]=>00000000000 [1,1,1,1,0,1,0,0,1,0,0,0]=>[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10)]=>[5,7,8,10,4,11,6,3,12,9,2,1]=>[10,7,5,8,11,12,4,6,3,9,2,1]=>00000000000 [1,1,1,1,0,1,0,1,0,0,0,0]=>[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)]=>[5,7,9,10,4,11,6,12,8,3,2,1]=>[9,7,5,10,11,12,4,6,8,3,2,1]=>00000000000 [1,1,1,1,0,1,1,0,0,0,0,0]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)]=>[5,8,9,10,4,11,12,7,6,3,2,1]=>[8,9,5,10,11,12,4,7,6,3,2,1]=>00000000000 [1,1,1,1,1,0,0,0,0,0,1,0]=>[(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]=>[6,7,8,9,10,5,4,3,2,1,12,11]=>[12,6,7,8,9,10,5,4,3,2,1,11]=>00000000000 [1,1,1,1,1,0,0,0,0,1,0,0]=>[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]=>[6,7,8,9,11,5,4,3,2,12,10,1]=>[11,6,7,8,9,12,5,4,3,2,10,1]=>00000000000 [1,1,1,1,1,0,0,0,1,0,0,0]=>[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10)]=>[6,7,8,10,11,5,4,3,12,9,2,1]=>[10,6,7,8,11,12,5,4,3,9,2,1]=>00000000000 [1,1,1,1,1,0,0,1,0,0,0,0]=>[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]=>[6,7,9,10,11,5,4,12,8,3,2,1]=>[9,6,7,10,11,12,5,4,8,3,2,1]=>00000000000 [1,1,1,1,1,0,1,0,0,0,0,0]=>[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)]=>[6,8,9,10,11,5,12,7,4,3,2,1]=>[8,6,9,10,11,12,5,7,4,3,2,1]=>00000000000 [1,1,1,1,1,1,0,0,0,0,0,0]=>[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]=>[7,8,9,10,11,12,6,5,4,3,2,1]=>[7,8,9,10,11,12,6,5,4,3,2,1]=>00000000000
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.