Identifier
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Images
=>
Cc0005;cc-rep-0
[1,0]=>[2,1]=>[2,1]=>1=>1
[1,0,1,0]=>[3,1,2]=>[3,2,1]=>11=>11
[1,1,0,0]=>[2,3,1]=>[3,1,2]=>01=>10
[1,0,1,0,1,0]=>[4,1,2,3]=>[4,3,2,1]=>111=>111
[1,0,1,1,0,0]=>[3,1,4,2]=>[4,2,1,3]=>101=>110
[1,1,0,0,1,0]=>[2,4,1,3]=>[4,3,1,2]=>011=>101
[1,1,0,1,0,0]=>[4,3,1,2]=>[4,2,3,1]=>011=>101
[1,1,1,0,0,0]=>[2,3,4,1]=>[4,1,2,3]=>001=>100
[1,0,1,0,1,0,1,0]=>[5,1,2,3,4]=>[5,4,3,2,1]=>1111=>1111
[1,0,1,0,1,1,0,0]=>[4,1,2,5,3]=>[5,3,2,1,4]=>1101=>1110
[1,0,1,1,0,0,1,0]=>[3,1,5,2,4]=>[5,4,2,1,3]=>1011=>1101
[1,0,1,1,0,1,0,0]=>[5,1,4,2,3]=>[5,3,4,2,1]=>1011=>1101
[1,0,1,1,1,0,0,0]=>[3,1,4,5,2]=>[5,2,1,3,4]=>1001=>1100
[1,1,0,0,1,0,1,0]=>[2,5,1,3,4]=>[5,4,3,1,2]=>0111=>1011
[1,1,0,0,1,1,0,0]=>[2,4,1,5,3]=>[5,3,1,2,4]=>0101=>1010
[1,1,0,1,0,0,1,0]=>[5,3,1,2,4]=>[5,4,2,3,1]=>0111=>1011
[1,1,0,1,0,1,0,0]=>[5,4,1,2,3]=>[4,2,5,3,1]=>0111=>1011
[1,1,0,1,1,0,0,0]=>[4,3,1,5,2]=>[5,2,3,1,4]=>0101=>1010
[1,1,1,0,0,0,1,0]=>[2,3,5,1,4]=>[5,4,1,2,3]=>0011=>1001
[1,1,1,0,0,1,0,0]=>[2,5,4,1,3]=>[5,3,4,1,2]=>0011=>1001
[1,1,1,0,1,0,0,0]=>[5,3,4,1,2]=>[5,2,3,4,1]=>0011=>1001
[1,1,1,1,0,0,0,0]=>[2,3,4,5,1]=>[5,1,2,3,4]=>0001=>1000
[1,0,1,0,1,0,1,0,1,0]=>[6,1,2,3,4,5]=>[6,5,4,3,2,1]=>11111=>11111
[1,0,1,0,1,0,1,1,0,0]=>[5,1,2,3,6,4]=>[6,4,3,2,1,5]=>11101=>11110
[1,0,1,0,1,1,0,0,1,0]=>[4,1,2,6,3,5]=>[6,5,3,2,1,4]=>11011=>11101
[1,0,1,0,1,1,0,1,0,0]=>[6,1,2,5,3,4]=>[6,4,5,3,2,1]=>11011=>11101
[1,0,1,0,1,1,1,0,0,0]=>[4,1,2,5,6,3]=>[6,3,2,1,4,5]=>11001=>11100
[1,0,1,1,0,0,1,0,1,0]=>[3,1,6,2,4,5]=>[6,5,4,2,1,3]=>10111=>11011
[1,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,6,4]=>[6,4,2,1,3,5]=>10101=>11010
[1,0,1,1,0,1,0,0,1,0]=>[6,1,4,2,3,5]=>[6,5,3,4,2,1]=>10111=>11011
[1,0,1,1,0,1,0,1,0,0]=>[6,1,5,2,3,4]=>[5,3,6,4,2,1]=>10111=>11011
[1,0,1,1,0,1,1,0,0,0]=>[5,1,4,2,6,3]=>[6,3,4,2,1,5]=>10101=>11010
[1,0,1,1,1,0,0,0,1,0]=>[3,1,4,6,2,5]=>[6,5,2,1,3,4]=>10011=>11001
[1,0,1,1,1,0,0,1,0,0]=>[3,1,6,5,2,4]=>[6,4,5,2,1,3]=>10011=>11001
[1,0,1,1,1,0,1,0,0,0]=>[6,1,4,5,2,3]=>[6,3,4,5,2,1]=>10011=>11001
[1,0,1,1,1,1,0,0,0,0]=>[3,1,4,5,6,2]=>[6,2,1,3,4,5]=>10001=>11000
[1,1,0,0,1,0,1,0,1,0]=>[2,6,1,3,4,5]=>[6,5,4,3,1,2]=>01111=>10111
[1,1,0,0,1,0,1,1,0,0]=>[2,5,1,3,6,4]=>[6,4,3,1,2,5]=>01101=>10110
[1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,5]=>[6,5,3,1,2,4]=>01011=>10101
[1,1,0,0,1,1,0,1,0,0]=>[2,6,1,5,3,4]=>[6,4,5,3,1,2]=>01011=>10101
[1,1,0,0,1,1,1,0,0,0]=>[2,4,1,5,6,3]=>[6,3,1,2,4,5]=>01001=>10100
[1,1,0,1,0,0,1,0,1,0]=>[6,3,1,2,4,5]=>[6,5,4,2,3,1]=>01111=>10111
[1,1,0,1,0,0,1,1,0,0]=>[5,3,1,2,6,4]=>[6,4,2,3,1,5]=>01101=>10110
[1,1,0,1,0,1,0,0,1,0]=>[6,4,1,2,3,5]=>[4,2,6,5,3,1]=>01111=>10111
[1,1,0,1,0,1,0,1,0,0]=>[5,6,1,2,3,4]=>[5,3,1,6,4,2]=>01111=>10111
[1,1,0,1,0,1,1,0,0,0]=>[5,4,1,2,6,3]=>[4,2,6,3,1,5]=>01101=>10110
[1,1,0,1,1,0,0,0,1,0]=>[4,3,1,6,2,5]=>[6,5,2,3,1,4]=>01011=>10101
[1,1,0,1,1,0,0,1,0,0]=>[6,3,1,5,2,4]=>[6,4,5,2,3,1]=>01011=>10101
[1,1,0,1,1,0,1,0,0,0]=>[6,4,1,5,2,3]=>[5,2,4,6,3,1]=>01011=>10101
[1,1,0,1,1,1,0,0,0,0]=>[4,3,1,5,6,2]=>[6,2,3,1,4,5]=>01001=>10100
[1,1,1,0,0,0,1,0,1,0]=>[2,3,6,1,4,5]=>[6,5,4,1,2,3]=>00111=>10011
[1,1,1,0,0,0,1,1,0,0]=>[2,3,5,1,6,4]=>[6,4,1,2,3,5]=>00101=>10010
[1,1,1,0,0,1,0,0,1,0]=>[2,6,4,1,3,5]=>[6,5,3,4,1,2]=>00111=>10011
[1,1,1,0,0,1,0,1,0,0]=>[2,6,5,1,3,4]=>[5,3,6,4,1,2]=>00111=>10011
[1,1,1,0,0,1,1,0,0,0]=>[2,5,4,1,6,3]=>[6,3,4,1,2,5]=>00101=>10010
[1,1,1,0,1,0,0,0,1,0]=>[6,3,4,1,2,5]=>[6,5,2,3,4,1]=>00111=>10011
[1,1,1,0,1,0,0,1,0,0]=>[6,3,5,1,2,4]=>[5,2,3,6,4,1]=>00111=>10011
[1,1,1,0,1,0,1,0,0,0]=>[6,5,4,1,2,3]=>[5,2,6,3,4,1]=>00111=>10011
[1,1,1,0,1,1,0,0,0,0]=>[5,3,4,1,6,2]=>[6,2,3,4,1,5]=>00101=>10010
[1,1,1,1,0,0,0,0,1,0]=>[2,3,4,6,1,5]=>[6,5,1,2,3,4]=>00011=>10001
[1,1,1,1,0,0,0,1,0,0]=>[2,3,6,5,1,4]=>[6,4,5,1,2,3]=>00011=>10001
[1,1,1,1,0,0,1,0,0,0]=>[2,6,4,5,1,3]=>[6,3,4,5,1,2]=>00011=>10001
[1,1,1,1,0,1,0,0,0,0]=>[6,3,4,5,1,2]=>[6,2,3,4,5,1]=>00011=>10001
[1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]=>00001=>10000
[1,0,1,0,1,0,1,0,1,0,1,0]=>[7,1,2,3,4,5,6]=>[7,6,5,4,3,2,1]=>111111=>111111
[1,0,1,0,1,0,1,0,1,1,0,0]=>[6,1,2,3,4,7,5]=>[7,5,4,3,2,1,6]=>111101=>111110
[1,0,1,0,1,0,1,1,0,0,1,0]=>[5,1,2,3,7,4,6]=>[7,6,4,3,2,1,5]=>111011=>111101
[1,0,1,0,1,0,1,1,0,1,0,0]=>[7,1,2,3,6,4,5]=>[7,5,6,4,3,2,1]=>111011=>111101
[1,0,1,0,1,0,1,1,1,0,0,0]=>[5,1,2,3,6,7,4]=>[7,4,3,2,1,5,6]=>111001=>111100
[1,0,1,0,1,1,0,0,1,0,1,0]=>[4,1,2,7,3,5,6]=>[7,6,5,3,2,1,4]=>110111=>111011
[1,0,1,0,1,1,0,0,1,1,0,0]=>[4,1,2,6,3,7,5]=>[7,5,3,2,1,4,6]=>110101=>111010
[1,0,1,0,1,1,0,1,0,0,1,0]=>[7,1,2,5,3,4,6]=>[7,6,4,5,3,2,1]=>110111=>111011
[1,0,1,0,1,1,0,1,0,1,0,0]=>[7,1,2,6,3,4,5]=>[6,4,7,5,3,2,1]=>110111=>111011
[1,0,1,0,1,1,0,1,1,0,0,0]=>[6,1,2,5,3,7,4]=>[7,4,5,3,2,1,6]=>110101=>111010
[1,0,1,0,1,1,1,0,0,0,1,0]=>[4,1,2,5,7,3,6]=>[7,6,3,2,1,4,5]=>110011=>111001
[1,0,1,0,1,1,1,0,0,1,0,0]=>[4,1,2,7,6,3,5]=>[7,5,6,3,2,1,4]=>110011=>111001
[1,0,1,0,1,1,1,0,1,0,0,0]=>[7,1,2,5,6,3,4]=>[7,4,5,6,3,2,1]=>110011=>111001
[1,0,1,0,1,1,1,1,0,0,0,0]=>[4,1,2,5,6,7,3]=>[7,3,2,1,4,5,6]=>110001=>111000
[1,0,1,1,0,0,1,0,1,0,1,0]=>[3,1,7,2,4,5,6]=>[7,6,5,4,2,1,3]=>101111=>110111
[1,0,1,1,0,0,1,0,1,1,0,0]=>[3,1,6,2,4,7,5]=>[7,5,4,2,1,3,6]=>101101=>110110
[1,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,5,2,7,4,6]=>[7,6,4,2,1,3,5]=>101011=>110101
[1,0,1,1,0,0,1,1,0,1,0,0]=>[3,1,7,2,6,4,5]=>[7,5,6,4,2,1,3]=>101011=>110101
[1,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,5,2,6,7,4]=>[7,4,2,1,3,5,6]=>101001=>110100
[1,0,1,1,0,1,0,0,1,0,1,0]=>[7,1,4,2,3,5,6]=>[7,6,5,3,4,2,1]=>101111=>110111
[1,0,1,1,0,1,0,0,1,1,0,0]=>[6,1,4,2,3,7,5]=>[7,5,3,4,2,1,6]=>101101=>110110
[1,0,1,1,0,1,0,1,0,0,1,0]=>[7,1,5,2,3,4,6]=>[5,3,7,6,4,2,1]=>101111=>110111
[1,0,1,1,0,1,0,1,0,1,0,0]=>[6,1,7,2,3,4,5]=>[6,4,2,1,7,5,3]=>101111=>110111
[1,0,1,1,0,1,0,1,1,0,0,0]=>[6,1,5,2,3,7,4]=>[5,3,7,4,2,1,6]=>101101=>110110
[1,0,1,1,0,1,1,0,0,0,1,0]=>[5,1,4,2,7,3,6]=>[7,6,3,4,2,1,5]=>101011=>110101
[1,0,1,1,0,1,1,0,0,1,0,0]=>[7,1,4,2,6,3,5]=>[7,5,6,3,4,2,1]=>101011=>110101
[1,0,1,1,0,1,1,0,1,0,0,0]=>[7,1,5,2,6,3,4]=>[6,3,5,7,4,2,1]=>101011=>110101
[1,0,1,1,0,1,1,1,0,0,0,0]=>[5,1,4,2,6,7,3]=>[7,3,4,2,1,5,6]=>101001=>110100
[1,0,1,1,1,0,0,0,1,0,1,0]=>[3,1,4,7,2,5,6]=>[7,6,5,2,1,3,4]=>100111=>110011
[1,0,1,1,1,0,0,0,1,1,0,0]=>[3,1,4,6,2,7,5]=>[7,5,2,1,3,4,6]=>100101=>110010
[1,0,1,1,1,0,0,1,0,0,1,0]=>[3,1,7,5,2,4,6]=>[7,6,4,5,2,1,3]=>100111=>110011
[1,0,1,1,1,0,0,1,0,1,0,0]=>[3,1,7,6,2,4,5]=>[6,4,7,5,2,1,3]=>100111=>110011
[1,0,1,1,1,0,0,1,1,0,0,0]=>[3,1,6,5,2,7,4]=>[7,4,5,2,1,3,6]=>100101=>110010
[1,0,1,1,1,0,1,0,0,0,1,0]=>[7,1,4,5,2,3,6]=>[7,6,3,4,5,2,1]=>100111=>110011
[1,0,1,1,1,0,1,0,0,1,0,0]=>[7,1,4,6,2,3,5]=>[6,3,4,7,5,2,1]=>100111=>110011
[1,0,1,1,1,0,1,0,1,0,0,0]=>[7,1,6,5,2,3,4]=>[6,3,7,4,5,2,1]=>100111=>110011
[1,0,1,1,1,0,1,1,0,0,0,0]=>[6,1,4,5,2,7,3]=>[7,3,4,5,2,1,6]=>100101=>110010
[1,0,1,1,1,1,0,0,0,0,1,0]=>[3,1,4,5,7,2,6]=>[7,6,2,1,3,4,5]=>100011=>110001
[1,0,1,1,1,1,0,0,0,1,0,0]=>[3,1,4,7,6,2,5]=>[7,5,6,2,1,3,4]=>100011=>110001
[1,0,1,1,1,1,0,0,1,0,0,0]=>[3,1,7,5,6,2,4]=>[7,4,5,6,2,1,3]=>100011=>110001
[1,0,1,1,1,1,0,1,0,0,0,0]=>[7,1,4,5,6,2,3]=>[7,3,4,5,6,2,1]=>100011=>110001
[1,0,1,1,1,1,1,0,0,0,0,0]=>[3,1,4,5,6,7,2]=>[7,2,1,3,4,5,6]=>100001=>110000
[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,7,1,3,4,5,6]=>[7,6,5,4,3,1,2]=>011111=>101111
[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,6,1,3,4,7,5]=>[7,5,4,3,1,2,6]=>011101=>101110
[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,5,1,3,7,4,6]=>[7,6,4,3,1,2,5]=>011011=>101101
[1,1,0,0,1,0,1,1,0,1,0,0]=>[2,7,1,3,6,4,5]=>[7,5,6,4,3,1,2]=>011011=>101101
[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,5,1,3,6,7,4]=>[7,4,3,1,2,5,6]=>011001=>101100
[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,4,1,7,3,5,6]=>[7,6,5,3,1,2,4]=>010111=>101011
[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,4,1,6,3,7,5]=>[7,5,3,1,2,4,6]=>010101=>101010
[1,1,0,0,1,1,0,1,0,0,1,0]=>[2,7,1,5,3,4,6]=>[7,6,4,5,3,1,2]=>010111=>101011
[1,1,0,0,1,1,0,1,0,1,0,0]=>[2,7,1,6,3,4,5]=>[6,4,7,5,3,1,2]=>010111=>101011
[1,1,0,0,1,1,0,1,1,0,0,0]=>[2,6,1,5,3,7,4]=>[7,4,5,3,1,2,6]=>010101=>101010
[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,4,1,5,7,3,6]=>[7,6,3,1,2,4,5]=>010011=>101001
[1,1,0,0,1,1,1,0,0,1,0,0]=>[2,4,1,7,6,3,5]=>[7,5,6,3,1,2,4]=>010011=>101001
[1,1,0,0,1,1,1,0,1,0,0,0]=>[2,7,1,5,6,3,4]=>[7,4,5,6,3,1,2]=>010011=>101001
[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,4,1,5,6,7,3]=>[7,3,1,2,4,5,6]=>010001=>101000
[1,1,0,1,0,0,1,0,1,0,1,0]=>[7,3,1,2,4,5,6]=>[7,6,5,4,2,3,1]=>011111=>101111
[1,1,0,1,0,0,1,0,1,1,0,0]=>[6,3,1,2,4,7,5]=>[7,5,4,2,3,1,6]=>011101=>101110
[1,1,0,1,0,0,1,1,0,0,1,0]=>[5,3,1,2,7,4,6]=>[7,6,4,2,3,1,5]=>011011=>101101
[1,1,0,1,0,0,1,1,0,1,0,0]=>[7,3,1,2,6,4,5]=>[7,5,6,4,2,3,1]=>011011=>101101
[1,1,0,1,0,0,1,1,1,0,0,0]=>[5,3,1,2,6,7,4]=>[7,4,2,3,1,5,6]=>011001=>101100
[1,1,0,1,0,1,0,0,1,0,1,0]=>[7,4,1,2,3,5,6]=>[4,2,7,6,5,3,1]=>011111=>101111
[1,1,0,1,0,1,0,0,1,1,0,0]=>[6,4,1,2,3,7,5]=>[4,2,7,5,3,1,6]=>011101=>101110
[1,1,0,1,0,1,0,1,0,0,1,0]=>[5,7,1,2,3,4,6]=>[5,3,1,7,6,4,2]=>011111=>101111
[1,1,0,1,0,1,0,1,0,1,0,0]=>[7,6,1,2,3,4,5]=>[6,4,2,7,5,3,1]=>011111=>101111
[1,1,0,1,0,1,0,1,1,0,0,0]=>[5,6,1,2,3,7,4]=>[5,3,1,7,4,2,6]=>011101=>101110
[1,1,0,1,0,1,1,0,0,0,1,0]=>[5,4,1,2,7,3,6]=>[4,2,7,6,3,1,5]=>011011=>101101
[1,1,0,1,0,1,1,0,0,1,0,0]=>[7,4,1,2,6,3,5]=>[4,2,7,5,6,3,1]=>011011=>101101
[1,1,0,1,0,1,1,0,1,0,0,0]=>[5,7,1,2,6,3,4]=>[6,3,1,5,7,4,2]=>011011=>101101
[1,1,0,1,0,1,1,1,0,0,0,0]=>[5,4,1,2,6,7,3]=>[4,2,7,3,1,5,6]=>011001=>101100
[1,1,0,1,1,0,0,0,1,0,1,0]=>[4,3,1,7,2,5,6]=>[7,6,5,2,3,1,4]=>010111=>101011
[1,1,0,1,1,0,0,0,1,1,0,0]=>[4,3,1,6,2,7,5]=>[7,5,2,3,1,4,6]=>010101=>101010
[1,1,0,1,1,0,0,1,0,0,1,0]=>[7,3,1,5,2,4,6]=>[7,6,4,5,2,3,1]=>010111=>101011
[1,1,0,1,1,0,0,1,0,1,0,0]=>[7,3,1,6,2,4,5]=>[6,4,7,5,2,3,1]=>010111=>101011
[1,1,0,1,1,0,0,1,1,0,0,0]=>[6,3,1,5,2,7,4]=>[7,4,5,2,3,1,6]=>010101=>101010
[1,1,0,1,1,0,1,0,0,0,1,0]=>[7,4,1,5,2,3,6]=>[5,2,4,7,6,3,1]=>010111=>101011
[1,1,0,1,1,0,1,0,0,1,0,0]=>[7,4,1,6,2,3,5]=>[7,5,2,4,6,3,1]=>010111=>101011
[1,1,0,1,1,0,1,0,1,0,0,0]=>[6,7,1,5,2,3,4]=>[6,3,1,7,4,5,2]=>010111=>101011
[1,1,0,1,1,0,1,1,0,0,0,0]=>[6,4,1,5,2,7,3]=>[5,2,4,7,3,1,6]=>010101=>101010
[1,1,0,1,1,1,0,0,0,0,1,0]=>[4,3,1,5,7,2,6]=>[7,6,2,3,1,4,5]=>010011=>101001
[1,1,0,1,1,1,0,0,0,1,0,0]=>[4,3,1,7,6,2,5]=>[7,5,6,2,3,1,4]=>010011=>101001
[1,1,0,1,1,1,0,0,1,0,0,0]=>[7,3,1,5,6,2,4]=>[7,4,5,6,2,3,1]=>010011=>101001
[1,1,0,1,1,1,0,1,0,0,0,0]=>[7,4,1,5,6,2,3]=>[6,2,4,5,7,3,1]=>010011=>101001
[1,1,0,1,1,1,1,0,0,0,0,0]=>[4,3,1,5,6,7,2]=>[7,2,3,1,4,5,6]=>010001=>101000
[1,1,1,0,0,0,1,0,1,0,1,0]=>[2,3,7,1,4,5,6]=>[7,6,5,4,1,2,3]=>001111=>100111
[1,1,1,0,0,0,1,0,1,1,0,0]=>[2,3,6,1,4,7,5]=>[7,5,4,1,2,3,6]=>001101=>100110
[1,1,1,0,0,0,1,1,0,0,1,0]=>[2,3,5,1,7,4,6]=>[7,6,4,1,2,3,5]=>001011=>100101
[1,1,1,0,0,0,1,1,0,1,0,0]=>[2,3,7,1,6,4,5]=>[7,5,6,4,1,2,3]=>001011=>100101
[1,1,1,0,0,0,1,1,1,0,0,0]=>[2,3,5,1,6,7,4]=>[7,4,1,2,3,5,6]=>001001=>100100
[1,1,1,0,0,1,0,0,1,0,1,0]=>[2,7,4,1,3,5,6]=>[7,6,5,3,4,1,2]=>001111=>100111
[1,1,1,0,0,1,0,0,1,1,0,0]=>[2,6,4,1,3,7,5]=>[7,5,3,4,1,2,6]=>001101=>100110
[1,1,1,0,0,1,0,1,0,0,1,0]=>[2,7,5,1,3,4,6]=>[5,3,7,6,4,1,2]=>001111=>100111
[1,1,1,0,0,1,0,1,0,1,0,0]=>[2,6,7,1,3,4,5]=>[6,4,1,2,7,5,3]=>001111=>100111
[1,1,1,0,0,1,0,1,1,0,0,0]=>[2,6,5,1,3,7,4]=>[5,3,7,4,1,2,6]=>001101=>100110
[1,1,1,0,0,1,1,0,0,0,1,0]=>[2,5,4,1,7,3,6]=>[7,6,3,4,1,2,5]=>001011=>100101
[1,1,1,0,0,1,1,0,0,1,0,0]=>[2,7,4,1,6,3,5]=>[7,5,6,3,4,1,2]=>001011=>100101
[1,1,1,0,0,1,1,0,1,0,0,0]=>[2,7,5,1,6,3,4]=>[6,3,5,7,4,1,2]=>001011=>100101
[1,1,1,0,0,1,1,1,0,0,0,0]=>[2,5,4,1,6,7,3]=>[7,3,4,1,2,5,6]=>001001=>100100
[1,1,1,0,1,0,0,0,1,0,1,0]=>[7,3,4,1,2,5,6]=>[7,6,5,2,3,4,1]=>001111=>100111
[1,1,1,0,1,0,0,0,1,1,0,0]=>[6,3,4,1,2,7,5]=>[7,5,2,3,4,1,6]=>001101=>100110
[1,1,1,0,1,0,0,1,0,0,1,0]=>[7,3,5,1,2,4,6]=>[5,2,3,7,6,4,1]=>001111=>100111
[1,1,1,0,1,0,0,1,0,1,0,0]=>[6,3,7,1,2,4,5]=>[6,4,1,7,5,2,3]=>001111=>100111
[1,1,1,0,1,0,0,1,1,0,0,0]=>[6,3,5,1,2,7,4]=>[5,2,3,7,4,1,6]=>001101=>100110
[1,1,1,0,1,0,1,0,0,0,1,0]=>[7,5,4,1,2,3,6]=>[5,2,7,6,3,4,1]=>001111=>100111
[1,1,1,0,1,0,1,0,0,1,0,0]=>[6,7,4,1,2,3,5]=>[6,3,4,1,7,5,2]=>001111=>100111
[1,1,1,0,1,0,1,0,1,0,0,0]=>[6,7,5,1,2,3,4]=>[7,4,1,6,3,5,2]=>001111=>100111
[1,1,1,0,1,0,1,1,0,0,0,0]=>[6,5,4,1,2,7,3]=>[5,2,7,3,4,1,6]=>001101=>100110
[1,1,1,0,1,1,0,0,0,0,1,0]=>[5,3,4,1,7,2,6]=>[7,6,2,3,4,1,5]=>001011=>100101
[1,1,1,0,1,1,0,0,0,1,0,0]=>[7,3,4,1,6,2,5]=>[7,5,6,2,3,4,1]=>001011=>100101
[1,1,1,0,1,1,0,0,1,0,0,0]=>[7,3,5,1,6,2,4]=>[6,2,3,5,7,4,1]=>001011=>100101
[1,1,1,0,1,1,0,1,0,0,0,0]=>[7,5,4,1,6,2,3]=>[6,2,5,7,3,4,1]=>001011=>100101
[1,1,1,0,1,1,1,0,0,0,0,0]=>[5,3,4,1,6,7,2]=>[7,2,3,4,1,5,6]=>001001=>100100
[1,1,1,1,0,0,0,0,1,0,1,0]=>[2,3,4,7,1,5,6]=>[7,6,5,1,2,3,4]=>000111=>100011
[1,1,1,1,0,0,0,0,1,1,0,0]=>[2,3,4,6,1,7,5]=>[7,5,1,2,3,4,6]=>000101=>100010
[1,1,1,1,0,0,0,1,0,0,1,0]=>[2,3,7,5,1,4,6]=>[7,6,4,5,1,2,3]=>000111=>100011
[1,1,1,1,0,0,0,1,0,1,0,0]=>[2,3,7,6,1,4,5]=>[6,4,7,5,1,2,3]=>000111=>100011
[1,1,1,1,0,0,0,1,1,0,0,0]=>[2,3,6,5,1,7,4]=>[7,4,5,1,2,3,6]=>000101=>100010
[1,1,1,1,0,0,1,0,0,0,1,0]=>[2,7,4,5,1,3,6]=>[7,6,3,4,5,1,2]=>000111=>100011
[1,1,1,1,0,0,1,0,0,1,0,0]=>[2,7,4,6,1,3,5]=>[6,3,4,7,5,1,2]=>000111=>100011
[1,1,1,1,0,0,1,0,1,0,0,0]=>[2,7,6,5,1,3,4]=>[6,3,7,4,5,1,2]=>000111=>100011
[1,1,1,1,0,0,1,1,0,0,0,0]=>[2,6,4,5,1,7,3]=>[7,3,4,5,1,2,6]=>000101=>100010
[1,1,1,1,0,1,0,0,0,0,1,0]=>[7,3,4,5,1,2,6]=>[7,6,2,3,4,5,1]=>000111=>100011
[1,1,1,1,0,1,0,0,0,1,0,0]=>[7,3,4,6,1,2,5]=>[6,2,3,4,7,5,1]=>000111=>100011
[1,1,1,1,0,1,0,0,1,0,0,0]=>[7,3,6,5,1,2,4]=>[6,2,3,7,4,5,1]=>000111=>100011
[1,1,1,1,0,1,0,1,0,0,0,0]=>[7,6,4,5,1,2,3]=>[6,2,7,3,4,5,1]=>000111=>100011
[1,1,1,1,0,1,1,0,0,0,0,0]=>[6,3,4,5,1,7,2]=>[7,2,3,4,5,1,6]=>000101=>100010
[1,1,1,1,1,0,0,0,0,0,1,0]=>[2,3,4,5,7,1,6]=>[7,6,1,2,3,4,5]=>000011=>100001
[1,1,1,1,1,0,0,0,0,1,0,0]=>[2,3,4,7,6,1,5]=>[7,5,6,1,2,3,4]=>000011=>100001
[1,1,1,1,1,0,0,0,1,0,0,0]=>[2,3,7,5,6,1,4]=>[7,4,5,6,1,2,3]=>000011=>100001
[1,1,1,1,1,0,0,1,0,0,0,0]=>[2,7,4,5,6,1,3]=>[7,3,4,5,6,1,2]=>000011=>100001
[1,1,1,1,1,0,1,0,0,0,0,0]=>[7,3,4,5,6,1,2]=>[7,2,3,4,5,6,1]=>000011=>100001
[1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,6,7,1]=>[7,1,2,3,4,5,6]=>000001=>100000
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,1,2,3,4,5,6,7]=>[8,7,6,5,4,3,2,1]=>1111111=>1111111
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,1,2,3,4,5,8,6]=>[8,6,5,4,3,2,1,7]=>1111101=>1111110
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[6,1,2,3,4,8,5,7]=>[8,7,5,4,3,2,1,6]=>1111011=>1111101
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[8,1,2,3,4,7,5,6]=>[8,6,7,5,4,3,2,1]=>1111011=>1111101
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[5,1,2,3,8,4,6,7]=>[8,7,6,4,3,2,1,5]=>1110111=>1111011
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[5,1,2,3,7,4,8,6]=>[8,6,4,3,2,1,5,7]=>1110101=>1111010
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[8,1,2,3,6,4,5,7]=>[8,7,5,6,4,3,2,1]=>1110111=>1111011
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[8,1,2,3,7,4,5,6]=>[7,5,8,6,4,3,2,1]=>1110111=>1111011
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]=>[7,1,2,3,6,4,8,5]=>[8,5,6,4,3,2,1,7]=>1110101=>1111010
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[5,1,2,3,6,8,4,7]=>[8,7,4,3,2,1,5,6]=>1110011=>1111001
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[5,1,2,3,8,7,4,6]=>[8,6,7,4,3,2,1,5]=>1110011=>1111001
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[8,1,2,3,6,7,4,5]=>[8,5,6,7,4,3,2,1]=>1110011=>1111001
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[5,1,2,3,6,7,8,4]=>[8,4,3,2,1,5,6,7]=>1110001=>1111000
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[4,1,2,8,3,5,6,7]=>[8,7,6,5,3,2,1,4]=>1101111=>1110111
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[4,1,2,7,3,5,8,6]=>[8,6,5,3,2,1,4,7]=>1101101=>1110110
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[4,1,2,6,3,8,5,7]=>[8,7,5,3,2,1,4,6]=>1101011=>1110101
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[4,1,2,8,3,7,5,6]=>[8,6,7,5,3,2,1,4]=>1101011=>1110101
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[4,1,2,6,3,7,8,5]=>[8,5,3,2,1,4,6,7]=>1101001=>1110100
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[8,1,2,5,3,4,6,7]=>[8,7,6,4,5,3,2,1]=>1101111=>1110111
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[7,1,2,8,3,4,5,6]=>[7,5,3,2,1,8,6,4]=>1101111=>1110111
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]=>[8,1,2,5,3,7,4,6]=>[8,6,7,4,5,3,2,1]=>1101011=>1110101
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]=>[8,1,2,6,3,7,4,5]=>[7,4,6,8,5,3,2,1]=>1101011=>1110101
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]=>[6,1,2,5,3,7,8,4]=>[8,4,5,3,2,1,6,7]=>1101001=>1110100
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[4,1,2,5,8,3,6,7]=>[8,7,6,3,2,1,4,5]=>1100111=>1110011
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[4,1,2,5,7,3,8,6]=>[8,6,3,2,1,4,5,7]=>1100101=>1110010
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]=>[4,1,2,8,6,3,5,7]=>[8,7,5,6,3,2,1,4]=>1100111=>1110011
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[4,1,2,8,7,3,5,6]=>[7,5,8,6,3,2,1,4]=>1100111=>1110011
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]=>[4,1,2,7,6,3,8,5]=>[8,5,6,3,2,1,4,7]=>1100101=>1110010
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[8,1,2,7,6,3,4,5]=>[7,4,8,5,6,3,2,1]=>1100111=>1110011
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[7,1,2,5,6,3,8,4]=>[8,4,5,6,3,2,1,7]=>1100101=>1110010
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[4,1,2,5,6,8,3,7]=>[8,7,3,2,1,4,5,6]=>1100011=>1110001
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]=>[4,1,2,8,6,7,3,5]=>[8,5,6,7,3,2,1,4]=>1100011=>1110001
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[4,1,2,5,6,7,8,3]=>[8,3,2,1,4,5,6,7]=>1100001=>1110000
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[3,1,8,2,4,5,6,7]=>[8,7,6,5,4,2,1,3]=>1011111=>1101111
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[3,1,7,2,4,5,8,6]=>[8,6,5,4,2,1,3,7]=>1011101=>1101110
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[3,1,6,2,4,7,8,5]=>[8,5,4,2,1,3,6,7]=>1011001=>1101100
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[3,1,5,2,8,4,6,7]=>[8,7,6,4,2,1,3,5]=>1010111=>1101011
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,7,4,8,6]=>[8,6,4,2,1,3,5,7]=>1010101=>1101010
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[3,1,8,2,6,4,5,7]=>[8,7,5,6,4,2,1,3]=>1010111=>1101011
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[3,1,8,2,7,4,5,6]=>[7,5,8,6,4,2,1,3]=>1010111=>1101011
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]=>[3,1,7,2,6,4,8,5]=>[8,5,6,4,2,1,3,7]=>1010101=>1101010
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]=>[3,1,5,2,8,7,4,6]=>[8,6,7,4,2,1,3,5]=>1010011=>1101001
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[3,1,5,2,6,7,8,4]=>[8,4,2,1,3,5,6,7]=>1010001=>1101000
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[8,1,4,2,3,5,6,7]=>[8,7,6,5,3,4,2,1]=>1011111=>1101111
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[7,1,4,2,3,5,8,6]=>[8,6,5,3,4,2,1,7]=>1011101=>1101110
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[6,1,4,2,3,8,5,7]=>[8,7,5,3,4,2,1,6]=>1011011=>1101101
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[6,1,8,2,3,4,5,7]=>[6,4,2,1,8,7,5,3]=>1011111=>1101111
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]=>[6,1,7,2,3,4,8,5]=>[6,4,2,1,8,5,3,7]=>1011101=>1101110
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]=>[6,1,5,2,3,8,4,7]=>[5,3,8,7,4,2,1,6]=>1011011=>1101101
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]=>[6,1,8,2,3,7,4,5]=>[7,4,2,1,6,8,5,3]=>1011011=>1101101
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]=>[6,1,5,2,3,7,8,4]=>[5,3,8,4,2,1,6,7]=>1011001=>1101100
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]=>[5,1,4,2,8,3,6,7]=>[8,7,6,3,4,2,1,5]=>1010111=>1101011
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]=>[5,1,4,2,7,3,8,6]=>[8,6,3,4,2,1,5,7]=>1010101=>1101010
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]=>[8,1,4,2,6,3,5,7]=>[8,7,5,6,3,4,2,1]=>1010111=>1101011
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]=>[8,1,4,2,7,3,5,6]=>[7,5,8,6,3,4,2,1]=>1010111=>1101011
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]=>[7,1,4,2,6,3,8,5]=>[8,5,6,3,4,2,1,7]=>1010101=>1101010
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]=>[8,1,5,2,7,3,4,6]=>[8,6,3,5,7,4,2,1]=>1010111=>1101011
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]=>[7,1,8,2,6,3,4,5]=>[7,4,2,1,8,5,6,3]=>1010111=>1101011
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]=>[7,1,5,2,6,3,8,4]=>[6,3,5,8,4,2,1,7]=>1010101=>1101010
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]=>[5,1,4,2,6,8,3,7]=>[8,7,3,4,2,1,5,6]=>1010011=>1101001
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]=>[8,1,4,2,6,7,3,5]=>[8,5,6,7,3,4,2,1]=>1010011=>1101001
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]=>[8,1,5,2,6,7,3,4]=>[7,3,5,6,8,4,2,1]=>1010011=>1101001
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,4,8,2,5,6,7]=>[8,7,6,5,2,1,3,4]=>1001111=>1100111
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,4,7,2,5,8,6]=>[8,6,5,2,1,3,4,7]=>1001101=>1100110
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[3,1,4,6,2,8,5,7]=>[8,7,5,2,1,3,4,6]=>1001011=>1100101
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]=>[3,1,8,6,2,4,5,7]=>[6,4,8,7,5,2,1,3]=>1001111=>1100111
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[3,1,7,8,2,4,5,6]=>[7,5,2,1,3,8,6,4]=>1001111=>1100111
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]=>[8,1,4,5,2,3,6,7]=>[8,7,6,3,4,5,2,1]=>1001111=>1100111
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[7,1,4,8,2,3,5,6]=>[7,5,2,1,8,6,3,4]=>1001111=>1100111
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]=>[8,1,6,5,2,3,4,7]=>[6,3,8,7,4,5,2,1]=>1001111=>1100111
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[7,1,8,5,2,3,4,6]=>[7,4,5,2,1,8,6,3]=>1001111=>1100111
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[7,1,8,6,2,3,4,5]=>[8,5,2,1,7,4,6,3]=>1001111=>1100111
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[7,1,6,5,2,3,8,4]=>[6,3,8,4,5,2,1,7]=>1001101=>1100110
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]=>[6,1,4,5,2,8,3,7]=>[8,7,3,4,5,2,1,6]=>1001011=>1100101
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[8,1,4,5,2,7,3,6]=>[8,6,7,3,4,5,2,1]=>1001011=>1100101
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[8,1,6,5,2,7,3,4]=>[7,3,6,8,4,5,2,1]=>1001011=>1100101
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[6,1,4,5,2,7,8,3]=>[8,3,4,5,2,1,6,7]=>1001001=>1100100
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[3,1,4,5,8,2,6,7]=>[8,7,6,2,1,3,4,5]=>1000111=>1100011
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[3,1,4,8,7,2,5,6]=>[7,5,8,6,2,1,3,4]=>1000111=>1100011
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]=>[3,1,4,7,6,2,8,5]=>[8,5,6,2,1,3,4,7]=>1000101=>1100010
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]=>[3,1,8,5,6,2,4,7]=>[8,7,4,5,6,2,1,3]=>1000111=>1100011
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]=>[3,1,8,5,7,2,4,6]=>[7,4,5,8,6,2,1,3]=>1000111=>1100011
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]=>[3,1,8,7,6,2,4,5]=>[7,4,8,5,6,2,1,3]=>1000111=>1100011
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]=>[8,1,4,5,6,2,3,7]=>[8,7,3,4,5,6,2,1]=>1000111=>1100011
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[8,1,4,5,7,2,3,6]=>[7,3,4,5,8,6,2,1]=>1000111=>1100011
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[8,1,7,5,6,2,3,4]=>[7,3,8,4,5,6,2,1]=>1000111=>1100011
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]=>[7,1,4,5,6,2,8,3]=>[8,3,4,5,6,2,1,7]=>1000101=>1100010
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[3,1,4,5,6,8,2,7]=>[8,7,2,1,3,4,5,6]=>1000011=>1100001
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]=>[3,1,4,8,6,7,2,5]=>[8,5,6,7,2,1,3,4]=>1000011=>1100001
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,8,1,3,4,5,6,7]=>[8,7,6,5,4,3,1,2]=>0111111=>1011111
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,5,1,3,8,4,6,7]=>[8,7,6,4,3,1,2,5]=>0110111=>1011011
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,4,1,8,3,5,6,7]=>[8,7,6,5,3,1,2,4]=>0101111=>1010111
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,8,5,7]=>[8,7,5,3,1,2,4,6]=>0101011=>1010101
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,4,1,6,3,7,8,5]=>[8,5,3,1,2,4,6,7]=>0101001=>1010100
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[2,8,1,6,3,4,5,7]=>[6,4,8,7,5,3,1,2]=>0101111=>1010111
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[2,7,1,8,3,4,5,6]=>[7,5,3,1,2,8,6,4]=>0101111=>1010111
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]=>[2,7,1,6,3,4,8,5]=>[6,4,8,5,3,1,2,7]=>0101101=>1010110
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]=>[2,8,1,5,3,7,4,6]=>[8,6,7,4,5,3,1,2]=>0101011=>1010101
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,4,1,5,8,3,6,7]=>[8,7,6,3,1,2,4,5]=>0100111=>1010011
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]=>[2,4,1,8,7,3,5,6]=>[7,5,8,6,3,1,2,4]=>0100111=>1010011
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]=>[2,4,1,7,6,3,8,5]=>[8,5,6,3,1,2,4,7]=>0100101=>1010010
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]=>[2,8,1,7,6,3,4,5]=>[7,4,8,5,6,3,1,2]=>0100111=>1010011
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,4,1,5,6,7,8,3]=>[8,3,1,2,4,5,6,7]=>0100001=>1010000
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[8,3,1,2,4,5,6,7]=>[8,7,6,5,4,2,3,1]=>0111111=>1011111
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[7,3,1,2,4,5,8,6]=>[8,6,5,4,2,3,1,7]=>0111101=>1011110
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[5,3,1,2,8,4,6,7]=>[8,7,6,4,2,3,1,5]=>0110111=>1011011
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[5,3,1,2,7,4,8,6]=>[8,6,4,2,3,1,5,7]=>0110101=>1011010
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]=>[7,3,1,2,6,4,8,5]=>[8,5,6,4,2,3,1,7]=>0110101=>1011010
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]=>[8,3,1,2,6,7,4,5]=>[8,5,6,7,4,2,3,1]=>0110011=>1011001
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[8,4,1,2,3,5,6,7]=>[4,2,8,7,6,5,3,1]=>0111111=>1011111
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[6,4,1,2,3,8,5,7]=>[4,2,8,7,5,3,1,6]=>0111011=>1011101
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[8,4,1,2,3,7,5,6]=>[4,2,8,6,7,5,3,1]=>0111011=>1011101
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[5,8,1,2,3,4,6,7]=>[5,3,1,8,7,6,4,2]=>0111111=>1011111
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[8,6,1,2,3,4,5,7]=>[6,4,2,8,7,5,3,1]=>0111111=>1011111
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,1,2,3,4,5,6]=>[8,6,4,2,7,5,3,1]=>0111111=>1011111
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]=>[7,6,1,2,3,4,8,5]=>[6,4,2,8,5,3,1,7]=>0111101=>1011110
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]=>[5,6,1,2,3,8,4,7]=>[5,3,1,8,7,4,2,6]=>0111011=>1011101
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]=>[5,4,1,2,8,3,6,7]=>[4,2,8,7,6,3,1,5]=>0110111=>1011011
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]=>[5,4,1,2,7,3,8,6]=>[4,2,8,6,3,1,5,7]=>0110101=>1011010
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]=>[8,4,1,2,7,3,5,6]=>[4,2,7,5,8,6,3,1]=>0110111=>1011011
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]=>[7,4,1,2,6,3,8,5]=>[4,2,8,5,6,3,1,7]=>0110101=>1011010
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]=>[5,8,1,2,7,3,4,6]=>[8,6,3,1,5,7,4,2]=>0110111=>1011011
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]=>[8,7,1,2,6,3,4,5]=>[7,4,2,8,5,6,3,1]=>0110111=>1011011
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]=>[5,4,1,2,6,8,3,7]=>[4,2,8,7,3,1,5,6]=>0110011=>1011001
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]=>[5,4,1,2,8,7,3,6]=>[4,2,8,6,7,3,1,5]=>0110011=>1011001
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]=>[8,4,1,2,6,7,3,5]=>[4,2,8,5,6,7,3,1]=>0110011=>1011001
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]=>[4,3,1,8,2,5,6,7]=>[8,7,6,5,2,3,1,4]=>0101111=>1010111
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]=>[4,3,1,7,2,5,8,6]=>[8,6,5,2,3,1,4,7]=>0101101=>1010110
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]=>[4,3,1,6,2,8,5,7]=>[8,7,5,2,3,1,4,6]=>0101011=>1010101
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]=>[4,3,1,8,2,7,5,6]=>[8,6,7,5,2,3,1,4]=>0101011=>1010101
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]=>[8,3,1,5,2,4,6,7]=>[8,7,6,4,5,2,3,1]=>0101111=>1010111
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]=>[8,3,1,6,2,4,5,7]=>[6,4,8,7,5,2,3,1]=>0101111=>1010111
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]=>[7,3,1,6,2,4,8,5]=>[6,4,8,5,2,3,1,7]=>0101101=>1010110
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]=>[6,3,1,5,2,8,4,7]=>[8,7,4,5,2,3,1,6]=>0101011=>1010101
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>[8,3,1,5,2,7,4,6]=>[8,6,7,4,5,2,3,1]=>0101011=>1010101
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]=>[6,3,1,5,2,7,8,4]=>[8,4,5,2,3,1,6,7]=>0101001=>1010100
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]=>[8,4,1,6,2,3,5,7]=>[8,7,5,2,4,6,3,1]=>0101111=>1010111
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]=>[7,4,1,8,2,3,5,6]=>[8,6,3,1,7,5,2,4]=>0101111=>1010111
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]=>[8,7,1,5,2,3,4,6]=>[7,4,5,2,8,6,3,1]=>0101111=>1010111
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]=>[8,7,1,6,2,3,4,5]=>[8,5,2,7,4,6,3,1]=>0101111=>1010111
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]=>[8,4,1,5,2,7,3,6]=>[5,2,4,8,6,7,3,1]=>0101011=>1010101
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]=>[4,3,1,8,7,2,5,6]=>[7,5,8,6,2,3,1,4]=>0100111=>1010011
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]=>[4,3,1,7,6,2,8,5]=>[8,5,6,2,3,1,4,7]=>0100101=>1010010
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]=>[8,3,1,5,6,2,4,7]=>[8,7,4,5,6,2,3,1]=>0100111=>1010011
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]=>[8,3,1,7,6,2,4,5]=>[7,4,8,5,6,2,3,1]=>0100111=>1010011
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]=>[8,4,1,5,6,2,3,7]=>[6,2,4,5,8,7,3,1]=>0100111=>1010011
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]=>[7,8,1,5,6,2,3,4]=>[7,3,1,8,4,5,6,2]=>0100111=>1010011
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]=>[7,4,1,5,6,2,8,3]=>[6,2,4,5,8,3,1,7]=>0100101=>1010010
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]=>[4,3,1,8,6,7,2,5]=>[8,5,6,7,2,3,1,4]=>0100011=>1010001
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]=>[8,4,1,5,6,7,2,3]=>[7,2,4,5,6,8,3,1]=>0100011=>1010001
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]=>[4,3,1,5,6,7,8,2]=>[8,2,3,1,4,5,6,7]=>0100001=>1010000
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[2,3,8,1,4,5,6,7]=>[8,7,6,5,4,1,2,3]=>0011111=>1001111
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]=>[2,3,8,1,7,4,5,6]=>[7,5,8,6,4,1,2,3]=>0010111=>1001011
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]=>[2,3,7,1,6,4,8,5]=>[8,5,6,4,1,2,3,7]=>0010101=>1001010
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]=>[2,3,8,1,6,7,4,5]=>[8,5,6,7,4,1,2,3]=>0010011=>1001001
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]=>[2,8,4,1,3,5,6,7]=>[8,7,6,5,3,4,1,2]=>0011111=>1001111
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]=>[2,6,4,1,3,8,5,7]=>[8,7,5,3,4,1,2,6]=>0011011=>1001101
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]=>[2,6,4,1,3,7,8,5]=>[8,5,3,4,1,2,6,7]=>0011001=>1001100
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]=>[2,6,8,1,3,4,5,7]=>[6,4,1,2,8,7,5,3]=>0011111=>1001111
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[2,8,7,1,3,4,5,6]=>[7,5,3,8,6,4,1,2]=>0011111=>1001111
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]=>[2,6,7,1,3,4,8,5]=>[6,4,1,2,8,5,3,7]=>0011101=>1001110
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]=>[2,6,8,1,3,7,4,5]=>[7,4,1,2,6,8,5,3]=>0011011=>1001101
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]=>[2,6,5,1,3,7,8,4]=>[5,3,8,4,1,2,6,7]=>0011001=>1001100
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]=>[2,8,4,1,6,3,5,7]=>[8,7,5,6,3,4,1,2]=>0010111=>1001011
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]=>[2,8,4,1,7,3,5,6]=>[7,5,8,6,3,4,1,2]=>0010111=>1001011
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[2,7,4,1,6,3,8,5]=>[8,5,6,3,4,1,2,7]=>0010101=>1001010
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]=>[2,7,5,1,6,3,8,4]=>[6,3,5,8,4,1,2,7]=>0010101=>1001010
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]=>[2,8,4,1,6,7,3,5]=>[8,5,6,7,3,4,1,2]=>0010011=>1001001
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]=>[2,8,5,1,6,7,3,4]=>[7,3,5,6,8,4,1,2]=>0010011=>1001001
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]=>[2,5,4,1,6,7,8,3]=>[8,3,4,1,2,5,6,7]=>0010001=>1001000
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]=>[8,3,4,1,2,5,6,7]=>[8,7,6,5,2,3,4,1]=>0011111=>1001111
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]=>[7,3,4,1,2,5,8,6]=>[8,6,5,2,3,4,1,7]=>0011101=>1001110
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]=>[8,3,4,1,2,7,5,6]=>[8,6,7,5,2,3,4,1]=>0011011=>1001101
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[8,3,7,1,2,4,5,6]=>[7,5,2,3,8,6,4,1]=>0011111=>1001111
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]=>[6,3,5,1,2,8,4,7]=>[5,2,3,8,7,4,1,6]=>0011011=>1001101
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]=>[8,3,5,1,2,7,4,6]=>[5,2,3,8,6,7,4,1]=>0011011=>1001101
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]=>[6,3,8,1,2,7,4,5]=>[7,4,1,6,8,5,2,3]=>0011011=>1001101
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]=>[6,8,4,1,2,3,5,7]=>[6,3,4,1,8,7,5,2]=>0011111=>1001111
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[8,7,4,1,2,3,5,6]=>[7,5,2,8,6,3,4,1]=>0011111=>1001111
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]=>[6,8,5,1,2,3,4,7]=>[8,7,4,1,6,3,5,2]=>0011111=>1001111
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[8,7,5,1,2,3,4,6]=>[8,6,3,5,2,7,4,1]=>0011111=>1001111
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[6,7,8,1,2,3,4,5]=>[8,5,2,7,4,1,6,3]=>0011111=>1001111
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[8,5,4,1,2,7,3,6]=>[5,2,8,6,7,3,4,1]=>0011011=>1001101
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]=>[8,3,4,1,6,2,5,7]=>[8,7,5,6,2,3,4,1]=>0010111=>1001011
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]=>[7,3,4,1,6,2,8,5]=>[8,5,6,2,3,4,1,7]=>0010101=>1001010
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]=>[7,8,4,1,6,2,3,5]=>[7,3,4,1,8,5,6,2]=>0010111=>1001011
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[7,8,5,1,6,2,3,4]=>[8,4,1,7,3,5,6,2]=>0010111=>1001011
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]=>[8,3,4,1,6,7,2,5]=>[8,5,6,7,2,3,4,1]=>0010011=>1001001
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[5,3,4,1,6,7,8,2]=>[8,2,3,4,1,5,6,7]=>0010001=>1001000
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[2,3,4,8,1,5,6,7]=>[8,7,6,5,1,2,3,4]=>0001111=>1000111
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[2,3,4,7,1,5,8,6]=>[8,6,5,1,2,3,4,7]=>0001101=>1000110
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[2,3,4,6,1,8,5,7]=>[8,7,5,1,2,3,4,6]=>0001011=>1000101
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]=>[2,3,4,8,1,7,5,6]=>[8,6,7,5,1,2,3,4]=>0001011=>1000101
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]=>[2,3,8,6,1,4,5,7]=>[6,4,8,7,5,1,2,3]=>0001111=>1000111
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]=>[2,3,6,5,1,8,4,7]=>[8,7,4,5,1,2,3,6]=>0001011=>1000101
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]=>[2,3,8,5,1,7,4,6]=>[8,6,7,4,5,1,2,3]=>0001011=>1000101
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]=>[2,3,8,6,1,7,4,5]=>[7,4,6,8,5,1,2,3]=>0001011=>1000101
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]=>[2,8,4,5,1,3,6,7]=>[8,7,6,3,4,5,1,2]=>0001111=>1000111
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]=>[2,8,4,6,1,3,5,7]=>[6,3,4,8,7,5,1,2]=>0001111=>1000111
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]=>[2,7,4,8,1,3,5,6]=>[7,5,1,2,8,6,3,4]=>0001111=>1000111
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]=>[2,7,8,5,1,3,4,6]=>[7,4,5,1,2,8,6,3]=>0001111=>1000111
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]=>[2,7,6,5,1,3,8,4]=>[6,3,8,4,5,1,2,7]=>0001101=>1000110
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]=>[7,3,8,5,1,2,4,6]=>[7,4,5,1,8,6,2,3]=>0001111=>1000111
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]=>[7,3,8,6,1,2,4,5]=>[8,5,1,7,4,6,2,3]=>0001111=>1000111
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[7,8,6,5,1,2,3,4]=>[8,4,5,1,7,3,6,2]=>0001111=>1000111
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]=>[8,3,4,5,1,7,2,6]=>[8,6,7,2,3,4,5,1]=>0001011=>1000101
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[2,3,4,5,8,1,6,7]=>[8,7,6,1,2,3,4,5]=>0000111=>1000011
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]=>[2,3,4,8,6,1,5,7]=>[8,7,5,6,1,2,3,4]=>0000111=>1000011
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[2,3,4,8,7,1,5,6]=>[7,5,8,6,1,2,3,4]=>0000111=>1000011
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]=>[2,3,4,7,6,1,8,5]=>[8,5,6,1,2,3,4,7]=>0000101=>1000010
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]=>[2,3,8,7,6,1,4,5]=>[7,4,8,5,6,1,2,3]=>0000111=>1000011
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]=>[2,3,7,5,6,1,8,4]=>[8,4,5,6,1,2,3,7]=>0000101=>1000010
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[2,8,4,5,7,1,3,6]=>[7,3,4,5,8,6,1,2]=>0000111=>1000011
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]=>[2,8,4,7,6,1,3,5]=>[7,3,4,8,5,6,1,2]=>0000111=>1000011
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[2,8,7,5,6,1,3,4]=>[7,3,8,4,5,6,1,2]=>0000111=>1000011
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]=>[2,7,4,5,6,1,8,3]=>[8,3,4,5,6,1,2,7]=>0000101=>1000010
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[8,3,4,5,7,1,2,6]=>[7,2,3,4,5,8,6,1]=>0000111=>1000011
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]=>[8,3,4,7,6,1,2,5]=>[7,2,3,4,8,5,6,1]=>0000111=>1000011
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[8,7,4,5,6,1,2,3]=>[7,2,8,3,4,5,6,1]=>0000111=>1000011
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[2,3,4,5,8,7,1,6]=>[8,6,7,1,2,3,4,5]=>0000011=>1000001
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]=>[2,3,4,8,6,7,1,5]=>[8,5,6,7,1,2,3,4]=>0000011=>1000001
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,1]=>[8,1,2,3,4,5,6,7]=>0000001=>1000000
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
descent tops
Description
The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.
Map
rotate back-to-front
Description
The rotation of a binary word, last letter first.
This is the word obtained by moving the last letter to the beginnig.
This is the word obtained by moving the last letter to the beginnig.
searching the database
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