Identifier
- St000325: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000470The number of runs in a permutation.)
Values
=>
[1]=>1
[1,2]=>1
[2,1]=>2
[1,2,3]=>1
[1,3,2]=>2
[2,1,3]=>2
[2,3,1]=>2
[3,1,2]=>2
[3,2,1]=>3
[1,2,3,4]=>1
[1,2,4,3]=>2
[1,3,2,4]=>2
[1,3,4,2]=>2
[1,4,2,3]=>2
[1,4,3,2]=>3
[2,1,3,4]=>2
[2,1,4,3]=>3
[2,3,1,4]=>2
[2,3,4,1]=>2
[2,4,1,3]=>2
[2,4,3,1]=>3
[3,1,2,4]=>2
[3,1,4,2]=>3
[3,2,1,4]=>3
[3,2,4,1]=>3
[3,4,1,2]=>2
[3,4,2,1]=>3
[4,1,2,3]=>2
[4,1,3,2]=>3
[4,2,1,3]=>3
[4,2,3,1]=>3
[4,3,1,2]=>3
[4,3,2,1]=>4
[1,2,3,4,5]=>1
[1,2,3,5,4]=>2
[1,2,4,3,5]=>2
[1,2,4,5,3]=>2
[1,2,5,3,4]=>2
[1,2,5,4,3]=>3
[1,3,2,4,5]=>2
[1,3,2,5,4]=>3
[1,3,4,2,5]=>2
[1,3,4,5,2]=>2
[1,3,5,2,4]=>2
[1,3,5,4,2]=>3
[1,4,2,3,5]=>2
[1,4,2,5,3]=>3
[1,4,3,2,5]=>3
[1,4,3,5,2]=>3
[1,4,5,2,3]=>2
[1,4,5,3,2]=>3
[1,5,2,3,4]=>2
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>3
[1,5,4,2,3]=>3
[1,5,4,3,2]=>4
[2,1,3,4,5]=>2
[2,1,3,5,4]=>3
[2,1,4,3,5]=>3
[2,1,4,5,3]=>3
[2,1,5,3,4]=>3
[2,1,5,4,3]=>4
[2,3,1,4,5]=>2
[2,3,1,5,4]=>3
[2,3,4,1,5]=>2
[2,3,4,5,1]=>2
[2,3,5,1,4]=>2
[2,3,5,4,1]=>3
[2,4,1,3,5]=>2
[2,4,1,5,3]=>3
[2,4,3,1,5]=>3
[2,4,3,5,1]=>3
[2,4,5,1,3]=>2
[2,4,5,3,1]=>3
[2,5,1,3,4]=>2
[2,5,1,4,3]=>3
[2,5,3,1,4]=>3
[2,5,3,4,1]=>3
[2,5,4,1,3]=>3
[2,5,4,3,1]=>4
[3,1,2,4,5]=>2
[3,1,2,5,4]=>3
[3,1,4,2,5]=>3
[3,1,4,5,2]=>3
[3,1,5,2,4]=>3
[3,1,5,4,2]=>4
[3,2,1,4,5]=>3
[3,2,1,5,4]=>4
[3,2,4,1,5]=>3
[3,2,4,5,1]=>3
[3,2,5,1,4]=>3
[3,2,5,4,1]=>4
[3,4,1,2,5]=>2
[3,4,1,5,2]=>3
[3,4,2,1,5]=>3
[3,4,2,5,1]=>3
[3,4,5,1,2]=>2
[3,4,5,2,1]=>3
[3,5,1,2,4]=>2
[3,5,1,4,2]=>3
[3,5,2,1,4]=>3
[3,5,2,4,1]=>3
[3,5,4,1,2]=>3
[3,5,4,2,1]=>4
[4,1,2,3,5]=>2
[4,1,2,5,3]=>3
[4,1,3,2,5]=>3
[4,1,3,5,2]=>3
[4,1,5,2,3]=>3
[4,1,5,3,2]=>4
[4,2,1,3,5]=>3
[4,2,1,5,3]=>4
[4,2,3,1,5]=>3
[4,2,3,5,1]=>3
[4,2,5,1,3]=>3
[4,2,5,3,1]=>4
[4,3,1,2,5]=>3
[4,3,1,5,2]=>4
[4,3,2,1,5]=>4
[4,3,2,5,1]=>4
[4,3,5,1,2]=>3
[4,3,5,2,1]=>4
[4,5,1,2,3]=>2
[4,5,1,3,2]=>3
[4,5,2,1,3]=>3
[4,5,2,3,1]=>3
[4,5,3,1,2]=>3
[4,5,3,2,1]=>4
[5,1,2,3,4]=>2
[5,1,2,4,3]=>3
[5,1,3,2,4]=>3
[5,1,3,4,2]=>3
[5,1,4,2,3]=>3
[5,1,4,3,2]=>4
[5,2,1,3,4]=>3
[5,2,1,4,3]=>4
[5,2,3,1,4]=>3
[5,2,3,4,1]=>3
[5,2,4,1,3]=>3
[5,2,4,3,1]=>4
[5,3,1,2,4]=>3
[5,3,1,4,2]=>4
[5,3,2,1,4]=>4
[5,3,2,4,1]=>4
[5,3,4,1,2]=>3
[5,3,4,2,1]=>4
[5,4,1,2,3]=>3
[5,4,1,3,2]=>4
[5,4,2,1,3]=>4
[5,4,2,3,1]=>4
[5,4,3,1,2]=>4
[5,4,3,2,1]=>5
[1,2,3,4,5,6]=>1
[1,2,3,4,6,5]=>2
[1,2,3,5,4,6]=>2
[1,2,3,5,6,4]=>2
[1,2,3,6,4,5]=>2
[1,2,3,6,5,4]=>3
[1,2,4,3,5,6]=>2
[1,2,4,3,6,5]=>3
[1,2,4,5,3,6]=>2
[1,2,4,5,6,3]=>2
[1,2,4,6,3,5]=>2
[1,2,4,6,5,3]=>3
[1,2,5,3,4,6]=>2
[1,2,5,3,6,4]=>3
[1,2,5,4,3,6]=>3
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>2
[1,2,5,6,4,3]=>3
[1,2,6,3,4,5]=>2
[1,2,6,3,5,4]=>3
[1,2,6,4,3,5]=>3
[1,2,6,4,5,3]=>3
[1,2,6,5,3,4]=>3
[1,2,6,5,4,3]=>4
[1,3,2,4,5,6]=>2
[1,3,2,4,6,5]=>3
[1,3,2,5,4,6]=>3
[1,3,2,5,6,4]=>3
[1,3,2,6,4,5]=>3
[1,3,2,6,5,4]=>4
[1,3,4,2,5,6]=>2
[1,3,4,2,6,5]=>3
[1,3,4,5,2,6]=>2
[1,3,4,5,6,2]=>2
[1,3,4,6,2,5]=>2
[1,3,4,6,5,2]=>3
[1,3,5,2,4,6]=>2
[1,3,5,2,6,4]=>3
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>3
[1,3,5,6,2,4]=>2
[1,3,5,6,4,2]=>3
[1,3,6,2,4,5]=>2
[1,3,6,2,5,4]=>3
[1,3,6,4,2,5]=>3
[1,3,6,4,5,2]=>3
[1,3,6,5,2,4]=>3
[1,3,6,5,4,2]=>4
[1,4,2,3,5,6]=>2
[1,4,2,3,6,5]=>3
[1,4,2,5,3,6]=>3
[1,4,2,5,6,3]=>3
[1,4,2,6,3,5]=>3
[1,4,2,6,5,3]=>4
[1,4,3,2,5,6]=>3
[1,4,3,2,6,5]=>4
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>3
[1,4,3,6,2,5]=>3
[1,4,3,6,5,2]=>4
[1,4,5,2,3,6]=>2
[1,4,5,2,6,3]=>3
[1,4,5,3,2,6]=>3
[1,4,5,3,6,2]=>3
[1,4,5,6,2,3]=>2
[1,4,5,6,3,2]=>3
[1,4,6,2,3,5]=>2
[1,4,6,2,5,3]=>3
[1,4,6,3,2,5]=>3
[1,4,6,3,5,2]=>3
[1,4,6,5,2,3]=>3
[1,4,6,5,3,2]=>4
[1,5,2,3,4,6]=>2
[1,5,2,3,6,4]=>3
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>3
[1,5,2,6,3,4]=>3
[1,5,2,6,4,3]=>4
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>4
[1,5,3,4,2,6]=>3
[1,5,3,4,6,2]=>3
[1,5,3,6,2,4]=>3
[1,5,3,6,4,2]=>4
[1,5,4,2,3,6]=>3
[1,5,4,2,6,3]=>4
[1,5,4,3,2,6]=>4
[1,5,4,3,6,2]=>4
[1,5,4,6,2,3]=>3
[1,5,4,6,3,2]=>4
[1,5,6,2,3,4]=>2
[1,5,6,2,4,3]=>3
[1,5,6,3,2,4]=>3
[1,5,6,3,4,2]=>3
[1,5,6,4,2,3]=>3
[1,5,6,4,3,2]=>4
[1,6,2,3,4,5]=>2
[1,6,2,3,5,4]=>3
[1,6,2,4,3,5]=>3
[1,6,2,4,5,3]=>3
[1,6,2,5,3,4]=>3
[1,6,2,5,4,3]=>4
[1,6,3,2,4,5]=>3
[1,6,3,2,5,4]=>4
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>3
[1,6,3,5,2,4]=>3
[1,6,3,5,4,2]=>4
[1,6,4,2,3,5]=>3
[1,6,4,2,5,3]=>4
[1,6,4,3,2,5]=>4
[1,6,4,3,5,2]=>4
[1,6,4,5,2,3]=>3
[1,6,4,5,3,2]=>4
[1,6,5,2,3,4]=>3
[1,6,5,2,4,3]=>4
[1,6,5,3,2,4]=>4
[1,6,5,3,4,2]=>4
[1,6,5,4,2,3]=>4
[1,6,5,4,3,2]=>5
[2,1,3,4,5,6]=>2
[2,1,3,4,6,5]=>3
[2,1,3,5,4,6]=>3
[2,1,3,5,6,4]=>3
[2,1,3,6,4,5]=>3
[2,1,3,6,5,4]=>4
[2,1,4,3,5,6]=>3
[2,1,4,3,6,5]=>4
[2,1,4,5,3,6]=>3
[2,1,4,5,6,3]=>3
[2,1,4,6,3,5]=>3
[2,1,4,6,5,3]=>4
[2,1,5,3,4,6]=>3
[2,1,5,3,6,4]=>4
[2,1,5,4,3,6]=>4
[2,1,5,4,6,3]=>4
[2,1,5,6,3,4]=>3
[2,1,5,6,4,3]=>4
[2,1,6,3,4,5]=>3
[2,1,6,3,5,4]=>4
[2,1,6,4,3,5]=>4
[2,1,6,4,5,3]=>4
[2,1,6,5,3,4]=>4
[2,1,6,5,4,3]=>5
[2,3,1,4,5,6]=>2
[2,3,1,4,6,5]=>3
[2,3,1,5,4,6]=>3
[2,3,1,5,6,4]=>3
[2,3,1,6,4,5]=>3
[2,3,1,6,5,4]=>4
[2,3,4,1,5,6]=>2
[2,3,4,1,6,5]=>3
[2,3,4,5,1,6]=>2
[2,3,4,5,6,1]=>2
[2,3,4,6,1,5]=>2
[2,3,4,6,5,1]=>3
[2,3,5,1,4,6]=>2
[2,3,5,1,6,4]=>3
[2,3,5,4,1,6]=>3
[2,3,5,4,6,1]=>3
[2,3,5,6,1,4]=>2
[2,3,5,6,4,1]=>3
[2,3,6,1,4,5]=>2
[2,3,6,1,5,4]=>3
[2,3,6,4,1,5]=>3
[2,3,6,4,5,1]=>3
[2,3,6,5,1,4]=>3
[2,3,6,5,4,1]=>4
[2,4,1,3,5,6]=>2
[2,4,1,3,6,5]=>3
[2,4,1,5,3,6]=>3
[2,4,1,5,6,3]=>3
[2,4,1,6,3,5]=>3
[2,4,1,6,5,3]=>4
[2,4,3,1,5,6]=>3
[2,4,3,1,6,5]=>4
[2,4,3,5,1,6]=>3
[2,4,3,5,6,1]=>3
[2,4,3,6,1,5]=>3
[2,4,3,6,5,1]=>4
[2,4,5,1,3,6]=>2
[2,4,5,1,6,3]=>3
[2,4,5,3,1,6]=>3
[2,4,5,3,6,1]=>3
[2,4,5,6,1,3]=>2
[2,4,5,6,3,1]=>3
[2,4,6,1,3,5]=>2
[2,4,6,1,5,3]=>3
[2,4,6,3,1,5]=>3
[2,4,6,3,5,1]=>3
[2,4,6,5,1,3]=>3
[2,4,6,5,3,1]=>4
[2,5,1,3,4,6]=>2
[2,5,1,3,6,4]=>3
[2,5,1,4,3,6]=>3
[2,5,1,4,6,3]=>3
[2,5,1,6,3,4]=>3
[2,5,1,6,4,3]=>4
[2,5,3,1,4,6]=>3
[2,5,3,1,6,4]=>4
[2,5,3,4,1,6]=>3
[2,5,3,4,6,1]=>3
[2,5,3,6,1,4]=>3
[2,5,3,6,4,1]=>4
[2,5,4,1,3,6]=>3
[2,5,4,1,6,3]=>4
[2,5,4,3,1,6]=>4
[2,5,4,3,6,1]=>4
[2,5,4,6,1,3]=>3
[2,5,4,6,3,1]=>4
[2,5,6,1,3,4]=>2
[2,5,6,1,4,3]=>3
[2,5,6,3,1,4]=>3
[2,5,6,3,4,1]=>3
[2,5,6,4,1,3]=>3
[2,5,6,4,3,1]=>4
[2,6,1,3,4,5]=>2
[2,6,1,3,5,4]=>3
[2,6,1,4,3,5]=>3
[2,6,1,4,5,3]=>3
[2,6,1,5,3,4]=>3
[2,6,1,5,4,3]=>4
[2,6,3,1,4,5]=>3
[2,6,3,1,5,4]=>4
[2,6,3,4,1,5]=>3
[2,6,3,4,5,1]=>3
[2,6,3,5,1,4]=>3
[2,6,3,5,4,1]=>4
[2,6,4,1,3,5]=>3
[2,6,4,1,5,3]=>4
[2,6,4,3,1,5]=>4
[2,6,4,3,5,1]=>4
[2,6,4,5,1,3]=>3
[2,6,4,5,3,1]=>4
[2,6,5,1,3,4]=>3
[2,6,5,1,4,3]=>4
[2,6,5,3,1,4]=>4
[2,6,5,3,4,1]=>4
[2,6,5,4,1,3]=>4
[2,6,5,4,3,1]=>5
[3,1,2,4,5,6]=>2
[3,1,2,4,6,5]=>3
[3,1,2,5,4,6]=>3
[3,1,2,5,6,4]=>3
[3,1,2,6,4,5]=>3
[3,1,2,6,5,4]=>4
[3,1,4,2,5,6]=>3
[3,1,4,2,6,5]=>4
[3,1,4,5,2,6]=>3
[3,1,4,5,6,2]=>3
[3,1,4,6,2,5]=>3
[3,1,4,6,5,2]=>4
[3,1,5,2,4,6]=>3
[3,1,5,2,6,4]=>4
[3,1,5,4,2,6]=>4
[3,1,5,4,6,2]=>4
[3,1,5,6,2,4]=>3
[3,1,5,6,4,2]=>4
[3,1,6,2,4,5]=>3
[3,1,6,2,5,4]=>4
[3,1,6,4,2,5]=>4
[3,1,6,4,5,2]=>4
[3,1,6,5,2,4]=>4
[3,1,6,5,4,2]=>5
[3,2,1,4,5,6]=>3
[3,2,1,4,6,5]=>4
[3,2,1,5,4,6]=>4
[3,2,1,5,6,4]=>4
[3,2,1,6,4,5]=>4
[3,2,1,6,5,4]=>5
[3,2,4,1,5,6]=>3
[3,2,4,1,6,5]=>4
[3,2,4,5,1,6]=>3
[3,2,4,5,6,1]=>3
[3,2,4,6,1,5]=>3
[3,2,4,6,5,1]=>4
[3,2,5,1,4,6]=>3
[3,2,5,1,6,4]=>4
[3,2,5,4,1,6]=>4
[3,2,5,4,6,1]=>4
[3,2,5,6,1,4]=>3
[3,2,5,6,4,1]=>4
[3,2,6,1,4,5]=>3
[3,2,6,1,5,4]=>4
[3,2,6,4,1,5]=>4
[3,2,6,4,5,1]=>4
[3,2,6,5,1,4]=>4
[3,2,6,5,4,1]=>5
[3,4,1,2,5,6]=>2
[3,4,1,2,6,5]=>3
[3,4,1,5,2,6]=>3
[3,4,1,5,6,2]=>3
[3,4,1,6,2,5]=>3
[3,4,1,6,5,2]=>4
[3,4,2,1,5,6]=>3
[3,4,2,1,6,5]=>4
[3,4,2,5,1,6]=>3
[3,4,2,5,6,1]=>3
[3,4,2,6,1,5]=>3
[3,4,2,6,5,1]=>4
[3,4,5,1,2,6]=>2
[3,4,5,1,6,2]=>3
[3,4,5,2,1,6]=>3
[3,4,5,2,6,1]=>3
[3,4,5,6,1,2]=>2
[3,4,5,6,2,1]=>3
[3,4,6,1,2,5]=>2
[3,4,6,1,5,2]=>3
[3,4,6,2,1,5]=>3
[3,4,6,2,5,1]=>3
[3,4,6,5,1,2]=>3
[3,4,6,5,2,1]=>4
[3,5,1,2,4,6]=>2
[3,5,1,2,6,4]=>3
[3,5,1,4,2,6]=>3
[3,5,1,4,6,2]=>3
[3,5,1,6,2,4]=>3
[3,5,1,6,4,2]=>4
[3,5,2,1,4,6]=>3
[3,5,2,1,6,4]=>4
[3,5,2,4,1,6]=>3
[3,5,2,4,6,1]=>3
[3,5,2,6,1,4]=>3
[3,5,2,6,4,1]=>4
[3,5,4,1,2,6]=>3
[3,5,4,1,6,2]=>4
[3,5,4,2,1,6]=>4
[3,5,4,2,6,1]=>4
[3,5,4,6,1,2]=>3
[3,5,4,6,2,1]=>4
[3,5,6,1,2,4]=>2
[3,5,6,1,4,2]=>3
[3,5,6,2,1,4]=>3
[3,5,6,2,4,1]=>3
[3,5,6,4,1,2]=>3
[3,5,6,4,2,1]=>4
[3,6,1,2,4,5]=>2
[3,6,1,2,5,4]=>3
[3,6,1,4,2,5]=>3
[3,6,1,4,5,2]=>3
[3,6,1,5,2,4]=>3
[3,6,1,5,4,2]=>4
[3,6,2,1,4,5]=>3
[3,6,2,1,5,4]=>4
[3,6,2,4,1,5]=>3
[3,6,2,4,5,1]=>3
[3,6,2,5,1,4]=>3
[3,6,2,5,4,1]=>4
[3,6,4,1,2,5]=>3
[3,6,4,1,5,2]=>4
[3,6,4,2,1,5]=>4
[3,6,4,2,5,1]=>4
[3,6,4,5,1,2]=>3
[3,6,4,5,2,1]=>4
[3,6,5,1,2,4]=>3
[3,6,5,1,4,2]=>4
[3,6,5,2,1,4]=>4
[3,6,5,2,4,1]=>4
[3,6,5,4,1,2]=>4
[3,6,5,4,2,1]=>5
[4,1,2,3,5,6]=>2
[4,1,2,3,6,5]=>3
[4,1,2,5,3,6]=>3
[4,1,2,5,6,3]=>3
[4,1,2,6,3,5]=>3
[4,1,2,6,5,3]=>4
[4,1,3,2,5,6]=>3
[4,1,3,2,6,5]=>4
[4,1,3,5,2,6]=>3
[4,1,3,5,6,2]=>3
[4,1,3,6,2,5]=>3
[4,1,3,6,5,2]=>4
[4,1,5,2,3,6]=>3
[4,1,5,2,6,3]=>4
[4,1,5,3,2,6]=>4
[4,1,5,3,6,2]=>4
[4,1,5,6,2,3]=>3
[4,1,5,6,3,2]=>4
[4,1,6,2,3,5]=>3
[4,1,6,2,5,3]=>4
[4,1,6,3,2,5]=>4
[4,1,6,3,5,2]=>4
[4,1,6,5,2,3]=>4
[4,1,6,5,3,2]=>5
[4,2,1,3,5,6]=>3
[4,2,1,3,6,5]=>4
[4,2,1,5,3,6]=>4
[4,2,1,5,6,3]=>4
[4,2,1,6,3,5]=>4
[4,2,1,6,5,3]=>5
[4,2,3,1,5,6]=>3
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>3
[4,2,3,5,6,1]=>3
[4,2,3,6,1,5]=>3
[4,2,3,6,5,1]=>4
[4,2,5,1,3,6]=>3
[4,2,5,1,6,3]=>4
[4,2,5,3,1,6]=>4
[4,2,5,3,6,1]=>4
[4,2,5,6,1,3]=>3
[4,2,5,6,3,1]=>4
[4,2,6,1,3,5]=>3
[4,2,6,1,5,3]=>4
[4,2,6,3,1,5]=>4
[4,2,6,3,5,1]=>4
[4,2,6,5,1,3]=>4
[4,2,6,5,3,1]=>5
[4,3,1,2,5,6]=>3
[4,3,1,2,6,5]=>4
[4,3,1,5,2,6]=>4
[4,3,1,5,6,2]=>4
[4,3,1,6,2,5]=>4
[4,3,1,6,5,2]=>5
[4,3,2,1,5,6]=>4
[4,3,2,1,6,5]=>5
[4,3,2,5,1,6]=>4
[4,3,2,5,6,1]=>4
[4,3,2,6,1,5]=>4
[4,3,2,6,5,1]=>5
[4,3,5,1,2,6]=>3
[4,3,5,1,6,2]=>4
[4,3,5,2,1,6]=>4
[4,3,5,2,6,1]=>4
[4,3,5,6,1,2]=>3
[4,3,5,6,2,1]=>4
[4,3,6,1,2,5]=>3
[4,3,6,1,5,2]=>4
[4,3,6,2,1,5]=>4
[4,3,6,2,5,1]=>4
[4,3,6,5,1,2]=>4
[4,3,6,5,2,1]=>5
[4,5,1,2,3,6]=>2
[4,5,1,2,6,3]=>3
[4,5,1,3,2,6]=>3
[4,5,1,3,6,2]=>3
[4,5,1,6,2,3]=>3
[4,5,1,6,3,2]=>4
[4,5,2,1,3,6]=>3
[4,5,2,1,6,3]=>4
[4,5,2,3,1,6]=>3
[4,5,2,3,6,1]=>3
[4,5,2,6,1,3]=>3
[4,5,2,6,3,1]=>4
[4,5,3,1,2,6]=>3
[4,5,3,1,6,2]=>4
[4,5,3,2,1,6]=>4
[4,5,3,2,6,1]=>4
[4,5,3,6,1,2]=>3
[4,5,3,6,2,1]=>4
[4,5,6,1,2,3]=>2
[4,5,6,1,3,2]=>3
[4,5,6,2,1,3]=>3
[4,5,6,2,3,1]=>3
[4,5,6,3,1,2]=>3
[4,5,6,3,2,1]=>4
[4,6,1,2,3,5]=>2
[4,6,1,2,5,3]=>3
[4,6,1,3,2,5]=>3
[4,6,1,3,5,2]=>3
[4,6,1,5,2,3]=>3
[4,6,1,5,3,2]=>4
[4,6,2,1,3,5]=>3
[4,6,2,1,5,3]=>4
[4,6,2,3,1,5]=>3
[4,6,2,3,5,1]=>3
[4,6,2,5,1,3]=>3
[4,6,2,5,3,1]=>4
[4,6,3,1,2,5]=>3
[4,6,3,1,5,2]=>4
[4,6,3,2,1,5]=>4
[4,6,3,2,5,1]=>4
[4,6,3,5,1,2]=>3
[4,6,3,5,2,1]=>4
[4,6,5,1,2,3]=>3
[4,6,5,1,3,2]=>4
[4,6,5,2,1,3]=>4
[4,6,5,2,3,1]=>4
[4,6,5,3,1,2]=>4
[4,6,5,3,2,1]=>5
[5,1,2,3,4,6]=>2
[5,1,2,3,6,4]=>3
[5,1,2,4,3,6]=>3
[5,1,2,4,6,3]=>3
[5,1,2,6,3,4]=>3
[5,1,2,6,4,3]=>4
[5,1,3,2,4,6]=>3
[5,1,3,2,6,4]=>4
[5,1,3,4,2,6]=>3
[5,1,3,4,6,2]=>3
[5,1,3,6,2,4]=>3
[5,1,3,6,4,2]=>4
[5,1,4,2,3,6]=>3
[5,1,4,2,6,3]=>4
[5,1,4,3,2,6]=>4
[5,1,4,3,6,2]=>4
[5,1,4,6,2,3]=>3
[5,1,4,6,3,2]=>4
[5,1,6,2,3,4]=>3
[5,1,6,2,4,3]=>4
[5,1,6,3,2,4]=>4
[5,1,6,3,4,2]=>4
[5,1,6,4,2,3]=>4
[5,1,6,4,3,2]=>5
[5,2,1,3,4,6]=>3
[5,2,1,3,6,4]=>4
[5,2,1,4,3,6]=>4
[5,2,1,4,6,3]=>4
[5,2,1,6,3,4]=>4
[5,2,1,6,4,3]=>5
[5,2,3,1,4,6]=>3
[5,2,3,1,6,4]=>4
[5,2,3,4,1,6]=>3
[5,2,3,4,6,1]=>3
[5,2,3,6,1,4]=>3
[5,2,3,6,4,1]=>4
[5,2,4,1,3,6]=>3
[5,2,4,1,6,3]=>4
[5,2,4,3,1,6]=>4
[5,2,4,3,6,1]=>4
[5,2,4,6,1,3]=>3
[5,2,4,6,3,1]=>4
[5,2,6,1,3,4]=>3
[5,2,6,1,4,3]=>4
[5,2,6,3,1,4]=>4
[5,2,6,3,4,1]=>4
[5,2,6,4,1,3]=>4
[5,2,6,4,3,1]=>5
[5,3,1,2,4,6]=>3
[5,3,1,2,6,4]=>4
[5,3,1,4,2,6]=>4
[5,3,1,4,6,2]=>4
[5,3,1,6,2,4]=>4
[5,3,1,6,4,2]=>5
[5,3,2,1,4,6]=>4
[5,3,2,1,6,4]=>5
[5,3,2,4,1,6]=>4
[5,3,2,4,6,1]=>4
[5,3,2,6,1,4]=>4
[5,3,2,6,4,1]=>5
[5,3,4,1,2,6]=>3
[5,3,4,1,6,2]=>4
[5,3,4,2,1,6]=>4
[5,3,4,2,6,1]=>4
[5,3,4,6,1,2]=>3
[5,3,4,6,2,1]=>4
[5,3,6,1,2,4]=>3
[5,3,6,1,4,2]=>4
[5,3,6,2,1,4]=>4
[5,3,6,2,4,1]=>4
[5,3,6,4,1,2]=>4
[5,3,6,4,2,1]=>5
[5,4,1,2,3,6]=>3
[5,4,1,2,6,3]=>4
[5,4,1,3,2,6]=>4
[5,4,1,3,6,2]=>4
[5,4,1,6,2,3]=>4
[5,4,1,6,3,2]=>5
[5,4,2,1,3,6]=>4
[5,4,2,1,6,3]=>5
[5,4,2,3,1,6]=>4
[5,4,2,3,6,1]=>4
[5,4,2,6,1,3]=>4
[5,4,2,6,3,1]=>5
[5,4,3,1,2,6]=>4
[5,4,3,1,6,2]=>5
[5,4,3,2,1,6]=>5
[5,4,3,2,6,1]=>5
[5,4,3,6,1,2]=>4
[5,4,3,6,2,1]=>5
[5,4,6,1,2,3]=>3
[5,4,6,1,3,2]=>4
[5,4,6,2,1,3]=>4
[5,4,6,2,3,1]=>4
[5,4,6,3,1,2]=>4
[5,4,6,3,2,1]=>5
[5,6,1,2,3,4]=>2
[5,6,1,2,4,3]=>3
[5,6,1,3,2,4]=>3
[5,6,1,3,4,2]=>3
[5,6,1,4,2,3]=>3
[5,6,1,4,3,2]=>4
[5,6,2,1,3,4]=>3
[5,6,2,1,4,3]=>4
[5,6,2,3,1,4]=>3
[5,6,2,3,4,1]=>3
[5,6,2,4,1,3]=>3
[5,6,2,4,3,1]=>4
[5,6,3,1,2,4]=>3
[5,6,3,1,4,2]=>4
[5,6,3,2,1,4]=>4
[5,6,3,2,4,1]=>4
[5,6,3,4,1,2]=>3
[5,6,3,4,2,1]=>4
[5,6,4,1,2,3]=>3
[5,6,4,1,3,2]=>4
[5,6,4,2,1,3]=>4
[5,6,4,2,3,1]=>4
[5,6,4,3,1,2]=>4
[5,6,4,3,2,1]=>5
[6,1,2,3,4,5]=>2
[6,1,2,3,5,4]=>3
[6,1,2,4,3,5]=>3
[6,1,2,4,5,3]=>3
[6,1,2,5,3,4]=>3
[6,1,2,5,4,3]=>4
[6,1,3,2,4,5]=>3
[6,1,3,2,5,4]=>4
[6,1,3,4,2,5]=>3
[6,1,3,4,5,2]=>3
[6,1,3,5,2,4]=>3
[6,1,3,5,4,2]=>4
[6,1,4,2,3,5]=>3
[6,1,4,2,5,3]=>4
[6,1,4,3,2,5]=>4
[6,1,4,3,5,2]=>4
[6,1,4,5,2,3]=>3
[6,1,4,5,3,2]=>4
[6,1,5,2,3,4]=>3
[6,1,5,2,4,3]=>4
[6,1,5,3,2,4]=>4
[6,1,5,3,4,2]=>4
[6,1,5,4,2,3]=>4
[6,1,5,4,3,2]=>5
[6,2,1,3,4,5]=>3
[6,2,1,3,5,4]=>4
[6,2,1,4,3,5]=>4
[6,2,1,4,5,3]=>4
[6,2,1,5,3,4]=>4
[6,2,1,5,4,3]=>5
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>4
[6,2,3,4,1,5]=>3
[6,2,3,4,5,1]=>3
[6,2,3,5,1,4]=>3
[6,2,3,5,4,1]=>4
[6,2,4,1,3,5]=>3
[6,2,4,1,5,3]=>4
[6,2,4,3,1,5]=>4
[6,2,4,3,5,1]=>4
[6,2,4,5,1,3]=>3
[6,2,4,5,3,1]=>4
[6,2,5,1,3,4]=>3
[6,2,5,1,4,3]=>4
[6,2,5,3,1,4]=>4
[6,2,5,3,4,1]=>4
[6,2,5,4,1,3]=>4
[6,2,5,4,3,1]=>5
[6,3,1,2,4,5]=>3
[6,3,1,2,5,4]=>4
[6,3,1,4,2,5]=>4
[6,3,1,4,5,2]=>4
[6,3,1,5,2,4]=>4
[6,3,1,5,4,2]=>5
[6,3,2,1,4,5]=>4
[6,3,2,1,5,4]=>5
[6,3,2,4,1,5]=>4
[6,3,2,4,5,1]=>4
[6,3,2,5,1,4]=>4
[6,3,2,5,4,1]=>5
[6,3,4,1,2,5]=>3
[6,3,4,1,5,2]=>4
[6,3,4,2,1,5]=>4
[6,3,4,2,5,1]=>4
[6,3,4,5,1,2]=>3
[6,3,4,5,2,1]=>4
[6,3,5,1,2,4]=>3
[6,3,5,1,4,2]=>4
[6,3,5,2,1,4]=>4
[6,3,5,2,4,1]=>4
[6,3,5,4,1,2]=>4
[6,3,5,4,2,1]=>5
[6,4,1,2,3,5]=>3
[6,4,1,2,5,3]=>4
[6,4,1,3,2,5]=>4
[6,4,1,3,5,2]=>4
[6,4,1,5,2,3]=>4
[6,4,1,5,3,2]=>5
[6,4,2,1,3,5]=>4
[6,4,2,1,5,3]=>5
[6,4,2,3,1,5]=>4
[6,4,2,3,5,1]=>4
[6,4,2,5,1,3]=>4
[6,4,2,5,3,1]=>5
[6,4,3,1,2,5]=>4
[6,4,3,1,5,2]=>5
[6,4,3,2,1,5]=>5
[6,4,3,2,5,1]=>5
[6,4,3,5,1,2]=>4
[6,4,3,5,2,1]=>5
[6,4,5,1,2,3]=>3
[6,4,5,1,3,2]=>4
[6,4,5,2,1,3]=>4
[6,4,5,2,3,1]=>4
[6,4,5,3,1,2]=>4
[6,4,5,3,2,1]=>5
[6,5,1,2,3,4]=>3
[6,5,1,2,4,3]=>4
[6,5,1,3,2,4]=>4
[6,5,1,3,4,2]=>4
[6,5,1,4,2,3]=>4
[6,5,1,4,3,2]=>5
[6,5,2,1,3,4]=>4
[6,5,2,1,4,3]=>5
[6,5,2,3,1,4]=>4
[6,5,2,3,4,1]=>4
[6,5,2,4,1,3]=>4
[6,5,2,4,3,1]=>5
[6,5,3,1,2,4]=>4
[6,5,3,1,4,2]=>5
[6,5,3,2,1,4]=>5
[6,5,3,2,4,1]=>5
[6,5,3,4,1,2]=>4
[6,5,3,4,2,1]=>5
[6,5,4,1,2,3]=>4
[6,5,4,1,3,2]=>5
[6,5,4,2,1,3]=>5
[6,5,4,2,3,1]=>5
[6,5,4,3,1,2]=>5
[6,5,4,3,2,1]=>6
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Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents St000021The number of descents of a permutation.. This also matches the number of runs in a permutation St000470The number of runs in a permutation..
See also St000308The height of the tree associated to a permutation. for the height of this tree.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents St000021The number of descents of a permutation.. This also matches the number of runs in a permutation St000470The number of runs in a permutation..
See also St000308The height of the tree associated to a permutation. for the height of this tree.
References
[1] Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows. OEIS:A123125
[2] http://oeis.org/wiki/User:Peter_Luschny/PermutationTrees
[2] http://oeis.org/wiki/User:Peter_Luschny/PermutationTrees
Code
def statistic(pi):
if pi == []: return 0
w, i, branch, next = 1, 0, [0], pi[0]
while true:
while next < branch[len(branch)-1]:
branch.pop()
current = 0
while next > current:
i += 1
if i == len(pi): return w
branch.append(next)
current, next = next, pi[i]
w += 1
Created
Dec 11, 2015 at 11:05 by Peter Luschny
Updated
Mar 14, 2023 at 13:31 by Tilman Möller
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