Identifier
Mp00245:
Signed permutations
—standardize⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Images
=>
Cc0010;cc-rep-2Cc0010;cc-rep-3
[1]=>[1]=>[.,.]=>[.,.]
[-1]=>[1]=>[.,.]=>[.,.]
[1,2]=>[1,2]=>[.,[.,.]]=>[[.,.],.]
[1,-2]=>[1,2]=>[.,[.,.]]=>[[.,.],.]
[-1,2]=>[2,1]=>[[.,.],.]=>[.,[.,.]]
[-1,-2]=>[2,1]=>[[.,.],.]=>[.,[.,.]]
[2,1]=>[2,1]=>[[.,.],.]=>[.,[.,.]]
[2,-1]=>[1,2]=>[.,[.,.]]=>[[.,.],.]
[-2,1]=>[2,1]=>[[.,.],.]=>[.,[.,.]]
[-2,-1]=>[1,2]=>[.,[.,.]]=>[[.,.],.]
[1,2,3]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[1,2,-3]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[1,-2,3]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[1,-2,-3]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[-1,2,3]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-1,2,-3]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-1,-2,3]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[-1,-2,-3]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[1,3,2]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[1,3,-2]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[1,-3,2]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[1,-3,-2]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[-1,3,2]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[-1,3,-2]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-1,-3,2]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[-1,-3,-2]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[2,1,3]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[2,1,-3]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[2,-1,3]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[2,-1,-3]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[-2,1,3]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-2,1,-3]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-2,-1,3]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[-2,-1,-3]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[2,3,1]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[2,3,-1]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[2,-3,1]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[2,-3,-1]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[-2,3,1]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[-2,3,-1]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-2,-3,1]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[-2,-3,-1]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[3,1,2]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[3,1,-2]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[3,-1,2]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[3,-1,-2]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[-3,1,2]=>[3,1,2]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-3,1,-2]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-3,-1,2]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[-3,-1,-2]=>[1,3,2]=>[.,[[.,.],.]]=>[[.,[.,.]],.]
[3,2,1]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[3,2,-1]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[3,-2,1]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[3,-2,-1]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[-3,2,1]=>[3,2,1]=>[[[.,.],.],.]=>[.,[.,[.,.]]]
[-3,2,-1]=>[2,1,3]=>[[.,.],[.,.]]=>[[.,.],[.,.]]
[-3,-2,1]=>[2,3,1]=>[[.,[.,.]],.]=>[.,[[.,.],.]]
[-3,-2,-1]=>[1,2,3]=>[.,[.,[.,.]]]=>[[[.,.],.],.]
[1,2,3,4]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,2,3,-4]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,2,-3,4]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,2,-3,-4]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,-2,3,4]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-2,3,-4]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-2,-3,4]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,-2,-3,-4]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[-1,2,3,4]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,2,3,-4]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,2,-3,4]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,2,-3,-4]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,-2,3,4]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-2,3,-4]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-2,-3,4]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,-2,-3,-4]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[1,2,4,3]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,2,4,-3]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,2,-4,3]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,2,-4,-3]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,-2,4,3]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,-2,4,-3]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-2,-4,3]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,-2,-4,-3]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[-1,2,4,3]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,2,4,-3]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,2,-4,3]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,2,-4,-3]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,-2,4,3]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,-2,4,-3]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-2,-4,3]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,-2,-4,-3]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[1,3,2,4]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,3,2,-4]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,3,-2,4]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,3,-2,-4]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,-3,2,4]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-3,2,-4]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-3,-2,4]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,-3,-2,-4]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[-1,3,2,4]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,3,2,-4]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,3,-2,4]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,3,-2,-4]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,-3,2,4]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-3,2,-4]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-3,-2,4]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,-3,-2,-4]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[1,3,4,2]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,3,4,-2]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,3,-4,2]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,3,-4,-2]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,-3,4,2]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,-3,4,-2]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-3,-4,2]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,-3,-4,-2]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[-1,3,4,2]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,3,4,-2]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,3,-4,2]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,3,-4,-2]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,-3,4,2]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,-3,4,-2]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-3,-4,2]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,-3,-4,-2]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[1,4,2,3]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,4,2,-3]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,4,-2,3]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,4,-2,-3]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[1,-4,2,3]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-4,2,-3]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-4,-2,3]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,-4,-2,-3]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[-1,4,2,3]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,4,2,-3]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,4,-2,3]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,4,-2,-3]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-1,-4,2,3]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-4,2,-3]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-4,-2,3]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,-4,-2,-3]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[1,4,3,2]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,4,3,-2]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,4,-3,2]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,4,-3,-2]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[1,-4,3,2]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[1,-4,3,-2]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[1,-4,-3,2]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[1,-4,-3,-2]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]
[-1,4,3,2]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,4,3,-2]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,4,-3,2]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,4,-3,-2]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-1,-4,3,2]=>[4,3,2,1]=>[[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]
[-1,-4,3,-2]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-1,-4,-3,2]=>[4,2,3,1]=>[[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]
[-1,-4,-3,-2]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[2,1,3,4]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[2,1,3,-4]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[2,1,-3,4]=>[2,1,4,3]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[2,1,-3,-4]=>[2,1,4,3]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[2,-1,3,4]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[2,-1,3,-4]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[2,-1,-3,4]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[2,-1,-3,-4]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[-2,1,3,4]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-2,1,3,-4]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-2,1,-3,4]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-2,1,-3,-4]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-2,-1,3,4]=>[3,4,1,2]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[-2,-1,3,-4]=>[3,4,1,2]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[-2,-1,-3,4]=>[3,4,2,1]=>[[[.,[.,.]],.],.]=>[.,[.,[[.,.],.]]]
[-2,-1,-3,-4]=>[3,4,2,1]=>[[[.,[.,.]],.],.]=>[.,[.,[[.,.],.]]]
[2,1,4,3]=>[2,1,4,3]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[2,1,4,-3]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[2,1,-4,3]=>[2,1,4,3]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[2,1,-4,-3]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[2,-1,4,3]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[2,-1,4,-3]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[2,-1,-4,3]=>[1,4,3,2]=>[.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]
[2,-1,-4,-3]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]
[-2,1,4,3]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-2,1,4,-3]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-2,1,-4,3]=>[4,1,3,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-2,1,-4,-3]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]
[-2,-1,4,3]=>[3,4,2,1]=>[[[.,[.,.]],.],.]=>[.,[.,[[.,.],.]]]
[-2,-1,4,-3]=>[3,4,1,2]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[-2,-1,-4,3]=>[3,4,2,1]=>[[[.,[.,.]],.],.]=>[.,[.,[[.,.],.]]]
[-2,-1,-4,-3]=>[3,4,1,2]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[2,3,1,4]=>[2,3,1,4]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[2,3,1,-4]=>[2,3,1,4]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[2,3,-1,4]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[2,3,-1,-4]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]
[2,-3,1,4]=>[2,4,1,3]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[2,-3,1,-4]=>[2,4,1,3]=>[[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]
[2,-3,-1,4]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[2,-3,-1,-4]=>[1,3,4,2]=>[.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]
[-2,3,1,4]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-2,3,1,-4]=>[4,2,1,3]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-2,3,-1,4]=>[3,1,4,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-2,3,-1,-4]=>[3,1,4,2]=>[[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]
[-2,-3,1,4]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
[-2,-3,1,-4]=>[4,3,1,2]=>[[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]
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[-5,-4,-2,1,3]=>[3,4,5,1,2]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[-5,-4,-2,1,-3]=>[2,3,5,1,4]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[-5,-4,-2,-1,3]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],.]]
[-5,-4,-2,-1,-3]=>[1,2,4,5,3]=>[.,[.,[[.,[.,.]],.]]]=>[[[.,[[.,.],.]],.],.]
[5,4,2,3,1]=>[5,4,2,3,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[5,4,2,3,-1]=>[4,3,1,2,5]=>[[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]
[5,4,2,-3,1]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[5,4,2,-3,-1]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]
[5,4,-2,3,1]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]=>[.,[.,[[.,.],[.,.]]]]
[5,4,-2,3,-1]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[5,4,-2,-3,1]=>[3,2,5,4,1]=>[[[.,.],[[.,.],.]],.]=>[.,[[.,[.,.]],[.,.]]]
[5,4,-2,-3,-1]=>[2,1,4,3,5]=>[[.,.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,.]]
[5,-4,2,3,1]=>[4,5,2,3,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[5,-4,2,3,-1]=>[3,4,1,2,5]=>[[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]
[5,-4,2,-3,1]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[5,-4,2,-3,-1]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]
[5,-4,-2,3,1]=>[3,4,5,2,1]=>[[[.,[.,[.,.]]],.],.]=>[.,[.,[[[.,.],.],.]]]
[5,-4,-2,3,-1]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[5,-4,-2,-3,1]=>[2,3,5,4,1]=>[[.,[.,[[.,.],.]]],.]=>[.,[[[.,[.,.]],.],.]]
[5,-4,-2,-3,-1]=>[1,2,4,3,5]=>[.,[.,[[.,.],[.,.]]]]=>[[[[.,.],[.,.]],.],.]
[-5,4,2,3,1]=>[5,4,2,3,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[-5,4,2,3,-1]=>[4,3,1,2,5]=>[[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]
[-5,4,2,-3,1]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[-5,4,2,-3,-1]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]
[-5,4,-2,3,1]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]=>[.,[.,[[.,.],[.,.]]]]
[-5,4,-2,3,-1]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[-5,4,-2,-3,1]=>[3,2,5,4,1]=>[[[.,.],[[.,.],.]],.]=>[.,[[.,[.,.]],[.,.]]]
[-5,4,-2,-3,-1]=>[2,1,4,3,5]=>[[.,.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,.]]
[-5,-4,2,3,1]=>[4,5,2,3,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[-5,-4,2,3,-1]=>[3,4,1,2,5]=>[[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]
[-5,-4,2,-3,1]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[-5,-4,2,-3,-1]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]
[-5,-4,-2,3,1]=>[3,4,5,2,1]=>[[[.,[.,[.,.]]],.],.]=>[.,[.,[[[.,.],.],.]]]
[-5,-4,-2,3,-1]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[-5,-4,-2,-3,1]=>[2,3,5,4,1]=>[[.,[.,[[.,.],.]]],.]=>[.,[[[.,[.,.]],.],.]]
[-5,-4,-2,-3,-1]=>[1,2,4,3,5]=>[.,[.,[[.,.],[.,.]]]]=>[[[[.,.],[.,.]],.],.]
[5,4,3,1,2]=>[5,4,3,1,2]=>[[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]
[5,4,3,1,-2]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]
[5,4,3,-1,2]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[5,4,3,-1,-2]=>[3,2,1,5,4]=>[[[.,.],.],[[.,.],.]]=>[[.,[.,.]],[.,[.,.]]]
[5,4,-3,1,2]=>[4,3,5,1,2]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[5,4,-3,1,-2]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[5,4,-3,-1,2]=>[3,2,4,5,1]=>[[[.,.],[.,[.,.]]],.]=>[.,[[[.,.],.],[.,.]]]
[5,4,-3,-1,-2]=>[2,1,3,5,4]=>[[.,.],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[.,.]]
[5,-4,3,1,2]=>[4,5,3,1,2]=>[[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]
[5,-4,3,1,-2]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]
[5,-4,3,-1,2]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[5,-4,3,-1,-2]=>[2,3,1,5,4]=>[[.,[.,.]],[[.,.],.]]=>[[.,[.,.]],[[.,.],.]]
[5,-4,-3,1,2]=>[3,4,5,1,2]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[5,-4,-3,1,-2]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[5,-4,-3,-1,2]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],.]]
[5,-4,-3,-1,-2]=>[1,2,3,5,4]=>[.,[.,[.,[[.,.],.]]]]=>[[[[.,[.,.]],.],.],.]
[-5,4,3,1,2]=>[5,4,3,1,2]=>[[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]
[-5,4,3,1,-2]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]
[-5,4,3,-1,2]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[-5,4,3,-1,-2]=>[3,2,1,5,4]=>[[[.,.],.],[[.,.],.]]=>[[.,[.,.]],[.,[.,.]]]
[-5,4,-3,1,2]=>[4,3,5,1,2]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[-5,4,-3,1,-2]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[-5,4,-3,-1,2]=>[3,2,4,5,1]=>[[[.,.],[.,[.,.]]],.]=>[.,[[[.,.],.],[.,.]]]
[-5,4,-3,-1,-2]=>[2,1,3,5,4]=>[[.,.],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[.,.]]
[-5,-4,3,1,2]=>[4,5,3,1,2]=>[[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]
[-5,-4,3,1,-2]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]
[-5,-4,3,-1,2]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[-5,-4,3,-1,-2]=>[2,3,1,5,4]=>[[.,[.,.]],[[.,.],.]]=>[[.,[.,.]],[[.,.],.]]
[-5,-4,-3,1,2]=>[3,4,5,1,2]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[-5,-4,-3,1,-2]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[-5,-4,-3,-1,2]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],.]]
[-5,-4,-3,-1,-2]=>[1,2,3,5,4]=>[.,[.,[.,[[.,.],.]]]]=>[[[[.,[.,.]],.],.],.]
[5,4,3,2,1]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.]=>[.,[.,[.,[.,[.,.]]]]]
[5,4,3,2,-1]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]
[5,4,3,-2,1]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[5,4,3,-2,-1]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]
[5,4,-3,2,1]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]=>[.,[.,[[.,.],[.,.]]]]
[5,4,-3,2,-1]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[5,4,-3,-2,1]=>[3,2,4,5,1]=>[[[.,.],[.,[.,.]]],.]=>[.,[[[.,.],.],[.,.]]]
[5,4,-3,-2,-1]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[.,.]]
[5,-4,3,2,1]=>[4,5,3,2,1]=>[[[[.,[.,.]],.],.],.]=>[.,[.,[.,[[.,.],.]]]]
[5,-4,3,2,-1]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]
[5,-4,3,-2,1]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[5,-4,3,-2,-1]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]
[5,-4,-3,2,1]=>[3,4,5,2,1]=>[[[.,[.,[.,.]]],.],.]=>[.,[.,[[[.,.],.],.]]]
[5,-4,-3,2,-1]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[5,-4,-3,-2,1]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],.]]
[5,-4,-3,-2,-1]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]]=>[[[[[.,.],.],.],.],.]
[-5,4,3,2,1]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.]=>[.,[.,[.,[.,[.,.]]]]]
[-5,4,3,2,-1]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]
[-5,4,3,-2,1]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]
[-5,4,3,-2,-1]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]
[-5,4,-3,2,1]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]=>[.,[.,[[.,.],[.,.]]]]
[-5,4,-3,2,-1]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]
[-5,4,-3,-2,1]=>[3,2,4,5,1]=>[[[.,.],[.,[.,.]]],.]=>[.,[[[.,.],.],[.,.]]]
[-5,4,-3,-2,-1]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[.,.]]
[-5,-4,3,2,1]=>[4,5,3,2,1]=>[[[[.,[.,.]],.],.],.]=>[.,[.,[.,[[.,.],.]]]]
[-5,-4,3,2,-1]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]
[-5,-4,3,-2,1]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]
[-5,-4,3,-2,-1]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]
[-5,-4,-3,2,1]=>[3,4,5,2,1]=>[[[.,[.,[.,.]]],.],.]=>[.,[.,[[[.,.],.],.]]]
[-5,-4,-3,2,-1]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]
[-5,-4,-3,-2,1]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],.]]
[-5,-4,-3,-2,-1]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]]=>[[[[[.,.],.],.],.],.]
Map
standardize
Description
Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let $\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order $1 < \dots < n < -n < \dots < -1$, this map returns the permutation in $\mathfrak S_n$ which is order isomorphic to $\pi(1),\dots,\pi(n)$.
Let $\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order $1 < \dots < n < -n < \dots < -1$, this map returns the permutation in $\mathfrak S_n$ which is order isomorphic to $\pi(1),\dots,\pi(n)$.
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.
Map
left-right symmetry
Description
Return the left-right symmetrized tree of a binary tree.
searching the database
Sorry, this map was not found in the database.