Identifier
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00114: Permutations connectivity setBinary words
Images
=>
Cc0010;cc-rep-0
[.,.]=>[1]=> [.,[.,.]]=>[2,1]=>0 [[.,.],.]=>[1,2]=>1 [.,[.,[.,.]]]=>[3,2,1]=>00 [.,[[.,.],.]]=>[2,3,1]=>00 [[.,.],[.,.]]=>[1,3,2]=>10 [[.,[.,.]],.]=>[2,1,3]=>01 [[[.,.],.],.]=>[1,2,3]=>11 [.,[.,[.,[.,.]]]]=>[4,3,2,1]=>000 [.,[.,[[.,.],.]]]=>[3,4,2,1]=>000 [.,[[.,.],[.,.]]]=>[2,4,3,1]=>000 [.,[[.,[.,.]],.]]=>[3,2,4,1]=>000 [.,[[[.,.],.],.]]=>[2,3,4,1]=>000 [[.,.],[.,[.,.]]]=>[1,4,3,2]=>100 [[.,.],[[.,.],.]]=>[1,3,4,2]=>100 [[.,[.,.]],[.,.]]=>[2,1,4,3]=>010 [[[.,.],.],[.,.]]=>[1,2,4,3]=>110 [[.,[.,[.,.]]],.]=>[3,2,1,4]=>001 [[.,[[.,.],.]],.]=>[2,3,1,4]=>001 [[[.,.],[.,.]],.]=>[1,3,2,4]=>101 [[[.,[.,.]],.],.]=>[2,1,3,4]=>011 [[[[.,.],.],.],.]=>[1,2,3,4]=>111 [.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>0000 [.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>0000 [.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>0000 [.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>0000 [.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>0000 [.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>0000 [.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>0000 [.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>0000 [.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>0000 [.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>0000 [.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>0000 [.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>0000 [.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>0000 [.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>0000 [[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>1000 [[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>1000 [[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>1000 [[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>1000 [[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>1000 [[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>0100 [[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>0100 [[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>1100 [[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>1100 [[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>0010 [[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>0010 [[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>1010 [[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>0110 [[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>1110 [[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>0001 [[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>0001 [[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>0001 [[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>0001 [[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>0001 [[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>1001 [[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>1001 [[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>0101 [[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>1101 [[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>0011 [[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>0011 [[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>1011 [[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>0111 [[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>1111 [.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>00000 [.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>00000 [.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>00000 [.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>00000 [.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>00000 [.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>00000 [.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>00000 [.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>00000 [.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>00000 [.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>00000 [.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>00000 [.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>00000 [.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>00000 [.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>00000 [.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>00000 [.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>00000 [.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>00000 [.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>00000 [.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>00000 [.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>00000 [.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>00000 [.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>00000 [.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>00000 [.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>00000 [.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>00000 [.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>00000 [.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>00000 [.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>00000 [.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>00000 [.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>00000 [.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>00000 [.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>00000 [.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>00000 [.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>00000 [.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>00000 [.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>00000 [.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>00000 [.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>00000 [.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>00000 [.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>00000 [.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>00000 [.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>00000 [[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>10000 [[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>10000 [[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>10000 [[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>10000 [[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>10000 [[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>10000 [[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>10000 [[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>10000 [[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>10000 [[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>10000 [[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>10000 [[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>10000 [[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>10000 [[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>10000 [[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>01000 [[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>01000 [[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>01000 [[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>01000 [[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>01000 [[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>11000 [[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>11000 [[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>11000 [[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>11000 [[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>11000 [[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>00100 [[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>00100 [[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>00100 [[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>00100 [[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>10100 [[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>10100 [[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>01100 [[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>01100 [[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>11100 [[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>11100 [[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>00010 [[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>00010 [[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>00010 [[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>00010 [[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>00010 [[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>10010 [[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>10010 [[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>01010 [[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>11010 [[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>00110 [[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>00110 [[[[.,.],[.,.]],.],[.,.]]=>[1,3,2,4,6,5]=>10110 [[[[.,[.,.]],.],.],[.,.]]=>[2,1,3,4,6,5]=>01110 [[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5]=>11110 [[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>00001 [[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>00001 [[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>00001 [[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>00001 [[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>00001 [[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>00001 [[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>00001 [[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>00001 [[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>00001 [[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>00001 [[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>00001 [[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>00001 [[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>00001 [[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>00001 [[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>10001 [[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6]=>10001 [[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6]=>10001 [[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6]=>10001 [[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6]=>10001 [[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>01001 [[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>01001 [[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>11001 [[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>11001 [[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>00101 [[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>00101 [[[[.,.],[.,.]],[.,.]],.]=>[1,3,2,5,4,6]=>10101 [[[[.,[.,.]],.],[.,.]],.]=>[2,1,3,5,4,6]=>01101 [[[[[.,.],.],.],[.,.]],.]=>[1,2,3,5,4,6]=>11101 [[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>00011 [[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>00011 [[[.,[[.,.],[.,.]]],.],.]=>[2,4,3,1,5,6]=>00011 [[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>00011 [[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>00011 [[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>10011 [[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>10011 [[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>01011 [[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>11011 [[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>00111 [[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>00111 [[[[[.,.],[.,.]],.],.],.]=>[1,3,2,4,5,6]=>10111 [[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>01111 [[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>11111 [.,[.,[.,[.,[.,[.,[.,.]]]]]]]=>[7,6,5,4,3,2,1]=>000000 [.,[.,[.,[.,[.,[[.,.],.]]]]]]=>[6,7,5,4,3,2,1]=>000000 [.,[.,[.,[.,[[.,.],[.,.]]]]]]=>[5,7,6,4,3,2,1]=>000000 [.,[.,[.,[.,[[.,[.,.]],.]]]]]=>[6,5,7,4,3,2,1]=>000000 [.,[.,[.,[.,[[[.,.],.],.]]]]]=>[5,6,7,4,3,2,1]=>000000 [.,[.,[.,[[.,.],[.,[.,.]]]]]]=>[4,7,6,5,3,2,1]=>000000 [.,[.,[.,[[.,.],[[.,.],.]]]]]=>[4,6,7,5,3,2,1]=>000000 [.,[.,[.,[[.,[.,.]],[.,.]]]]]=>[5,4,7,6,3,2,1]=>000000 [.,[.,[.,[[[.,.],.],[.,.]]]]]=>[4,5,7,6,3,2,1]=>000000 [.,[.,[.,[[.,[.,[.,.]]],.]]]]=>[6,5,4,7,3,2,1]=>000000 [.,[.,[.,[[.,[[.,.],.]],.]]]]=>[5,6,4,7,3,2,1]=>000000 [.,[.,[.,[[[.,.],[.,.]],.]]]]=>[4,6,5,7,3,2,1]=>000000 [.,[.,[.,[[[.,[.,.]],.],.]]]]=>[5,4,6,7,3,2,1]=>000000 [.,[.,[.,[[[[.,.],.],.],.]]]]=>[4,5,6,7,3,2,1]=>000000 [.,[.,[[.,.],[.,[.,[.,.]]]]]]=>[3,7,6,5,4,2,1]=>000000 [.,[.,[[.,.],[.,[[.,.],.]]]]]=>[3,6,7,5,4,2,1]=>000000 [.,[.,[[.,.],[[.,.],[.,.]]]]]=>[3,5,7,6,4,2,1]=>000000 [.,[.,[[.,.],[[.,[.,.]],.]]]]=>[3,6,5,7,4,2,1]=>000000 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[.,[[.,.],[.,[.,[[[.,.],.],.]]]]]=>[2,6,7,8,5,4,3,1]=>0000000 [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]=>[2,5,8,7,6,4,3,1]=>0000000 [.,[[.,.],[.,[[[[.,.],.],.],.]]]]=>[2,5,6,7,8,4,3,1]=>0000000 [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]=>[2,4,8,7,6,5,3,1]=>0000000 [.,[[.,.],[[.,.],[.,[[.,.],.]]]]]=>[2,4,7,8,6,5,3,1]=>0000000 [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]=>[2,4,6,8,7,5,3,1]=>0000000 [.,[[.,.],[[.,.],[[[.,.],.],.]]]]=>[2,4,6,7,8,5,3,1]=>0000000 [.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]=>[2,5,4,8,7,6,3,1]=>0000000 [.,[[.,.],[[[.,.],.],[[.,.],.]]]]=>[2,4,5,7,8,6,3,1]=>0000000 [.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]=>[2,6,5,4,8,7,3,1]=>0000000 [.,[[.,.],[[.,[[.,.],.]],[.,.]]]]=>[2,5,6,4,8,7,3,1]=>0000000 [.,[[.,.],[[[.,.],[.,.]],[.,.]]]]=>[2,4,6,5,8,7,3,1]=>0000000 [.,[[.,.],[[[[.,.],.],.],[.,.]]]]=>[2,4,5,6,8,7,3,1]=>0000000 [.,[[.,.],[[[.,.],[.,[.,.]]],.]]]=>[2,4,7,6,5,8,3,1]=>0000000 [.,[[.,.],[[[.,.],[[.,.],.]],.]]]=>[2,4,6,7,5,8,3,1]=>0000000 [.,[[.,.],[[[.,[.,[.,.]]],.],.]]]=>[2,6,5,4,7,8,3,1]=>0000000 [.,[[.,.],[[[[.,.],[.,.]],.],.]]]=>[2,4,6,5,7,8,3,1]=>0000000 [.,[[.,.],[[[[[.,.],.],.],.],.]]]=>[2,4,5,6,7,8,3,1]=>0000000 [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]=>[3,2,8,7,6,5,4,1]=>0000000 [.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]=>[3,2,6,8,7,5,4,1]=>0000000 [.,[[.,[.,.]],[[[.,.],[.,.]],.]]]=>[3,2,5,7,6,8,4,1]=>0000000 [.,[[.,[.,.]],[[[[.,.],.],.],.]]]=>[3,2,5,6,7,8,4,1]=>0000000 [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]=>[2,3,8,7,6,5,4,1]=>0000000 [.,[[[.,.],.],[.,[[[.,.],.],.]]]]=>[2,3,6,7,8,5,4,1]=>0000000 [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]=>[2,3,5,8,7,6,4,1]=>0000000 [.,[[[.,.],.],[[[.,.],.],[.,.]]]]=>[2,3,5,6,8,7,4,1]=>0000000 [.,[[[.,.],.],[[[[.,.],.],.],.]]]=>[2,3,5,6,7,8,4,1]=>0000000 [.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]=>[4,3,2,8,7,6,5,1]=>0000000 [.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]=>[4,3,2,7,8,6,5,1]=>0000000 [.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]=>[4,3,2,6,8,7,5,1]=>0000000 [.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]=>[3,4,2,8,7,6,5,1]=>0000000 [.,[[.,[[.,.],.]],[.,[[.,.],.]]]]=>[3,4,2,7,8,6,5,1]=>0000000 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[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[1,9,8,7,6,5,4,3,2]=>10000000 [[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]=>[7,8,6,5,4,3,2,1,9]=>00000001 [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[1,10,9,8,7,6,5,4,3,2]=>100000000 [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]=>[10,9,8,7,6,5,4,3,2,1,11]=>0000000001 [[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]=>[8,9,7,6,5,4,3,2,1,10]=>000000001 [[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]=>[7,6,8,5,4,3,2,1,9]=>00000001 [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[1,11,10,9,8,7,6,5,4,3,2]=>1000000000 [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]=>[3,2,5,4,7,6,9,8,10,1]=>000000000 [[.,.],[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]]=>[1,4,3,6,5,8,7,10,9,2]=>100000000 [[.,.],[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]]=>[1,4,3,6,5,8,7,10,9,12,11,2]=>10000000000 [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[2,1,9,8,7,6,5,4,3]=>01000000 [[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]=>[3,2,1,9,8,7,6,5,4]=>00100000 [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]=>[4,3,2,1,9,8,7,6,5]=>00010000 [[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]=>[5,4,3,2,1,9,8,7,6]=>00001000 [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]=>[6,5,4,3,2,1,9,8,7]=>00000100 [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]=>[7,6,5,4,3,2,1,9,8]=>00000010 [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[1,2,9,8,7,6,5,4,3]=>11000000 [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[2,1,10,9,8,7,6,5,4,3]=>010000000 [[.,[[.,.],.]],[.,[.,[.,[.,[.,.]]]]]]=>[2,3,1,9,8,7,6,5,4]=>00100000 [[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[3,2,1,10,9,8,7,6,5,4]=>001000000 [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]=>[4,3,2,1,10,9,8,7,6,5]=>000100000 [[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1,10,9,8,7,6]=>000010000 [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]=>[6,5,4,3,2,1,10,9,8,7]=>000001000 [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]=>[7,6,5,4,3,2,1,10,9,8]=>000000100 [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]=>[8,7,6,5,4,3,2,1,10,9]=>000000010 [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]=>[7,6,5,4,3,2,1,8,9]=>00000011 [[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]=>[6,5,4,3,2,1,7,8,9]=>00000111 [[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]=>[5,4,3,2,1,6,7,8,9]=>00001111 [[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9]=>00011111 [[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9]=>00111111 [[[[[[[[.,[.,.]],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9]=>01111111 [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]=>[8,7,6,5,4,3,2,1,9,10]=>000000011 [[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]=>[7,6,5,4,3,2,1,8,9,10]=>000000111 [[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]=>[6,5,4,3,2,1,7,8,9,10]=>000001111 [[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]=>[5,4,3,2,1,6,7,8,9,10]=>000011111 [[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9,10]=>000111111 [[[[[[[.,.],[.,[.,.]]],.],.],.],.],.]=>[1,4,3,2,5,6,7,8,9]=>10011111 [[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9,10]=>001111111 [[[[[[[[.,.],[.,.]],.],.],.],.],.],.]=>[1,3,2,4,5,6,7,8,9]=>10111111 [[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9,10]=>011111111 [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]=>[4,9,8,7,6,5,3,2,1]=>00000000 [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]=>[3,9,8,7,6,5,4,2,1]=>00000000 [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[2,9,8,7,6,5,4,3,1]=>00000000 [.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]]=>[6,10,9,8,7,5,4,3,2,1]=>000000000 [.,[[[[[[[.,.],.],.],.],.],.],[.,.]]]=>[2,3,4,5,6,7,9,8,1]=>00000000 [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9,10,11]=>0111111111 [[[[[[[.,[.,.]],.],.],.],.],[.,.]],.]=>[2,1,3,4,5,6,8,7,9]=>01111101 [[[[[[.,[.,.]],.],.],.],.],[[.,.],.]]=>[2,1,3,4,5,6,8,9,7]=>01111100 [.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]=>[2,3,4,5,6,8,9,7,1]=>00000000 [.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]=>[2,3,4,5,6,7,8,10,9,1]=>000000000 [[[[[[[.,[.,.]],[.,.]],.],.],.],.],.]=>[2,1,4,3,5,6,7,8,9]=>01011111 [[[[[[.,[.,[.,.]]],[.,.]],.],.],.],.]=>[3,2,1,5,4,6,7,8,9]=>00101111 [[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],.]=>[6,5,4,3,2,1,8,7,9]=>00000101 [.,[[.,[[[[[[.,.],.],.],.],.],.]],.]]=>[3,4,5,6,7,8,2,9,1]=>00000000 [[.,[[[[[[[.,.],.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,8,1,9]=>00000001 [[[[[[[.,[[.,.],.]],.],.],.],.],.],.]=>[2,3,1,4,5,6,7,8,9]=>00111111 [[[.,[[[[[[.,.],.],.],.],.],.]],.],.]=>[2,3,4,5,6,7,1,8,9]=>00000011 [[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.]=>[2,3,1,4,5,6,7,8,9,10]=>001111111 [[[[[[.,[[.,.],[.,.]]],.],.],.],.],.]=>[2,4,3,1,5,6,7,8,9]=>00011111 [[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,8,9,1,10]=>000000001 [[[[[[[[.,.],.],[.,.]],.],.],.],.],.]=>[1,2,4,3,5,6,7,8,9]=>11011111 [[[[[[[[.,.],.],.],[.,.]],.],.],.],.]=>[1,2,3,5,4,6,7,8,9]=>11101111 [[[[[[[[.,.],.],.],.],[.,.]],.],.],.]=>[1,2,3,4,6,5,7,8,9]=>11110111 [[[[[[[[.,.],.],.],.],.],[.,.]],.],.]=>[1,2,3,4,5,7,6,8,9]=>11111011 [[[[[[[[.,.],.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,6,8,7,9]=>11111101 [[[[[[[[[.,.],[.,.]],.],.],.],.],.],.],.]=>[1,3,2,4,5,6,7,8,9,10]=>101111111 [[[[[[[[[.,.],.],[.,.]],.],.],.],.],.],.]=>[1,2,4,3,5,6,7,8,9,10]=>110111111 [[[[[[[[[.,.],.],.],[.,.]],.],.],.],.],.]=>[1,2,3,5,4,6,7,8,9,10]=>111011111 [[[[[[[[[.,.],.],.],.],[.,.]],.],.],.],.]=>[1,2,3,4,6,5,7,8,9,10]=>111101111 [[[[[[[[[.,.],.],.],.],.],[.,.]],.],.],.]=>[1,2,3,4,5,7,6,8,9,10]=>111110111 [[[[[[[[[.,.],.],.],.],.],.],[.,.]],.],.]=>[1,2,3,4,5,6,8,7,9,10]=>111111011 [[[[[[[[[.,.],.],.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,6,7,9,8,10]=>111111101 [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]=>[7,6,5,4,3,2,9,8,1]=>00000000 [.,[[[[[[.,[.,[.,.]]],.],.],.],.],.]]=>[4,3,2,5,6,7,8,9,1]=>00000000 [.,[[[[[.,[.,[.,[.,.]]]],.],.],.],.]]=>[5,4,3,2,6,7,8,9,1]=>00000000 [.,[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]]=>[6,5,4,3,2,7,8,9,1]=>00000000 [.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]]=>[7,6,5,4,3,2,8,9,1]=>00000000 [.,[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]]=>[4,3,2,5,6,7,8,9,10,1]=>000000000 [.,[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]]=>[5,4,3,2,6,7,8,9,10,1]=>000000000 [.,[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]]=>[6,5,4,3,2,7,8,9,10,1]=>000000000 [.,[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]]=>[7,6,5,4,3,2,8,9,10,1]=>000000000 [.,[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]]=>[8,7,6,5,4,3,2,9,10,1]=>000000000 [[[.,.],[.,[.,[.,[.,[.,.]]]]]],[.,.]]=>[1,7,6,5,4,3,2,9,8]=>10000010 [[.,[.,[.,[.,[.,[.,.]]]]]],[[.,.],.]]=>[6,5,4,3,2,1,8,9,7]=>00000100 [[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8,9,10,11]=>1111111111 [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[1,2,10,9,8,7,6,5,4,3]=>110000000 [[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[1,3,2,9,8,7,6,5,4]=>10100000 [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]=>[2,1,8,7,6,5,4,3,9]=>01000001 [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]=>[1,9,8,7,6,5,4,3,2,10]=>100000001 [[[[[[.,.],[[[.,.],.],.]],.],.],.],.]=>[1,3,4,5,2,6,7,8,9]=>10001111 [[[[[[[.,.],.],[[.,.],.]],.],.],.],.]=>[1,2,4,5,3,6,7,8,9]=>11001111 [[[[[[.,[.,.]],[[.,.],.]],.],.],.],.]=>[2,1,4,5,3,6,7,8,9]=>01001111 [[[[[[[.,.],[.,.]],[.,.]],.],.],.],.]=>[1,3,2,5,4,6,7,8,9]=>10101111 [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]=>[8,7,6,5,4,3,2,10,9,1]=>000000000 [[.,[.,.]],[[[[[[.,.],.],.],.],.],.]]=>[2,1,4,5,6,7,8,9,3]=>01000000 [[[.,.],[[[[[[.,.],.],.],.],.],.]],.]=>[1,3,4,5,6,7,8,2,9]=>10000001 [[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]=>[1,2,3,4,5,6,7,10,9,8]=>111111100 [[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],.]=>[6,7,8,5,4,3,2,1,9]=>00000001 [[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]],.]=>[6,5,7,8,4,3,2,1,9]=>00000001 [[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]],.]=>[6,5,4,7,8,3,2,1,9]=>00000001 [[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]],.]=>[6,5,4,3,7,8,2,1,9]=>00000001 [[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]],.]=>[6,5,4,3,2,7,8,1,9]=>00000001 [[.,[.,[.,[.,[.,.]]]]],[[[.,.],.],.]]=>[5,4,3,2,1,7,8,9,6]=>00001000 [[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]],.]=>[7,8,9,6,5,4,3,2,1,10]=>000000001 [[.,[.,[.,[.,.]]]],[[[[.,.],.],.],.]]=>[4,3,2,1,6,7,8,9,5]=>00010000 [[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]],.]=>[7,6,8,9,5,4,3,2,1,10]=>000000001 [[.,[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]],.]=>[7,6,5,8,9,4,3,2,1,10]=>000000001 [[.,[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]],.]=>[7,6,5,4,8,9,3,2,1,10]=>000000001 [[.,[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]],.]=>[7,6,5,4,3,8,9,2,1,10]=>000000001 [[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]],.]=>[7,6,5,4,3,2,8,9,1,10]=>000000001 [[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]=>[1,2,3,4,5,9,8,7,6]=>11111000 [[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]=>[1,2,3,4,9,8,7,6,5]=>11110000 [[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]=>[1,2,3,9,8,7,6,5,4]=>11100000 [[[[[[[.,.],.],.],.],.],[.,.]],[.,.]]=>[1,2,3,4,5,7,6,9,8]=>11111010 [[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]=>[1,2,3,4,5,6,10,9,8,7]=>111111000 [[[[[[.,.],.],.],.],[.,.]],[.,[.,.]]]=>[1,2,3,4,6,5,9,8,7]=>11110100 [[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]=>[1,2,3,4,5,10,9,8,7,6]=>111110000 [[[[[.,.],.],.],[.,.]],[.,[.,[.,.]]]]=>[1,2,3,5,4,9,8,7,6]=>11101000 [[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]=>[1,2,3,4,10,9,8,7,6,5]=>111100000 [[[[.,.],.],[.,.]],[.,[.,[.,[.,.]]]]]=>[1,2,4,3,9,8,7,6,5]=>11010000 [[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[1,2,3,10,9,8,7,6,5,4]=>111000000 [[[[[[[[.,.],.],.],.],.],.],[.,.]],[.,.]]=>[1,2,3,4,5,6,8,7,10,9]=>111111010 [.,[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]]=>[10,9,11,8,7,6,5,4,3,2,1]=>0000000000 [.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]=>[10,11,9,8,7,6,5,4,3,2,1]=>0000000000 [.,[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]]=>[7,8,6,5,4,3,2,9,1]=>00000000 [[[[[[.,[.,[[.,.],.]]],.],.],.],.],.]=>[3,4,2,1,5,6,7,8,9]=>00011111 [[[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.],.]=>[2,3,1,4,5,6,7,8,9,10,11]=>0011111111 [[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.],.]=>[3,4,2,1,5,6,7,8,9,10]=>000111111 [[[[[.,[[.,[[.,.],.]],.]],.],.],.],.]=>[3,4,2,5,1,6,7,8,9]=>00001111 [[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],.]=>[7,6,5,4,3,2,8,1,9]=>00000001 [[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]],.]=>[6,5,8,7,4,3,2,1,9]=>00000001 [[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]],.]=>[7,6,5,8,4,3,2,1,9]=>00000001 [[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]],.]=>[6,7,5,8,4,3,2,1,9]=>00000001 [[[[[[.,[[.,.],.]],[.,.]],.],.],.],.]=>[2,3,1,5,4,6,7,8,9]=>00101111 [[[[[.,[.,[.,[[.,.],.]]]],.],.],.],.]=>[4,5,3,2,1,6,7,8,9]=>00001111 [[.,[.,[.,[.,.]]]],[[[[[.,.],.],.],.],.]]=>[4,3,2,1,6,7,8,9,10,5]=>000100000 [[.,[[[[[[.,[.,.]],.],.],.],.],.]],.]=>[3,2,4,5,6,7,8,1,9]=>00000001 [[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]=>[5,4,3,2,1,8,7,6,9]=>00001001 [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]],.]=>[7,6,5,4,3,2,1,9,8,10]=>000000101 [[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]],.]=>[2,8,7,6,5,4,3,1,9]=>00000001 [[.,[[[[[[.,.],.],.],.],.],.]],[.,.]]=>[2,3,4,5,6,7,1,9,8]=>00000010 [[[[[[[.,.],[[.,.],.]],.],.],.],.],.]=>[1,3,4,2,5,6,7,8,9]=>10011111 [[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]],.]=>[8,7,6,5,4,3,2,9,1,10]=>000000001 [[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,6,7,8,10,9,11]=>1111111101 [[[.,[.,.]],.],[[[[[.,.],.],.],.],.]]=>[2,1,3,5,6,7,8,9,4]=>01100000 [[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.]=>[5,6,4,3,2,1,7,8,9]=>00000111 [[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.],.]=>[5,6,4,3,2,1,7,8,9,10]=>000001111 [[[[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.],.],.]=>[6,7,5,4,3,2,1,8,9,10,11]=>0000001111 [[[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.],.]=>[6,7,5,4,3,2,1,8,9,10]=>000000111 [[[.,[[.,[.,[.,[[.,.],.]]]],.]],.],.]=>[5,6,4,3,2,7,1,8,9]=>00000011 [[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.]=>[6,7,5,4,3,2,1,8,9]=>00000011 [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[[.,.],.]]=>[7,6,5,4,3,2,1,9,10,8]=>000000100
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
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.