Identifier
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
=>
Cc0005;cc-rep-1Cc0010;cc-rep-3
[1]=>[1,0]=>[1]=>[.,.]
[1,1]=>[1,0,1,0]=>[1,2]=>[.,[.,.]]
[2]=>[1,1,0,0]=>[2,1]=>[[.,.],.]
[1,1,1]=>[1,0,1,0,1,0]=>[1,2,3]=>[.,[.,[.,.]]]
[1,2]=>[1,0,1,1,0,0]=>[1,3,2]=>[.,[[.,.],.]]
[2,1]=>[1,1,0,0,1,0]=>[2,1,3]=>[[.,.],[.,.]]
[3]=>[1,1,1,0,0,0]=>[3,2,1]=>[[[.,.],.],.]
[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]
[1,1,2]=>[1,0,1,0,1,1,0,0]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]
[1,2,1]=>[1,0,1,1,0,0,1,0]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]
[1,3]=>[1,0,1,1,1,0,0,0]=>[1,4,3,2]=>[.,[[[.,.],.],.]]
[2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]
[2,2]=>[1,1,0,0,1,1,0,0]=>[2,1,4,3]=>[[.,.],[[.,.],.]]
[3,1]=>[1,1,1,0,0,0,1,0]=>[3,2,1,4]=>[[[.,.],.],[.,.]]
[4]=>[1,1,1,1,0,0,0,0]=>[4,3,2,1]=>[[[[.,.],.],.],.]
[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]]
[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4]=>[.,[.,[.,[[.,.],.]]]]
[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5]=>[.,[.,[[.,.],[.,.]]]]
[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[1,2,5,4,3]=>[.,[.,[[[.,.],.],.]]]
[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,5]=>[.,[[.,.],[.,[.,.]]]]
[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4]=>[.,[[.,.],[[.,.],.]]]
[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[1,4,3,2,5]=>[.,[[[.,.],.],[.,.]]]
[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[1,5,4,3,2]=>[.,[[[[.,.],.],.],.]]
[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]]
[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,1,3,5,4]=>[[.,.],[.,[[.,.],.]]]
[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,5]=>[[.,.],[[.,.],[.,.]]]
[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,1,5,4,3]=>[[.,.],[[[.,.],.],.]]
[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]
[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[3,2,1,5,4]=>[[[.,.],.],[[.,.],.]]
[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]
[5]=>[1,1,1,1,1,0,0,0,0,0]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.]
[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6]=>[.,[.,[.,[.,[.,[.,.]]]]]]
[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,6,5]=>[.,[.,[.,[.,[[.,.],.]]]]]
[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,4,6]=>[.,[.,[.,[[.,.],[.,.]]]]]
[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,6,5,4]=>[.,[.,[.,[[[.,.],.],.]]]]
[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,5,6]=>[.,[.,[[.,.],[.,[.,.]]]]]
[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5]=>[.,[.,[[.,.],[[.,.],.]]]]
[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,5,4,3,6]=>[.,[.,[[[.,.],.],[.,.]]]]
[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,6,5,4,3]=>[.,[.,[[[[.,.],.],.],.]]]
[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,4,5,6]=>[.,[[.,.],[.,[.,[.,.]]]]]
[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,4,6,5]=>[.,[[.,.],[.,[[.,.],.]]]]
[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,6]=>[.,[[.,.],[[.,.],[.,.]]]]
[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,6,5,4]=>[.,[[.,.],[[[.,.],.],.]]]
[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,4,3,2,5,6]=>[.,[[[.,.],.],[.,[.,.]]]]
[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,4,3,2,6,5]=>[.,[[[.,.],.],[[.,.],.]]]
[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,5,4,3,2,6]=>[.,[[[[.,.],.],.],[.,.]]]
[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,6,5,4,3,2]=>[.,[[[[[.,.],.],.],.],.]]
[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]]
[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,6,5]=>[[.,.],[.,[.,[[.,.],.]]]]
[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,3,5,4,6]=>[[.,.],[.,[[.,.],[.,.]]]]
[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,3,6,5,4]=>[[.,.],[.,[[[.,.],.],.]]]
[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,5,6]=>[[.,.],[[.,.],[.,[.,.]]]]
[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5]=>[[.,.],[[.,.],[[.,.],.]]]
[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,5,4,3,6]=>[[.,.],[[[.,.],.],[.,.]]]
[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,6,5,4,3]=>[[.,.],[[[[.,.],.],.],.]]
[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[3,2,1,4,5,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[3,2,1,4,6,5]=>[[[.,.],.],[.,[[.,.],.]]]
[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[3,2,1,5,4,6]=>[[[.,.],.],[[.,.],[.,.]]]
[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,2,1,6,5,4]=>[[[.,.],.],[[[.,.],.],.]]
[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[4,3,2,1,5,6]=>[[[[.,.],.],.],[.,[.,.]]]
[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,3,2,1,6,5]=>[[[[.,.],.],.],[[.,.],.]]
[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[5,4,3,2,1,6]=>[[[[[.,.],.],.],.],[.,.]]
[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[6,5,4,3,2,1]=>[[[[[[.,.],.],.],.],.],.]
[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,6]=>[.,[.,[.,[.,[.,[[.,.],.]]]]]]
[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,7]=>[.,[.,[.,[.,[[.,.],[.,.]]]]]]
[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,7,6,5]=>[.,[.,[.,[.,[[[.,.],.],.]]]]]
[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,6,7]=>[.,[.,[.,[[.,.],[.,[.,.]]]]]]
[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6]=>[.,[.,[.,[[.,.],[[.,.],.]]]]]
[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,6,5,4,7]=>[.,[.,[.,[[[.,.],.],[.,.]]]]]
[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,7,6,5,4]=>[.,[.,[.,[[[[.,.],.],.],.]]]]
[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7]=>[.,[.,[[.,.],[.,[.,[.,.]]]]]]
[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,4,3,5,7,6]=>[.,[.,[[.,.],[.,[[.,.],.]]]]]
[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,7]=>[.,[.,[[.,.],[[.,.],[.,.]]]]]
[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,7,6,5]=>[.,[.,[[.,.],[[[.,.],.],.]]]]
[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,5,4,3,6,7]=>[.,[.,[[[.,.],.],[.,[.,.]]]]]
[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,5,4,3,7,6]=>[.,[.,[[[.,.],.],[[.,.],.]]]]
[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,6,5,4,3,7]=>[.,[.,[[[[.,.],.],.],[.,.]]]]
[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,7,6,5,4,3]=>[.,[.,[[[[[.,.],.],.],.],.]]]
[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7]=>[.,[[.,.],[.,[.,[.,[.,.]]]]]]
[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,7,6]=>[.,[[.,.],[.,[.,[[.,.],.]]]]]
[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,6,5,7]=>[.,[[.,.],[.,[[.,.],[.,.]]]]]
[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,7,6,5]=>[.,[[.,.],[.,[[[.,.],.],.]]]]
[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,6,7]=>[.,[[.,.],[[.,.],[.,[.,.]]]]]
[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6]=>[.,[[.,.],[[.,.],[[.,.],.]]]]
[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,6,5,4,7]=>[.,[[.,.],[[[.,.],.],[.,.]]]]
[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,7,6,5,4]=>[.,[[.,.],[[[[.,.],.],.],.]]]
[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,4,3,2,5,6,7]=>[.,[[[.,.],.],[.,[.,[.,.]]]]]
[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,4,3,2,5,7,6]=>[.,[[[.,.],.],[.,[[.,.],.]]]]
[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,4,3,2,6,5,7]=>[.,[[[.,.],.],[[.,.],[.,.]]]]
[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,4,3,2,7,6,5]=>[.,[[[.,.],.],[[[.,.],.],.]]]
[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,5,4,3,2,6,7]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,5,4,3,2,7,6]=>[.,[[[[.,.],.],.],[[.,.],.]]]
[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,6,5,4,3,2,7]=>[.,[[[[[.,.],.],.],.],[.,.]]]
[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,7,6,5,4,3,2]=>[.,[[[[[[.,.],.],.],.],.],.]]
[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,7,6]=>[[.,.],[.,[.,[.,[[.,.],.]]]]]
[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,6,5,7]=>[[.,.],[.,[.,[[.,.],[.,.]]]]]
[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,7,6,5]=>[[.,.],[.,[.,[[[.,.],.],.]]]]
[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,5,4,6,7]=>[[.,.],[.,[[.,.],[.,[.,.]]]]]
[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,5,4,7,6]=>[[.,.],[.,[[.,.],[[.,.],.]]]]
[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,6,5,4,7]=>[[.,.],[.,[[[.,.],.],[.,.]]]]
[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,7,6,5,4]=>[[.,.],[.,[[[[.,.],.],.],.]]]
[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7]=>[[.,.],[[.,.],[.,[.,[.,.]]]]]
[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,3,5,7,6]=>[[.,.],[[.,.],[.,[[.,.],.]]]]
[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,7]=>[[.,.],[[.,.],[[.,.],[.,.]]]]
[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,7,6,5]=>[[.,.],[[.,.],[[[.,.],.],.]]]
[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,5,4,3,6,7]=>[[.,.],[[[.,.],.],[.,[.,.]]]]
[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,5,4,3,7,6]=>[[.,.],[[[.,.],.],[[.,.],.]]]
[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,6,5,4,3,7]=>[[.,.],[[[[.,.],.],.],[.,.]]]
[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,7,6,5,4,3]=>[[.,.],[[[[[.,.],.],.],.],.]]
[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,2,1,4,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[3,2,1,4,5,7,6]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[3,2,1,4,6,5,7]=>[[[.,.],.],[.,[[.,.],[.,.]]]]
[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[3,2,1,4,7,6,5]=>[[[.,.],.],[.,[[[.,.],.],.]]]
[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[3,2,1,5,4,6,7]=>[[[.,.],.],[[.,.],[.,[.,.]]]]
[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[3,2,1,5,4,7,6]=>[[[.,.],.],[[.,.],[[.,.],.]]]
[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[3,2,1,6,5,4,7]=>[[[.,.],.],[[[.,.],.],[.,.]]]
[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,2,1,7,6,5,4]=>[[[.,.],.],[[[[.,.],.],.],.]]
[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[4,3,2,1,5,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[4,3,2,1,5,7,6]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[4,3,2,1,6,5,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[4,3,2,1,7,6,5]=>[[[[.,.],.],.],[[[.,.],.],.]]
[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[5,4,3,2,1,6,7]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[5,4,3,2,1,7,6]=>[[[[[.,.],.],.],.],[[.,.],.]]
[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[6,5,4,3,2,1,7]=>[[[[[[.,.],.],.],.],.],[.,.]]
[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7,6,5,4,3,2,1]=>[[[[[[[.,.],.],.],.],.],.],.]
[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,6,5,7,8]=>[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,5,7,6,8]=>[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,4,3,2,7,6,5,8]=>[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,6,8,7]=>[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,8,7,6,5]=>[[.,.],[[.,.],[[[[.,.],.],.],.]]]
[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[2,1,6,5,4,3,8,7]=>[[.,.],[[[[.,.],.],.],[[.,.],.]]]
[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,1,8,7,6,5,4,3]=>[[.,.],[[[[[[.,.],.],.],.],.],.]]
[3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[3,2,1,4,5,8,7,6]=>[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
[4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[4,3,2,1,6,5,8,7]=>[[[[.,.],.],.],[[.,.],[[.,.],.]]]
[4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[4,3,2,1,8,7,6,5]=>[[[[.,.],.],.],[[[[.,.],.],.],.]]
[6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[6,5,4,3,2,1,8,7]=>[[[[[[.,.],.],.],.],.],[[.,.],.]]
[8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[8,7,6,5,4,3,2,1]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
searching the database
Sorry, this map was not found in the database.