Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
=>
Cc0005;cc-rep-1
[1]=>[1,0]=>[1]=> [1,1]=>[1,0,1,0]=>[2,1]=>1 [2]=>[1,1,0,0]=>[1,2]=>0 [1,1,1]=>[1,0,1,0,1,0]=>[2,3,1]=>01 [1,2]=>[1,0,1,1,0,0]=>[2,1,3]=>10 [2,1]=>[1,1,0,0,1,0]=>[1,3,2]=>01 [3]=>[1,1,1,0,0,0]=>[1,2,3]=>00 [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[2,3,4,1]=>001 [1,1,2]=>[1,0,1,0,1,1,0,0]=>[2,3,1,4]=>010 [1,2,1]=>[1,0,1,1,0,0,1,0]=>[2,1,4,3]=>101 [1,3]=>[1,0,1,1,1,0,0,0]=>[2,1,3,4]=>100 [2,1,1]=>[1,1,0,0,1,0,1,0]=>[1,3,4,2]=>001 [2,2]=>[1,1,0,0,1,1,0,0]=>[1,3,2,4]=>010 [3,1]=>[1,1,1,0,0,0,1,0]=>[1,2,4,3]=>001 [4]=>[1,1,1,1,0,0,0,0]=>[1,2,3,4]=>000 [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,1]=>0001 [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[2,3,4,1,5]=>0010 [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[2,3,1,5,4]=>0101 [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[2,3,1,4,5]=>0100 [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[2,1,4,5,3]=>1001 [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,5]=>1010 [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[2,1,3,5,4]=>1001 [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[2,1,3,4,5]=>1000 [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[1,3,4,5,2]=>0001 [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[1,3,4,2,5]=>0010 [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4]=>0101 [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[1,3,2,4,5]=>0100 [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[1,2,4,5,3]=>0001 [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[1,2,4,3,5]=>0010 [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[1,2,3,5,4]=>0001 [5]=>[1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,5]=>0000 [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,1]=>00001 [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,1,6]=>00010 [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,1,6,5]=>00101 [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,1,5,6]=>00100 [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,1,5,6,4]=>01001 [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,1,5,4,6]=>01010 [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[2,3,1,4,6,5]=>01001 [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,1,4,5,6]=>01000 [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,5,6,3]=>10001 [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,5,3,6]=>10010 [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5]=>10101 [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,5,6]=>10100 [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,3,5,6,4]=>10001 [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,3,5,4,6]=>10010 [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,3,4,6,5]=>10001 [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,3,4,5,6]=>10000 [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,2]=>00001 [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,2,6]=>00010 [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,4,2,6,5]=>00101 [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,4,2,5,6]=>00100 [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,6,4]=>01001 [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,6]=>01010 [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,4,6,5]=>01001 [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,4,5,6]=>01000 [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[1,2,4,5,6,3]=>00001 [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[1,2,4,5,3,6]=>00010 [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5]=>00101 [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[1,2,4,3,5,6]=>00100 [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[1,2,3,5,6,4]=>00001 [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[1,2,3,5,4,6]=>00010 [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[1,2,3,4,6,5]=>00001 [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,4,5,6]=>00000 [1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,7,1]=>000001 [1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,6,1,7]=>000010 [1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,5,1,7,6]=>000101 [1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,5,1,6,7]=>000100 [1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,4,1,6,7,5]=>001001 [1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,4,1,6,5,7]=>001010 [1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[2,3,4,1,5,7,6]=>001001 [1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,4,1,5,6,7]=>001000 [1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,3,1,5,6,7,4]=>010001 [1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[2,3,1,5,6,4,7]=>010010 [1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,1,5,4,7,6]=>010101 [1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[2,3,1,5,4,6,7]=>010100 [1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[2,3,1,4,6,7,5]=>010001 [1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[2,3,1,4,6,5,7]=>010010 [1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[2,3,1,4,5,7,6]=>010001 [1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,1,4,5,6,7]=>010000 [1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,4,5,6,7,3]=>100001 [1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,4,5,6,3,7]=>100010 [1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,4,5,3,7,6]=>100101 [1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,4,5,3,6,7]=>100100 [1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,6,7,5]=>101001 [1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,7]=>101010 [1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,4,3,5,7,6]=>101001 [1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,5,6,7]=>101000 [1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,1,3,5,6,7,4]=>100001 [1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[2,1,3,5,6,4,7]=>100010 [1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[2,1,3,5,4,7,6]=>100101 [1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[2,1,3,5,4,6,7]=>100100 [1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[2,1,3,4,6,7,5]=>100001 [1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[2,1,3,4,6,5,7]=>100010 [1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[2,1,3,4,5,7,6]=>100001 [1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,1,3,4,5,6,7]=>100000 [2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,7,2]=>000001 [2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,6,2,7]=>000010 [2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,4,5,2,7,6]=>000101 [2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,4,5,2,6,7]=>000100 [2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,3,4,2,6,7,5]=>001001 [2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,3,4,2,6,5,7]=>001010 [2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[1,3,4,2,5,7,6]=>001001 [2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[1,3,4,2,5,6,7]=>001000 [2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,5,6,7,4]=>010001 [2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,5,6,4,7]=>010010 [2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,6]=>010101 [2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,5,4,6,7]=>010100 [2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,3,2,4,6,7,5]=>010001 [2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,3,2,4,6,5,7]=>010010 [2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,3,2,4,5,7,6]=>010001 [2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,2,4,5,6,7]=>010000 [3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,4,5,6,7,3]=>000001 [3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[1,2,4,5,6,3,7]=>000010 [3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[1,2,4,5,3,7,6]=>000101 [3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[1,2,4,5,3,6,7]=>000100 [3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,6,7,5]=>001001 [3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5,7]=>001010 [3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,2,4,3,5,7,6]=>001001 [3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[1,2,4,3,5,6,7]=>001000 [4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,2,3,5,6,7,4]=>000001 [4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[1,2,3,5,6,4,7]=>000010 [4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[1,2,3,5,4,7,6]=>000101 [4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[1,2,3,5,4,6,7]=>000100 [5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,2,3,4,6,7,5]=>000001 [5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[1,2,3,4,6,5,7]=>000010 [6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,2,3,4,5,7,6]=>000001 [7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7]=>000000 [1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,7,8,1]=>0000001 [1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,6,7,1,8]=>0000010 [1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,5,6,1,8,7]=>0000101 [1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,5,6,1,7,8]=>0000100 [1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,4,5,1,7,8,6]=>0001001 [1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,4,5,1,7,6,8]=>0001010 [1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[2,3,4,5,1,6,8,7]=>0001001 [1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,4,5,1,6,7,8]=>0001000 [1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,3,4,1,6,7,8,5]=>0010001 [1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[2,3,4,1,6,7,5,8]=>0010010 [1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,4,1,6,5,8,7]=>0010101 [1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[2,3,4,1,6,5,7,8]=>0010100 [1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[2,3,4,1,5,7,8,6]=>0010001 [1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[2,3,4,1,5,7,6,8]=>0010010 [1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[2,3,4,1,5,6,8,7]=>0010001 [1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,4,1,5,6,7,8]=>0010000 [1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,3,1,5,6,7,8,4]=>0100001 [1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[2,3,1,5,6,7,4,8]=>0100010 [1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[2,3,1,5,6,4,7,8]=>0100100 [1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,3,1,5,4,7,6,8]=>0101010 [1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,3,1,5,4,6,7,8]=>0101000 [1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,3,1,4,6,7,8,5]=>0100001 [1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[2,3,1,4,6,7,5,8]=>0100010 [1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[2,3,1,4,6,5,8,7]=>0100101 [1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[2,3,1,4,5,7,8,6]=>0100001 [1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[2,3,1,4,5,7,6,8]=>0100010 [1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[2,3,1,4,5,6,8,7]=>0100001 [1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,1,4,5,6,7,8]=>0100000 [1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,4,5,6,7,8,3]=>1000001 [1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,4,5,6,7,3,8]=>1000010 [1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,4,5,6,3,8,7]=>1000101 [1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,4,5,3,6,8,7]=>1001001 [1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,4,5,3,6,7,8]=>1001000 [1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,6,7,8,5]=>1010001 [1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,8,7]=>1010101 [1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,6,5,7,8]=>1010100 [1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,4,3,5,7,8,6]=>1010001 [1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,4,3,5,6,8,7]=>1010001 [1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,4,3,5,6,7,8]=>1010000 [1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,5,6,7,8,4]=>1000001 [1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[2,1,3,5,6,4,8,7]=>1000101 [1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[2,1,3,5,4,7,8,6]=>1001001 [1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[2,1,3,5,4,6,8,7]=>1001001 [1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[2,1,3,4,6,7,8,5]=>1000001 [1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[2,1,3,4,6,7,5,8]=>1000010 [1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[2,1,3,4,6,5,8,7]=>1000101 [1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[2,1,3,4,6,5,7,8]=>1000100 [1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[2,1,3,4,5,7,8,6]=>1000001 [1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[2,1,3,4,5,7,6,8]=>1000010 [1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,1,3,4,5,6,8,7]=>1000001 [1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8]=>1000000 [2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,7,8,2]=>0000001 [2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,6,7,2,8]=>0000010 [2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,4,5,6,2,8,7]=>0000101 [2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,4,5,6,2,7,8]=>0000100 [2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,3,4,5,2,7,8,6]=>0001001 [2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,3,4,5,2,7,6,8]=>0001010 [2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,3,4,5,2,6,8,7]=>0001001 [2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,3,4,5,2,6,7,8]=>0001000 [2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,4,2,6,7,8,5]=>0010001 [2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,4,2,6,7,5,8]=>0010010 [2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,4,2,6,5,8,7]=>0010101 [2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,3,4,2,5,7,8,6]=>0010001 [2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,3,4,2,5,7,6,8]=>0010010 [2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,3,4,2,5,6,8,7]=>0010001 [2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,4,2,5,6,7,8]=>0010000 [2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,5,6,7,8,4]=>0100001 [2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,5,6,7,4,8]=>0100010 [2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,5,6,4,8,7]=>0100101 [2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,2,5,6,4,7,8]=>0100100 [2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,7,8,6]=>0101001 [2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6,8]=>0101010 [2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,5,4,6,7,8]=>0101000 [2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,3,2,4,6,7,8,5]=>0100001 [2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,3,2,4,6,5,8,7]=>0100101 [2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,3,2,4,6,5,7,8]=>0100100 [2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,3,2,4,5,7,8,6]=>0100001 [2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,3,2,4,5,7,6,8]=>0100010 [2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,3,2,4,5,6,8,7]=>0100001 [2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,3,2,4,5,6,7,8]=>0100000 [3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,4,5,6,7,8,3]=>0000001 [3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,4,5,6,7,3,8]=>0000010 [3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,4,5,6,3,8,7]=>0000101 [3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,4,5,6,3,7,8]=>0000100 [3,1,2,1,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,4,5,3,7,8,6]=>0001001 [3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,5,3,7,6,8]=>0001010 [3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,6,7,8,5]=>0010001 [3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,4,3,6,7,5,8]=>0010010 [3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,8,7]=>0010101 [3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,6,5,7,8]=>0010100 [3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,4,3,5,7,8,6]=>0010001 [3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,4,3,5,7,6,8]=>0010010 [3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,4,3,5,6,8,7]=>0010001 [3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,4,3,5,6,7,8]=>0010000 [4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,3,5,6,7,8,4]=>0000001 [4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,6,7,4,8]=>0000010 [4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,6,4,8,7]=>0000101 [4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[1,2,3,5,6,4,7,8]=>0000100 [4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,7,8,6]=>0001001 [4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6,8]=>0001010 [4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[1,2,3,5,4,6,8,7]=>0001001 [4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,2,3,5,4,6,7,8]=>0001000 [5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[1,2,3,4,6,7,8,5]=>0000001 [5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[1,2,3,4,6,7,5,8]=>0000010 [5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,8,7]=>0000101 [5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[1,2,3,4,6,5,7,8]=>0000100 [6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[1,2,3,4,5,7,8,6]=>0000001 [6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[1,2,3,4,5,7,6,8]=>0000010 [7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[1,2,3,4,5,6,8,7]=>0000001 [8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7,8]=>0000000 [1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,7,8,9,1]=>00000001 [1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,6,7,8,1,9]=>00000010 [1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,5,6,7,1,9,8]=>00000101 [1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,5,6,7,1,8,9]=>00000100 [1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,4,5,6,1,8,9,7]=>00001001 [1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,4,5,6,1,8,7,9]=>00001010 [1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,4,5,6,1,7,8,9]=>00001000 [1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,3,4,5,1,7,8,9,6]=>00010001 [1,1,1,1,2,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,4,5,1,7,6,9,8]=>00010101 [1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[2,3,4,5,1,7,6,8,9]=>00010100 [1,1,1,1,4,1]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[2,3,4,5,1,6,7,9,8]=>00010001 [1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,1,6,7,8,9]=>00010000 [1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,3,4,1,6,7,8,9,5]=>00100001 [1,1,1,2,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,3,4,1,6,5,8,7,9]=>00101010 [1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,3,4,1,6,5,7,8,9]=>00101000 [1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,1,5,6,7,8,9]=>00100000 [1,1,2,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,1,5,6,7,8,9,4]=>01000001 [1,1,2,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,1,5,4,7,6,9,8]=>01010101 [1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,1,5,4,6,7,8,9]=>01010000 [1,1,3,2,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[2,3,1,4,6,5,8,7,9]=>01001010 [1,1,5,1,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[2,3,1,4,5,6,8,9,7]=>01000001 [1,1,6,1]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,3,1,4,5,6,7,9,8]=>01000001 [1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,1,4,5,6,7,8,9]=>01000000 [1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,4,5,6,7,8,9,3]=>10000001 [1,2,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,4,5,6,7,8,3,9]=>10000010 [1,2,1,5]=>[1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,4,5,3,6,7,8,9]=>10010000 [1,2,2,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,4,3,6,7,8,9,5]=>10100001 [1,2,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7,9]=>10101010 [1,2,2,4]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,6,5,7,8,9]=>10101000 [1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,1,4,3,5,6,7,8,9]=>10100000 [1,3,1,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,5,6,7,8,9,4]=>10000001 [1,6,1,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[2,1,3,4,5,6,8,9,7]=>10000001 [1,6,2]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[2,1,3,4,5,6,8,7,9]=>10000010 [1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[2,1,3,4,5,6,7,9,8]=>10000001 [1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8,9]=>10000000 [2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,7,8,9,2]=>00000001 [2,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,6,7,8,2,9]=>00000010 [2,1,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,4,5,6,7,2,8,9]=>00000100 [2,1,1,5]=>[1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,4,5,2,6,7,8,9]=>00010000 [2,1,2,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,4,2,6,5,8,7,9]=>00101010 [2,1,6]=>[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,3,4,2,5,6,7,8,9]=>00100000 [2,2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,6,9,8]=>01010101 [2,2,5]=>[1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,2,5,4,6,7,8,9]=>01010000 [2,5,1,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,3,2,4,5,6,8,9,7]=>01000001 [2,6,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,3,2,4,5,6,7,9,8]=>01000001 [2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,3,2,4,5,6,7,8,9]=>01000000 [3,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,4,5,6,7,8,9,3]=>00000001 [3,1,5]=>[1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,4,5,3,6,7,8,9]=>00010000 [3,2,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5,8,7,9]=>00101010 [3,3,1,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,2,4,3,5,7,8,9,6]=>00100001 [3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,4,3,5,6,7,8,9]=>00100000 [4,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,5,6,7,8,9,4]=>00000001 [4,1,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,6,7,8,4,9]=>00000010 [4,2,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,3,5,4,7,6,9,8]=>00010101 [4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,5,4,6,7,8,9]=>00010000 [5,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,6,7,8,9,5]=>00000001 [5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,6,5,7,8,9]=>00001000 [6,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>[1,2,3,4,5,7,8,9,6]=>00000001 [6,1,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,8,6,9]=>00000010 [6,2,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]=>[1,2,3,4,5,7,6,9,8]=>00000101 [6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[1,2,3,4,5,7,6,8,9]=>00000100 [7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[1,2,3,4,5,6,8,9,7]=>00000001 [7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[1,2,3,4,5,6,8,7,9]=>00000010 [8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[1,2,3,4,5,6,7,9,8]=>00000001 [9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7,8,9]=>00000000 [1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,7,8,9,10,1]=>000000001 [1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,6,7,8,9,1,10]=>000000010 [1,1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,5,6,7,8,1,10,9]=>000000101 [1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,5,6,7,8,1,9,10]=>000000100 [1,1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,4,5,6,7,1,9,10,8]=>000001001 [1,1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,4,5,6,7,1,9,8,10]=>000001010 [1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,4,5,6,7,1,8,9,10]=>000001000 [1,1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,3,4,5,6,1,8,9,10,7]=>000010001 [1,1,1,1,1,2,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,4,5,6,1,8,7,10,9]=>000010101 [1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,6,1,7,8,9,10]=>000010000 [1,1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,1,7,8,9,10,6]=>000100001 [1,1,1,1,2,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,3,4,5,1,7,6,9,8,10]=>000101010 [1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,1,6,7,8,9,10]=>000100000 [1,1,1,2,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,1,6,7,8,9,10,5]=>001000001 [1,1,1,2,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,3,4,1,6,5,8,9,10,7]=>001010001 [1,1,1,2,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,4,1,6,5,8,7,10,9]=>001010101 [1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,4,1,5,6,7,8,9,10]=>001000000 [1,1,2,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,1,5,6,7,8,9,10,4]=>010000001 [1,1,2,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,3,1,5,4,7,6,9,8,10]=>010101010 [1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,3,1,4,5,6,7,8,9,10]=>010000000 [1,2,1,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,4,5,6,7,8,9,10,3]=>100000001 [1,2,1,1,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,4,5,6,7,8,3,10,9]=>100000101 [1,2,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,4,3,6,7,8,9,10,5]=>101000001 [1,2,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,6,5,8,9,10,7]=>101010001 [1,2,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,8,7,10,9]=>101010101 [1,2,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,6,5,8,7,9,10]=>101010100 [1,2,4,1,1,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,8,9,10,7]=>101000001 [1,2,7]=>[1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,1,4,3,5,6,7,8,9,10]=>101000000 [1,5,1,1,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,7,8,9,10,6]=>100000001 [1,7,1,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[2,1,3,4,5,6,7,9,10,8]=>100000001 [1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[2,1,3,4,5,6,7,8,10,9]=>100000001 [1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8,9,10]=>100000000 [2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,7,8,9,10,2]=>000000001 [2,1,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,6,7,8,9,2,10]=>000000010 [2,1,7]=>[1,1,0,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,3,4,2,5,6,7,8,9,10]=>001000000 [2,2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,8,6,10,9]=>010100101 [2,2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,7,6,9,10,8]=>010101001 [2,2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6,9,8,10]=>010101010 [2,2,3,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,3,2,5,4,6,8,9,10,7]=>010100001 [2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,3,2,4,5,6,7,8,9,10]=>010000000 [3,1,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,4,5,6,7,8,9,10,3]=>000000001 [3,2,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,8,7,10,9]=>001010101 [3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,2,4,3,5,6,7,8,9,10]=>001000000 [4,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,5,6,7,8,9,10,4]=>000000001 [4,2,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6,9,8,10]=>000101010 [4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,5,4,6,7,8,9,10]=>000100000 [5,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,6,7,8,9,10,5]=>000000001 [5,2,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,8,7,10,9]=>000010101 [5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,6,5,7,8,9,10]=>000010000 [6,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,7,8,9,10,6]=>000000001 [6,1,1,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,8,9,6,10]=>000000010 [6,1,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,7,8,6,9,10]=>000000100 [6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,5,7,6,8,9,10]=>000001000 [7,1,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,8,9,10,7]=>000000001 [7,1,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0]=>[1,2,3,4,5,6,8,9,7,10]=>000000010 [7,2,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]=>[1,2,3,4,5,6,8,7,10,9]=>000000101 [7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[1,2,3,4,5,6,8,7,9,10]=>000000100 [8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>[1,2,3,4,5,6,7,9,10,8]=>000000001 [8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[1,2,3,4,5,6,7,9,8,10]=>000000010 [9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[1,2,3,4,5,6,7,8,10,9]=>000000001 [10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7,8,9,10]=>000000000 [1,10]=>[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8,9,10,11]=>1000000000 [10,1]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]=>[1,2,3,4,5,6,7,8,9,11,10]=>0000000001 [1,2,2,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>10101010101 [1,1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,6,7,8,9,10,1,11]=>0000000010
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.