Identifier
Mp00235:
Permutations
—descent views to invisible inversion bottoms⟶
Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Images
=>
[1]=>[1]=>[1]=>[-1]
[1,2]=>[1,2]=>[1,2]=>[-1,-2]
[2,1]=>[2,1]=>[2,1]=>[-2,-1]
[1,2,3]=>[1,2,3]=>[1,2,3]=>[-1,-2,-3]
[1,3,2]=>[1,3,2]=>[1,3,2]=>[-1,-3,-2]
[2,1,3]=>[2,1,3]=>[2,1,3]=>[-2,-1,-3]
[2,3,1]=>[3,2,1]=>[3,2,1]=>[-3,-2,-1]
[3,1,2]=>[3,1,2]=>[3,1,2]=>[-3,-1,-2]
[3,2,1]=>[2,3,1]=>[2,3,1]=>[-2,-3,-1]
[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4]=>[-1,-2,-3,-4]
[1,2,4,3]=>[1,2,4,3]=>[1,2,4,3]=>[-1,-2,-4,-3]
[1,3,2,4]=>[1,3,2,4]=>[1,3,2,4]=>[-1,-3,-2,-4]
[1,3,4,2]=>[1,4,3,2]=>[1,4,3,2]=>[-1,-4,-3,-2]
[1,4,2,3]=>[1,4,2,3]=>[1,4,2,3]=>[-1,-4,-2,-3]
[1,4,3,2]=>[1,3,4,2]=>[1,3,4,2]=>[-1,-3,-4,-2]
[2,1,3,4]=>[2,1,3,4]=>[2,1,3,4]=>[-2,-1,-3,-4]
[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3]=>[-2,-1,-4,-3]
[2,3,1,4]=>[3,2,1,4]=>[3,2,1,4]=>[-3,-2,-1,-4]
[2,3,4,1]=>[4,2,3,1]=>[4,2,3,1]=>[-4,-2,-3,-1]
[2,4,1,3]=>[4,2,1,3]=>[4,2,1,3]=>[-4,-2,-1,-3]
[2,4,3,1]=>[3,2,4,1]=>[3,2,4,1]=>[-3,-2,-4,-1]
[3,1,2,4]=>[3,1,2,4]=>[3,1,2,4]=>[-3,-1,-2,-4]
[3,1,4,2]=>[3,4,1,2]=>[3,4,1,2]=>[-3,-4,-1,-2]
[3,2,1,4]=>[2,3,1,4]=>[2,3,1,4]=>[-2,-3,-1,-4]
[3,2,4,1]=>[4,3,2,1]=>[4,3,2,1]=>[-4,-3,-2,-1]
[3,4,1,2]=>[4,1,3,2]=>[4,1,3,2]=>[-4,-1,-3,-2]
[3,4,2,1]=>[2,4,3,1]=>[2,4,3,1]=>[-2,-4,-3,-1]
[4,1,2,3]=>[4,1,2,3]=>[4,1,2,3]=>[-4,-1,-2,-3]
[4,1,3,2]=>[4,3,1,2]=>[4,3,1,2]=>[-4,-3,-1,-2]
[4,2,1,3]=>[2,4,1,3]=>[2,4,1,3]=>[-2,-4,-1,-3]
[4,2,3,1]=>[3,4,2,1]=>[3,4,2,1]=>[-3,-4,-2,-1]
[4,3,1,2]=>[3,1,4,2]=>[3,1,4,2]=>[-3,-1,-4,-2]
[4,3,2,1]=>[2,3,4,1]=>[2,3,4,1]=>[-2,-3,-4,-1]
[1,2,3,4,5]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[-1,-2,-3,-4,-5]
[1,2,3,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4]=>[-1,-2,-3,-5,-4]
[1,2,4,3,5]=>[1,2,4,3,5]=>[1,2,4,3,5]=>[-1,-2,-4,-3,-5]
[1,2,4,5,3]=>[1,2,5,4,3]=>[1,2,5,4,3]=>[-1,-2,-5,-4,-3]
[1,2,5,3,4]=>[1,2,5,3,4]=>[1,2,5,3,4]=>[-1,-2,-5,-3,-4]
[1,2,5,4,3]=>[1,2,4,5,3]=>[1,2,4,5,3]=>[-1,-2,-4,-5,-3]
[1,3,2,4,5]=>[1,3,2,4,5]=>[1,3,2,4,5]=>[-1,-3,-2,-4,-5]
[1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4]=>[-1,-3,-2,-5,-4]
[1,3,4,2,5]=>[1,4,3,2,5]=>[1,4,3,2,5]=>[-1,-4,-3,-2,-5]
[1,3,4,5,2]=>[1,5,3,4,2]=>[1,5,3,4,2]=>[-1,-5,-3,-4,-2]
[1,3,5,2,4]=>[1,5,3,2,4]=>[1,5,3,2,4]=>[-1,-5,-3,-2,-4]
[1,3,5,4,2]=>[1,4,3,5,2]=>[1,4,3,5,2]=>[-1,-4,-3,-5,-2]
[1,4,2,3,5]=>[1,4,2,3,5]=>[1,4,2,3,5]=>[-1,-4,-2,-3,-5]
[1,4,2,5,3]=>[1,4,5,2,3]=>[1,4,5,2,3]=>[-1,-4,-5,-2,-3]
[1,4,3,2,5]=>[1,3,4,2,5]=>[1,3,4,2,5]=>[-1,-3,-4,-2,-5]
[1,4,3,5,2]=>[1,5,4,3,2]=>[1,5,4,3,2]=>[-1,-5,-4,-3,-2]
[1,4,5,2,3]=>[1,5,2,4,3]=>[1,5,2,4,3]=>[-1,-5,-2,-4,-3]
[1,4,5,3,2]=>[1,3,5,4,2]=>[1,3,5,4,2]=>[-1,-3,-5,-4,-2]
[1,5,2,3,4]=>[1,5,2,3,4]=>[1,5,2,3,4]=>[-1,-5,-2,-3,-4]
[1,5,2,4,3]=>[1,5,4,2,3]=>[1,5,4,2,3]=>[-1,-5,-4,-2,-3]
[1,5,3,2,4]=>[1,3,5,2,4]=>[1,3,5,2,4]=>[-1,-3,-5,-2,-4]
[1,5,3,4,2]=>[1,4,5,3,2]=>[1,4,5,3,2]=>[-1,-4,-5,-3,-2]
[1,5,4,2,3]=>[1,4,2,5,3]=>[1,4,2,5,3]=>[-1,-4,-2,-5,-3]
[1,5,4,3,2]=>[1,3,4,5,2]=>[1,3,4,5,2]=>[-1,-3,-4,-5,-2]
[2,1,3,4,5]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[-2,-1,-3,-4,-5]
[2,1,3,5,4]=>[2,1,3,5,4]=>[2,1,3,5,4]=>[-2,-1,-3,-5,-4]
[2,1,4,3,5]=>[2,1,4,3,5]=>[2,1,4,3,5]=>[-2,-1,-4,-3,-5]
[2,1,4,5,3]=>[2,1,5,4,3]=>[2,1,5,4,3]=>[-2,-1,-5,-4,-3]
[2,1,5,3,4]=>[2,1,5,3,4]=>[2,1,5,3,4]=>[-2,-1,-5,-3,-4]
[2,1,5,4,3]=>[2,1,4,5,3]=>[2,1,4,5,3]=>[-2,-1,-4,-5,-3]
[2,3,1,4,5]=>[3,2,1,4,5]=>[3,2,1,4,5]=>[-3,-2,-1,-4,-5]
[2,3,1,5,4]=>[3,2,1,5,4]=>[3,2,1,5,4]=>[-3,-2,-1,-5,-4]
[2,3,4,1,5]=>[4,2,3,1,5]=>[4,2,3,1,5]=>[-4,-2,-3,-1,-5]
[2,3,4,5,1]=>[5,2,3,4,1]=>[5,2,3,4,1]=>[-5,-2,-3,-4,-1]
[2,3,5,1,4]=>[5,2,3,1,4]=>[5,2,3,1,4]=>[-5,-2,-3,-1,-4]
[2,3,5,4,1]=>[4,2,3,5,1]=>[4,2,3,5,1]=>[-4,-2,-3,-5,-1]
[2,4,1,3,5]=>[4,2,1,3,5]=>[4,2,1,3,5]=>[-4,-2,-1,-3,-5]
[2,4,1,5,3]=>[4,2,5,1,3]=>[4,2,5,1,3]=>[-4,-2,-5,-1,-3]
[2,4,3,1,5]=>[3,2,4,1,5]=>[3,2,4,1,5]=>[-3,-2,-4,-1,-5]
[2,4,3,5,1]=>[5,2,4,3,1]=>[5,2,4,3,1]=>[-5,-2,-4,-3,-1]
[2,4,5,1,3]=>[5,2,1,4,3]=>[5,2,1,4,3]=>[-5,-2,-1,-4,-3]
[2,4,5,3,1]=>[3,2,5,4,1]=>[3,2,5,4,1]=>[-3,-2,-5,-4,-1]
[2,5,1,3,4]=>[5,2,1,3,4]=>[5,2,1,3,4]=>[-5,-2,-1,-3,-4]
[2,5,1,4,3]=>[5,2,4,1,3]=>[5,2,4,1,3]=>[-5,-2,-4,-1,-3]
[2,5,3,1,4]=>[3,2,5,1,4]=>[3,2,5,1,4]=>[-3,-2,-5,-1,-4]
[2,5,3,4,1]=>[4,2,5,3,1]=>[4,2,5,3,1]=>[-4,-2,-5,-3,-1]
[2,5,4,1,3]=>[4,2,1,5,3]=>[4,2,1,5,3]=>[-4,-2,-1,-5,-3]
[2,5,4,3,1]=>[3,2,4,5,1]=>[3,2,4,5,1]=>[-3,-2,-4,-5,-1]
[3,1,2,4,5]=>[3,1,2,4,5]=>[3,1,2,4,5]=>[-3,-1,-2,-4,-5]
[3,1,2,5,4]=>[3,1,2,5,4]=>[3,1,2,5,4]=>[-3,-1,-2,-5,-4]
[3,1,4,2,5]=>[3,4,1,2,5]=>[3,4,1,2,5]=>[-3,-4,-1,-2,-5]
[3,1,4,5,2]=>[3,5,1,4,2]=>[3,5,1,4,2]=>[-3,-5,-1,-4,-2]
[3,1,5,2,4]=>[3,5,1,2,4]=>[3,5,1,2,4]=>[-3,-5,-1,-2,-4]
[3,1,5,4,2]=>[3,4,1,5,2]=>[3,4,1,5,2]=>[-3,-4,-1,-5,-2]
[3,2,1,4,5]=>[2,3,1,4,5]=>[2,3,1,4,5]=>[-2,-3,-1,-4,-5]
[3,2,1,5,4]=>[2,3,1,5,4]=>[2,3,1,5,4]=>[-2,-3,-1,-5,-4]
[3,2,4,1,5]=>[4,3,2,1,5]=>[4,3,2,1,5]=>[-4,-3,-2,-1,-5]
[3,2,4,5,1]=>[5,3,2,4,1]=>[5,3,2,4,1]=>[-5,-3,-2,-4,-1]
[3,2,5,1,4]=>[5,3,2,1,4]=>[5,3,2,1,4]=>[-5,-3,-2,-1,-4]
[3,2,5,4,1]=>[4,3,2,5,1]=>[4,3,2,5,1]=>[-4,-3,-2,-5,-1]
[3,4,1,2,5]=>[4,1,3,2,5]=>[4,1,3,2,5]=>[-4,-1,-3,-2,-5]
[3,4,1,5,2]=>[4,5,3,1,2]=>[4,5,3,1,2]=>[-4,-5,-3,-1,-2]
[3,4,2,1,5]=>[2,4,3,1,5]=>[2,4,3,1,5]=>[-2,-4,-3,-1,-5]
[3,4,2,5,1]=>[5,4,3,2,1]=>[5,4,3,2,1]=>[-5,-4,-3,-2,-1]
[3,4,5,1,2]=>[5,1,3,4,2]=>[5,1,3,4,2]=>[-5,-1,-3,-4,-2]
[3,4,5,2,1]=>[2,5,3,4,1]=>[2,5,3,4,1]=>[-2,-5,-3,-4,-1]
[3,5,1,2,4]=>[5,1,3,2,4]=>[5,1,3,2,4]=>[-5,-1,-3,-2,-4]
[3,5,1,4,2]=>[5,4,3,1,2]=>[5,4,3,1,2]=>[-5,-4,-3,-1,-2]
[3,5,2,1,4]=>[2,5,3,1,4]=>[2,5,3,1,4]=>[-2,-5,-3,-1,-4]
[3,5,2,4,1]=>[4,5,3,2,1]=>[4,5,3,2,1]=>[-4,-5,-3,-2,-1]
[3,5,4,1,2]=>[4,1,3,5,2]=>[4,1,3,5,2]=>[-4,-1,-3,-5,-2]
[3,5,4,2,1]=>[2,4,3,5,1]=>[2,4,3,5,1]=>[-2,-4,-3,-5,-1]
[4,1,2,3,5]=>[4,1,2,3,5]=>[4,1,2,3,5]=>[-4,-1,-2,-3,-5]
[4,1,2,5,3]=>[4,1,5,2,3]=>[4,1,5,2,3]=>[-4,-1,-5,-2,-3]
[4,1,3,2,5]=>[4,3,1,2,5]=>[4,3,1,2,5]=>[-4,-3,-1,-2,-5]
[4,1,3,5,2]=>[4,5,1,3,2]=>[4,5,1,3,2]=>[-4,-5,-1,-3,-2]
[4,1,5,2,3]=>[4,5,2,1,3]=>[4,5,2,1,3]=>[-4,-5,-2,-1,-3]
[4,1,5,3,2]=>[4,3,5,1,2]=>[4,3,5,1,2]=>[-4,-3,-5,-1,-2]
[4,2,1,3,5]=>[2,4,1,3,5]=>[2,4,1,3,5]=>[-2,-4,-1,-3,-5]
[4,2,1,5,3]=>[2,4,5,1,3]=>[2,4,5,1,3]=>[-2,-4,-5,-1,-3]
[4,2,3,1,5]=>[3,4,2,1,5]=>[3,4,2,1,5]=>[-3,-4,-2,-1,-5]
[4,2,3,5,1]=>[5,4,2,3,1]=>[5,4,2,3,1]=>[-5,-4,-2,-3,-1]
[4,2,5,1,3]=>[5,4,1,2,3]=>[5,4,1,2,3]=>[-5,-4,-1,-2,-3]
[4,2,5,3,1]=>[3,5,4,1,2]=>[3,5,4,1,2]=>[-3,-5,-4,-1,-2]
[4,3,1,2,5]=>[3,1,4,2,5]=>[3,1,4,2,5]=>[-3,-1,-4,-2,-5]
[4,3,1,5,2]=>[3,4,5,2,1]=>[3,4,5,2,1]=>[-3,-4,-5,-2,-1]
[4,3,2,1,5]=>[2,3,4,1,5]=>[2,3,4,1,5]=>[-2,-3,-4,-1,-5]
[4,3,2,5,1]=>[5,3,4,2,1]=>[5,3,4,2,1]=>[-5,-3,-4,-2,-1]
[4,3,5,1,2]=>[5,1,4,3,2]=>[5,1,4,3,2]=>[-5,-1,-4,-3,-2]
[4,3,5,2,1]=>[2,5,4,3,1]=>[2,5,4,3,1]=>[-2,-5,-4,-3,-1]
[4,5,1,2,3]=>[5,1,2,4,3]=>[5,1,2,4,3]=>[-5,-1,-2,-4,-3]
[4,5,1,3,2]=>[5,3,1,4,2]=>[5,3,1,4,2]=>[-5,-3,-1,-4,-2]
[4,5,2,1,3]=>[2,5,1,4,3]=>[2,5,1,4,3]=>[-2,-5,-1,-4,-3]
[4,5,2,3,1]=>[3,5,2,4,1]=>[3,5,2,4,1]=>[-3,-5,-2,-4,-1]
[4,5,3,1,2]=>[3,1,5,4,2]=>[3,1,5,4,2]=>[-3,-1,-5,-4,-2]
[4,5,3,2,1]=>[2,3,5,4,1]=>[2,3,5,4,1]=>[-2,-3,-5,-4,-1]
[5,1,2,3,4]=>[5,1,2,3,4]=>[5,1,2,3,4]=>[-5,-1,-2,-3,-4]
[5,1,2,4,3]=>[5,1,4,2,3]=>[5,1,4,2,3]=>[-5,-1,-4,-2,-3]
[5,1,3,2,4]=>[5,3,1,2,4]=>[5,3,1,2,4]=>[-5,-3,-1,-2,-4]
[5,1,3,4,2]=>[5,4,1,3,2]=>[5,4,1,3,2]=>[-5,-4,-1,-3,-2]
[5,1,4,2,3]=>[5,4,2,1,3]=>[5,4,2,1,3]=>[-5,-4,-2,-1,-3]
[5,1,4,3,2]=>[5,3,4,1,2]=>[5,3,4,1,2]=>[-5,-3,-4,-1,-2]
[5,2,1,3,4]=>[2,5,1,3,4]=>[2,5,1,3,4]=>[-2,-5,-1,-3,-4]
[5,2,1,4,3]=>[2,5,4,1,3]=>[2,5,4,1,3]=>[-2,-5,-4,-1,-3]
[5,2,3,1,4]=>[3,5,2,1,4]=>[3,5,2,1,4]=>[-3,-5,-2,-1,-4]
[5,2,3,4,1]=>[4,5,2,3,1]=>[4,5,2,3,1]=>[-4,-5,-2,-3,-1]
[5,2,4,1,3]=>[4,5,1,2,3]=>[4,5,1,2,3]=>[-4,-5,-1,-2,-3]
[5,2,4,3,1]=>[3,4,5,1,2]=>[3,4,5,1,2]=>[-3,-4,-5,-1,-2]
[5,3,1,2,4]=>[3,1,5,2,4]=>[3,1,5,2,4]=>[-3,-1,-5,-2,-4]
[5,3,1,4,2]=>[3,5,4,2,1]=>[3,5,4,2,1]=>[-3,-5,-4,-2,-1]
[5,3,2,1,4]=>[2,3,5,1,4]=>[2,3,5,1,4]=>[-2,-3,-5,-1,-4]
[5,3,2,4,1]=>[4,3,5,2,1]=>[4,3,5,2,1]=>[-4,-3,-5,-2,-1]
[5,3,4,1,2]=>[4,1,5,3,2]=>[4,1,5,3,2]=>[-4,-1,-5,-3,-2]
[5,3,4,2,1]=>[2,4,5,3,1]=>[2,4,5,3,1]=>[-2,-4,-5,-3,-1]
[5,4,1,2,3]=>[4,1,2,5,3]=>[4,1,2,5,3]=>[-4,-1,-2,-5,-3]
[5,4,1,3,2]=>[4,3,1,5,2]=>[4,3,1,5,2]=>[-4,-3,-1,-5,-2]
[5,4,2,1,3]=>[2,4,1,5,3]=>[2,4,1,5,3]=>[-2,-4,-1,-5,-3]
[5,4,2,3,1]=>[3,4,2,5,1]=>[3,4,2,5,1]=>[-3,-4,-2,-5,-1]
[5,4,3,1,2]=>[3,1,4,5,2]=>[3,1,4,5,2]=>[-3,-1,-4,-5,-2]
[5,4,3,2,1]=>[2,3,4,5,1]=>[2,3,4,5,1]=>[-2,-3,-4,-5,-1]
[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[-2,-1,-4,-3,-6,-5]
[2,1,5,3,6,4]=>[2,1,5,6,3,4]=>[2,1,5,6,3,4]=>[-2,-1,-5,-6,-3,-4]
[2,1,6,4,5,3]=>[2,1,5,6,4,3]=>[2,1,5,6,4,3]=>[-2,-1,-5,-6,-4,-3]
[3,1,4,2,6,5]=>[3,4,1,2,6,5]=>[3,4,1,2,6,5]=>[-3,-4,-1,-2,-6,-5]
[3,1,5,2,6,4]=>[3,5,1,6,2,4]=>[3,5,1,6,2,4]=>[-3,-5,-1,-6,-2,-4]
[3,1,6,4,5,2]=>[3,5,1,6,4,2]=>[3,5,1,6,4,2]=>[-3,-5,-1,-6,-4,-2]
[4,1,5,2,6,3]=>[4,5,6,1,2,3]=>[4,5,6,1,2,3]=>[-4,-5,-6,-1,-2,-3]
[4,1,6,3,5,2]=>[4,5,6,1,3,2]=>[4,5,6,1,3,2]=>[-4,-5,-6,-1,-3,-2]
[4,2,3,1,6,5]=>[3,4,2,1,6,5]=>[3,4,2,1,6,5]=>[-3,-4,-2,-1,-6,-5]
[5,2,3,1,6,4]=>[3,5,2,6,1,4]=>[3,5,2,6,1,4]=>[-3,-5,-2,-6,-1,-4]
[5,2,4,1,6,3]=>[4,5,6,2,1,3]=>[4,5,6,2,1,3]=>[-4,-5,-6,-2,-1,-3]
[5,2,6,3,4,1]=>[4,5,6,3,2,1]=>[4,5,6,3,2,1]=>[-4,-5,-6,-3,-2,-1]
[6,3,4,1,5,2]=>[4,5,6,3,1,2]=>[4,5,6,3,1,2]=>[-4,-5,-6,-3,-1,-2]
[6,3,5,2,4,1]=>[4,5,6,2,3,1]=>[4,5,6,2,3,1]=>[-4,-5,-6,-2,-3,-1]
[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>[3,5,2,6,4,1]=>[-3,-5,-2,-6,-4,-1]
[8,6,7,4,5,2,3,1]=>[3,5,2,7,4,8,6,1]=>[3,5,2,7,4,8,6,1]=>[-3,-5,-2,-7,-4,-8,-6,-1]
[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[-2,-1,-4,-3,-6,-5,-8,-7]
[2,1,4,3,8,6,7,5]=>[2,1,4,3,7,8,6,5]=>[2,1,4,3,7,8,6,5]=>[-2,-1,-4,-3,-7,-8,-6,-5]
[2,1,6,4,5,3,8,7]=>[2,1,5,6,4,3,8,7]=>[2,1,5,6,4,3,8,7]=>[-2,-1,-5,-6,-4,-3,-8,-7]
[4,2,3,1,6,5,8,7]=>[3,4,2,1,6,5,8,7]=>[3,4,2,1,6,5,8,7]=>[-3,-4,-2,-1,-6,-5,-8,-7]
[4,2,3,1,8,6,7,5]=>[3,4,2,1,7,8,6,5]=>[3,4,2,1,7,8,6,5]=>[-3,-4,-2,-1,-7,-8,-6,-5]
[2,1,4,3,7,5,8,6]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,8,5,6]=>[-2,-1,-4,-3,-7,-8,-5,-6]
[2,1,5,3,6,4,8,7]=>[2,1,5,6,3,4,8,7]=>[2,1,5,6,3,4,8,7]=>[-2,-1,-5,-6,-3,-4,-8,-7]
[2,1,5,3,7,4,8,6]=>[2,1,5,7,3,8,4,6]=>[2,1,5,7,3,8,4,6]=>[-2,-1,-5,-7,-3,-8,-4,-6]
[2,1,6,3,7,4,8,5]=>[2,1,6,7,8,3,4,5]=>[2,1,6,7,8,3,4,5]=>[-2,-1,-6,-7,-8,-3,-4,-5]
[3,1,4,2,6,5,8,7]=>[3,4,1,2,6,5,8,7]=>[3,4,1,2,6,5,8,7]=>[-3,-4,-1,-2,-6,-5,-8,-7]
[3,1,4,2,7,5,8,6]=>[3,4,1,2,7,8,5,6]=>[3,4,1,2,7,8,5,6]=>[-3,-4,-1,-2,-7,-8,-5,-6]
[3,1,5,2,6,4,8,7]=>[3,5,1,6,2,4,8,7]=>[3,5,1,6,2,4,8,7]=>[-3,-5,-1,-6,-2,-4,-8,-7]
[3,1,5,2,7,4,8,6]=>[3,5,1,7,2,8,4,6]=>[3,5,1,7,2,8,4,6]=>[-3,-5,-1,-7,-2,-8,-4,-6]
[3,1,6,2,7,4,8,5]=>[3,6,1,7,8,2,4,5]=>[3,6,1,7,8,2,4,5]=>[-3,-6,-1,-7,-8,-2,-4,-5]
[4,1,5,2,6,3,8,7]=>[4,5,6,1,2,3,8,7]=>[4,5,6,1,2,3,8,7]=>[-4,-5,-6,-1,-2,-3,-8,-7]
[4,1,5,2,7,3,8,6]=>[4,5,7,1,2,8,3,6]=>[4,5,7,1,2,8,3,6]=>[-4,-5,-7,-1,-2,-8,-3,-6]
[4,1,6,2,7,3,8,5]=>[4,6,7,1,8,2,3,5]=>[4,6,7,1,8,2,3,5]=>[-4,-6,-7,-1,-8,-2,-3,-5]
[5,1,6,2,7,3,8,4]=>[5,6,7,8,1,2,3,4]=>[5,6,7,8,1,2,3,4]=>[-5,-6,-7,-8,-1,-2,-3,-4]
[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>[3,5,2,6,4,1,8,7]=>[-3,-5,-2,-6,-4,-1,-8,-7]
[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>[2,1,5,7,4,8,6,3]=>[-2,-1,-5,-7,-4,-8,-6,-3]
[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>[5,6,7,8,3,4,2,1]=>[-5,-6,-7,-8,-3,-4,-2,-1]
[7,3,4,1,6,2,8,5]=>[4,6,7,3,8,1,2,5]=>[4,6,7,3,8,1,2,5]=>[-4,-6,-7,-3,-8,-1,-2,-5]
[7,3,5,1,6,2,8,4]=>[5,6,7,8,3,1,2,4]=>[5,6,7,8,3,1,2,4]=>[-5,-6,-7,-8,-3,-1,-2,-4]
[6,2,8,4,5,1,7,3]=>[5,6,7,8,4,2,1,3]=>[5,6,7,8,4,2,1,3]=>[-5,-6,-7,-8,-4,-2,-1,-3]
[7,3,6,2,4,1,8,5]=>[4,6,7,2,8,3,1,5]=>[4,6,7,2,8,3,1,5]=>[-4,-6,-7,-2,-8,-3,-1,-5]
[5,1,7,3,8,4,6,2]=>[5,6,7,8,1,4,3,2]=>[5,6,7,8,1,4,3,2]=>[-5,-6,-7,-8,-1,-4,-3,-2]
[7,3,5,1,8,4,6,2]=>[5,6,7,8,3,4,1,2]=>[5,6,7,8,3,4,1,2]=>[-5,-6,-7,-8,-3,-4,-1,-2]
[7,3,4,1,8,5,6,2]=>[4,6,7,3,8,5,1,2]=>[4,6,7,3,8,5,1,2]=>[-4,-6,-7,-3,-8,-5,-1,-2]
[5,2,3,1,8,6,7,4]=>[3,5,2,7,1,8,6,4]=>[3,5,2,7,1,8,6,4]=>[-3,-5,-2,-7,-1,-8,-6,-4]
[6,2,4,1,8,5,7,3]=>[4,6,7,2,8,1,5,3]=>[4,6,7,2,8,1,5,3]=>[-4,-6,-7,-2,-8,-1,-5,-3]
[6,2,5,1,8,4,7,3]=>[5,6,7,8,2,1,4,3]=>[5,6,7,8,2,1,4,3]=>[-5,-6,-7,-8,-2,-1,-4,-3]
[5,1,8,4,6,2,7,3]=>[5,6,7,8,1,4,2,3]=>[5,6,7,8,1,4,2,3]=>[-5,-6,-7,-8,-1,-4,-2,-3]
[8,4,7,3,6,2,5,1]=>[5,6,7,8,2,3,4,1]=>[5,6,7,8,2,3,4,1]=>[-5,-6,-7,-8,-2,-3,-4,-1]
[6,2,7,3,5,1,8,4]=>[5,6,7,8,3,2,1,4]=>[5,6,7,8,3,2,1,4]=>[-5,-6,-7,-8,-3,-2,-1,-4]
[7,3,8,4,5,1,6,2]=>[5,6,7,8,4,1,3,2]=>[5,6,7,8,4,1,3,2]=>[-5,-6,-7,-8,-4,-1,-3,-2]
[4,1,7,3,8,5,6,2]=>[4,6,7,1,8,5,3,2]=>[4,6,7,1,8,5,3,2]=>[-4,-6,-7,-1,-8,-5,-3,-2]
[8,5,7,3,6,2,4,1]=>[4,6,7,2,8,3,5,1]=>[4,6,7,2,8,3,5,1]=>[-4,-6,-7,-2,-8,-3,-5,-1]
[8,6,7,3,5,2,4,1]=>[4,5,7,2,3,8,6,1]=>[4,5,7,2,3,8,6,1]=>[-4,-5,-7,-2,-3,-8,-6,-1]
[8,5,7,4,6,2,3,1]=>[3,6,2,7,8,4,5,1]=>[3,6,2,7,8,4,5,1]=>[-3,-6,-2,-7,-8,-4,-5,-1]
[5,2,6,3,4,1,8,7]=>[4,5,6,3,2,1,8,7]=>[4,5,6,3,2,1,8,7]=>[-4,-5,-6,-3,-2,-1,-8,-7]
[2,1,7,4,8,5,6,3]=>[2,1,6,7,8,5,4,3]=>[2,1,6,7,8,5,4,3]=>[-2,-1,-6,-7,-8,-5,-4,-3]
[4,1,8,5,7,3,6,2]=>[4,6,7,1,8,3,5,2]=>[4,6,7,1,8,3,5,2]=>[-4,-6,-7,-1,-8,-3,-5,-2]
[2,1,6,3,8,5,7,4]=>[2,1,6,7,8,3,5,4]=>[2,1,6,7,8,3,5,4]=>[-2,-1,-6,-7,-8,-3,-5,-4]
[4,1,6,2,8,5,7,3]=>[4,6,7,1,8,2,5,3]=>[4,6,7,1,8,2,5,3]=>[-4,-6,-7,-1,-8,-2,-5,-3]
[5,1,7,3,6,2,8,4]=>[5,6,7,8,1,3,2,4]=>[5,6,7,8,1,3,2,4]=>[-5,-6,-7,-8,-1,-3,-2,-4]
[3,1,7,4,5,2,8,6]=>[3,5,1,7,4,8,2,6]=>[3,5,1,7,4,8,2,6]=>[-3,-5,-1,-7,-4,-8,-2,-6]
[5,2,8,6,7,3,4,1]=>[4,5,7,3,2,8,6,1]=>[4,5,7,3,2,8,6,1]=>[-4,-5,-7,-3,-2,-8,-6,-1]
[8,4,5,1,7,3,6,2]=>[5,6,7,8,4,3,1,2]=>[5,6,7,8,4,3,1,2]=>[-5,-6,-7,-8,-4,-3,-1,-2]
[7,4,8,5,6,2,3,1]=>[3,6,2,7,8,5,4,1]=>[3,6,2,7,8,5,4,1]=>[-3,-6,-2,-7,-8,-5,-4,-1]
[6,2,5,1,7,3,8,4]=>[5,6,7,8,2,1,3,4]=>[5,6,7,8,2,1,3,4]=>[-5,-6,-7,-8,-2,-1,-3,-4]
[3,1,5,2,8,6,7,4]=>[3,5,1,7,2,8,6,4]=>[3,5,1,7,2,8,6,4]=>[-3,-5,-1,-7,-2,-8,-6,-4]
[8,6,7,3,4,1,5,2]=>[4,5,7,3,1,8,6,2]=>[4,5,7,3,1,8,6,2]=>[-4,-5,-7,-3,-1,-8,-6,-2]
[5,2,4,1,6,3,8,7]=>[4,5,6,2,1,3,8,7]=>[4,5,6,2,1,3,8,7]=>[-4,-5,-6,-2,-1,-3,-8,-7]
[2,1,7,4,6,3,8,5]=>[2,1,6,7,8,4,3,5]=>[2,1,6,7,8,4,3,5]=>[-2,-1,-6,-7,-8,-4,-3,-5]
[3,1,6,4,5,2,8,7]=>[3,5,1,6,4,2,8,7]=>[3,5,1,6,4,2,8,7]=>[-3,-5,-1,-6,-4,-2,-8,-7]
[2,1,5,3,8,6,7,4]=>[2,1,5,7,3,8,6,4]=>[2,1,5,7,3,8,6,4]=>[-2,-1,-5,-7,-3,-8,-6,-4]
[5,2,3,1,6,4,8,7]=>[3,5,2,6,1,4,8,7]=>[3,5,2,6,1,4,8,7]=>[-3,-5,-2,-6,-1,-4,-8,-7]
[7,4,5,2,3,1,8,6]=>[3,5,2,7,4,8,1,6]=>[3,5,2,7,4,8,1,6]=>[-3,-5,-2,-7,-4,-8,-1,-6]
[7,3,5,2,4,1,8,6]=>[4,5,7,2,3,8,1,6]=>[4,5,7,2,3,8,1,6]=>[-4,-5,-7,-2,-3,-8,-1,-6]
[6,3,5,2,4,1,8,7]=>[4,5,6,2,3,1,8,7]=>[4,5,6,2,3,1,8,7]=>[-4,-5,-6,-2,-3,-1,-8,-7]
[4,1,6,3,5,2,8,7]=>[4,5,6,1,3,2,8,7]=>[4,5,6,1,3,2,8,7]=>[-4,-5,-6,-1,-3,-2,-8,-7]
[6,3,4,1,5,2,8,7]=>[4,5,6,3,1,2,8,7]=>[4,5,6,3,1,2,8,7]=>[-4,-5,-6,-3,-1,-2,-8,-7]
[5,2,7,3,4,1,8,6]=>[4,5,7,3,2,8,1,6]=>[4,5,7,3,2,8,1,6]=>[-4,-5,-7,-3,-2,-8,-1,-6]
[6,2,8,5,7,3,4,1]=>[4,6,7,3,8,2,5,1]=>[4,6,7,3,8,2,5,1]=>[-4,-6,-7,-3,-8,-2,-5,-1]
[6,2,7,3,4,1,8,5]=>[4,6,7,3,8,2,1,5]=>[4,6,7,3,8,2,1,5]=>[-4,-6,-7,-3,-8,-2,-1,-5]
[7,3,4,1,5,2,8,6]=>[4,5,7,3,1,8,2,6]=>[4,5,7,3,1,8,2,6]=>[-4,-5,-7,-3,-1,-8,-2,-6]
[4,1,7,3,5,2,8,6]=>[4,5,7,1,3,8,2,6]=>[4,5,7,1,3,8,2,6]=>[-4,-5,-7,-1,-3,-8,-2,-6]
[2,1,7,4,5,3,8,6]=>[2,1,5,7,4,8,3,6]=>[2,1,5,7,4,8,3,6]=>[-2,-1,-5,-7,-4,-8,-3,-6]
[3,1,8,6,7,4,5,2]=>[3,5,1,7,4,8,6,2]=>[3,5,1,7,4,8,6,2]=>[-3,-5,-1,-7,-4,-8,-6,-2]
[4,1,8,6,7,3,5,2]=>[4,5,7,1,3,8,6,2]=>[4,5,7,1,3,8,6,2]=>[-4,-5,-7,-1,-3,-8,-6,-2]
[8,5,7,3,4,1,6,2]=>[4,6,7,3,8,1,5,2]=>[4,6,7,3,8,1,5,2]=>[-4,-6,-7,-3,-8,-1,-5,-2]
[8,5,6,2,7,3,4,1]=>[4,6,7,3,8,5,2,1]=>[4,6,7,3,8,5,2,1]=>[-4,-6,-7,-3,-8,-5,-2,-1]
[8,4,7,3,5,1,6,2]=>[5,6,7,8,3,1,4,2]=>[5,6,7,8,3,1,4,2]=>[-5,-6,-7,-8,-3,-1,-4,-2]
[5,1,8,4,7,3,6,2]=>[5,6,7,8,1,3,4,2]=>[5,6,7,8,1,3,4,2]=>[-5,-6,-7,-8,-1,-3,-4,-2]
[3,1,8,5,7,4,6,2]=>[3,6,1,7,8,4,5,2]=>[3,6,1,7,8,4,5,2]=>[-3,-6,-1,-7,-8,-4,-5,-2]
[2,1,8,5,7,4,6,3]=>[2,1,6,7,8,4,5,3]=>[2,1,6,7,8,4,5,3]=>[-2,-1,-6,-7,-8,-4,-5,-3]
[3,1,7,4,6,2,8,5]=>[3,6,1,7,8,4,2,5]=>[3,6,1,7,8,4,2,5]=>[-3,-6,-1,-7,-8,-4,-2,-5]
[4,1,7,3,6,2,8,5]=>[4,6,7,1,8,3,2,5]=>[4,6,7,1,8,3,2,5]=>[-4,-6,-7,-1,-8,-3,-2,-5]
[6,2,8,4,7,3,5,1]=>[5,6,7,8,3,2,4,1]=>[5,6,7,8,3,2,4,1]=>[-5,-6,-7,-8,-3,-2,-4,-1]
[7,3,6,2,5,1,8,4]=>[5,6,7,8,2,3,1,4]=>[5,6,7,8,2,3,1,4]=>[-5,-6,-7,-8,-2,-3,-1,-4]
[5,2,4,1,7,3,8,6]=>[4,5,7,2,1,8,3,6]=>[4,5,7,2,1,8,3,6]=>[-4,-5,-7,-2,-1,-8,-3,-6]
[6,2,4,1,7,3,8,5]=>[4,6,7,2,8,1,3,5]=>[4,6,7,2,8,1,3,5]=>[-4,-6,-7,-2,-8,-1,-3,-5]
[7,4,6,2,3,1,8,5]=>[3,6,2,7,8,4,1,5]=>[3,6,2,7,8,4,1,5]=>[-3,-6,-2,-7,-8,-4,-1,-5]
[6,2,3,1,7,4,8,5]=>[3,6,2,7,8,1,4,5]=>[3,6,2,7,8,1,4,5]=>[-3,-6,-2,-7,-8,-1,-4,-5]
[5,2,3,1,7,4,8,6]=>[3,5,2,7,1,8,4,6]=>[3,5,2,7,1,8,4,6]=>[-3,-5,-2,-7,-1,-8,-4,-6]
[4,2,3,1,7,5,8,6]=>[3,4,2,1,7,8,5,6]=>[3,4,2,1,7,8,5,6]=>[-3,-4,-2,-1,-7,-8,-5,-6]
[3,1,4,2,8,6,7,5]=>[3,4,1,2,7,8,6,5]=>[3,4,1,2,7,8,6,5]=>[-3,-4,-1,-2,-7,-8,-6,-5]
[6,2,3,1,8,5,7,4]=>[3,6,2,7,8,1,5,4]=>[3,6,2,7,8,1,5,4]=>[-3,-6,-2,-7,-8,-1,-5,-4]
[8,5,6,2,3,1,7,4]=>[3,6,2,7,8,5,1,4]=>[3,6,2,7,8,5,1,4]=>[-3,-6,-2,-7,-8,-5,-1,-4]
[7,3,8,5,6,2,4,1]=>[4,6,7,2,8,5,3,1]=>[4,6,7,2,8,5,3,1]=>[-4,-6,-7,-2,-8,-5,-3,-1]
[8,5,6,2,4,1,7,3]=>[4,6,7,2,8,5,1,3]=>[4,6,7,2,8,5,1,3]=>[-4,-6,-7,-2,-8,-5,-1,-3]
[5,2,4,1,8,6,7,3]=>[4,5,7,2,1,8,6,3]=>[4,5,7,2,1,8,6,3]=>[-4,-5,-7,-2,-1,-8,-6,-3]
[4,1,5,2,8,6,7,3]=>[4,5,7,1,2,8,6,3]=>[4,5,7,1,2,8,6,3]=>[-4,-5,-7,-1,-2,-8,-6,-3]
[3,1,6,2,8,5,7,4]=>[3,6,1,7,8,2,5,4]=>[3,6,1,7,8,2,5,4]=>[-3,-6,-1,-7,-8,-2,-5,-4]
[5,1,6,2,8,4,7,3]=>[5,6,7,8,1,2,4,3]=>[5,6,7,8,1,2,4,3]=>[-5,-6,-7,-8,-1,-2,-4,-3]
[8,4,6,2,5,1,7,3]=>[5,6,7,8,2,4,1,3]=>[5,6,7,8,2,4,1,3]=>[-5,-6,-7,-8,-2,-4,-1,-3]
[7,3,8,4,6,2,5,1]=>[5,6,7,8,2,4,3,1]=>[5,6,7,8,2,4,3,1]=>[-5,-6,-7,-8,-2,-4,-3,-1]
[6,2,7,3,8,4,5,1]=>[5,6,7,8,4,2,3,1]=>[5,6,7,8,4,2,3,1]=>[-5,-6,-7,-8,-4,-2,-3,-1]
[8,4,5,1,6,2,7,3]=>[5,6,7,8,4,1,2,3]=>[5,6,7,8,4,1,2,3]=>[-5,-6,-7,-8,-4,-1,-2,-3]
[4,1,8,5,6,2,7,3]=>[4,6,7,1,8,5,2,3]=>[4,6,7,1,8,5,2,3]=>[-4,-6,-7,-1,-8,-5,-2,-3]
[3,1,8,5,6,2,7,4]=>[3,6,1,7,8,5,2,4]=>[3,6,1,7,8,5,2,4]=>[-3,-6,-1,-7,-8,-5,-2,-4]
[2,1,8,5,6,3,7,4]=>[2,1,6,7,8,5,3,4]=>[2,1,6,7,8,5,3,4]=>[-2,-1,-6,-7,-8,-5,-3,-4]
[3,1,7,4,8,5,6,2]=>[3,6,1,7,8,5,4,2]=>[3,6,1,7,8,5,4,2]=>[-3,-6,-1,-7,-8,-5,-4,-2]
[7,3,6,2,8,4,5,1]=>[5,6,7,8,4,3,2,1]=>[5,6,7,8,4,3,2,1]=>[-5,-6,-7,-8,-4,-3,-2,-1]
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
bar
Description
Return the signed permutation with all signs reversed.
searching the database
Sorry, this map was not found in the database.