Identifier
- St000056: Permutations ⟶ ℤ (values match St000234The number of global ascents of a permutation.)
Values
=>
[1]=>1
[1,2]=>2
[2,1]=>1
[1,2,3]=>3
[1,3,2]=>2
[2,1,3]=>2
[2,3,1]=>1
[3,1,2]=>1
[3,2,1]=>1
[1,2,3,4]=>4
[1,2,4,3]=>3
[1,3,2,4]=>3
[1,3,4,2]=>2
[1,4,2,3]=>2
[1,4,3,2]=>2
[2,1,3,4]=>3
[2,1,4,3]=>2
[2,3,1,4]=>2
[2,3,4,1]=>1
[2,4,1,3]=>1
[2,4,3,1]=>1
[3,1,2,4]=>2
[3,1,4,2]=>1
[3,2,1,4]=>2
[3,2,4,1]=>1
[3,4,1,2]=>1
[3,4,2,1]=>1
[4,1,2,3]=>1
[4,1,3,2]=>1
[4,2,1,3]=>1
[4,2,3,1]=>1
[4,3,1,2]=>1
[4,3,2,1]=>1
[1,2,3,4,5]=>5
[1,2,3,5,4]=>4
[1,2,4,3,5]=>4
[1,2,4,5,3]=>3
[1,2,5,3,4]=>3
[1,2,5,4,3]=>3
[1,3,2,4,5]=>4
[1,3,2,5,4]=>3
[1,3,4,2,5]=>3
[1,3,4,5,2]=>2
[1,3,5,2,4]=>2
[1,3,5,4,2]=>2
[1,4,2,3,5]=>3
[1,4,2,5,3]=>2
[1,4,3,2,5]=>3
[1,4,3,5,2]=>2
[1,4,5,2,3]=>2
[1,4,5,3,2]=>2
[1,5,2,3,4]=>2
[1,5,2,4,3]=>2
[1,5,3,2,4]=>2
[1,5,3,4,2]=>2
[1,5,4,2,3]=>2
[1,5,4,3,2]=>2
[2,1,3,4,5]=>4
[2,1,3,5,4]=>3
[2,1,4,3,5]=>3
[2,1,4,5,3]=>2
[2,1,5,3,4]=>2
[2,1,5,4,3]=>2
[2,3,1,4,5]=>3
[2,3,1,5,4]=>2
[2,3,4,1,5]=>2
[2,3,4,5,1]=>1
[2,3,5,1,4]=>1
[2,3,5,4,1]=>1
[2,4,1,3,5]=>2
[2,4,1,5,3]=>1
[2,4,3,1,5]=>2
[2,4,3,5,1]=>1
[2,4,5,1,3]=>1
[2,4,5,3,1]=>1
[2,5,1,3,4]=>1
[2,5,1,4,3]=>1
[2,5,3,1,4]=>1
[2,5,3,4,1]=>1
[2,5,4,1,3]=>1
[2,5,4,3,1]=>1
[3,1,2,4,5]=>3
[3,1,2,5,4]=>2
[3,1,4,2,5]=>2
[3,1,4,5,2]=>1
[3,1,5,2,4]=>1
[3,1,5,4,2]=>1
[3,2,1,4,5]=>3
[3,2,1,5,4]=>2
[3,2,4,1,5]=>2
[3,2,4,5,1]=>1
[3,2,5,1,4]=>1
[3,2,5,4,1]=>1
[3,4,1,2,5]=>2
[3,4,1,5,2]=>1
[3,4,2,1,5]=>2
[3,4,2,5,1]=>1
[3,4,5,1,2]=>1
[3,4,5,2,1]=>1
[3,5,1,2,4]=>1
[3,5,1,4,2]=>1
[3,5,2,1,4]=>1
[3,5,2,4,1]=>1
[3,5,4,1,2]=>1
[3,5,4,2,1]=>1
[4,1,2,3,5]=>2
[4,1,2,5,3]=>1
[4,1,3,2,5]=>2
[4,1,3,5,2]=>1
[4,1,5,2,3]=>1
[4,1,5,3,2]=>1
[4,2,1,3,5]=>2
[4,2,1,5,3]=>1
[4,2,3,1,5]=>2
[4,2,3,5,1]=>1
[4,2,5,1,3]=>1
[4,2,5,3,1]=>1
[4,3,1,2,5]=>2
[4,3,1,5,2]=>1
[4,3,2,1,5]=>2
[4,3,2,5,1]=>1
[4,3,5,1,2]=>1
[4,3,5,2,1]=>1
[4,5,1,2,3]=>1
[4,5,1,3,2]=>1
[4,5,2,1,3]=>1
[4,5,2,3,1]=>1
[4,5,3,1,2]=>1
[4,5,3,2,1]=>1
[5,1,2,3,4]=>1
[5,1,2,4,3]=>1
[5,1,3,2,4]=>1
[5,1,3,4,2]=>1
[5,1,4,2,3]=>1
[5,1,4,3,2]=>1
[5,2,1,3,4]=>1
[5,2,1,4,3]=>1
[5,2,3,1,4]=>1
[5,2,3,4,1]=>1
[5,2,4,1,3]=>1
[5,2,4,3,1]=>1
[5,3,1,2,4]=>1
[5,3,1,4,2]=>1
[5,3,2,1,4]=>1
[5,3,2,4,1]=>1
[5,3,4,1,2]=>1
[5,3,4,2,1]=>1
[5,4,1,2,3]=>1
[5,4,1,3,2]=>1
[5,4,2,1,3]=>1
[5,4,2,3,1]=>1
[5,4,3,1,2]=>1
[5,4,3,2,1]=>1
[1,2,3,4,5,6]=>6
[1,2,3,4,6,5]=>5
[1,2,3,5,4,6]=>5
[1,2,3,5,6,4]=>4
[1,2,3,6,4,5]=>4
[1,2,3,6,5,4]=>4
[1,2,4,3,5,6]=>5
[1,2,4,3,6,5]=>4
[1,2,4,5,3,6]=>4
[1,2,4,5,6,3]=>3
[1,2,4,6,3,5]=>3
[1,2,4,6,5,3]=>3
[1,2,5,3,4,6]=>4
[1,2,5,3,6,4]=>3
[1,2,5,4,3,6]=>4
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>3
[1,2,5,6,4,3]=>3
[1,2,6,3,4,5]=>3
[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]=>3
[1,3,2,4,5,6]=>5
[1,3,2,4,6,5]=>4
[1,3,2,5,4,6]=>4
[1,3,2,5,6,4]=>3
[1,3,2,6,4,5]=>3
[1,3,2,6,5,4]=>3
[1,3,4,2,5,6]=>4
[1,3,4,2,6,5]=>3
[1,3,4,5,2,6]=>3
[1,3,4,5,6,2]=>2
[1,3,4,6,2,5]=>2
[1,3,4,6,5,2]=>2
[1,3,5,2,4,6]=>3
[1,3,5,2,6,4]=>2
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>2
[1,3,5,6,2,4]=>2
[1,3,5,6,4,2]=>2
[1,3,6,2,4,5]=>2
[1,3,6,2,5,4]=>2
[1,3,6,4,2,5]=>2
[1,3,6,4,5,2]=>2
[1,3,6,5,2,4]=>2
[1,3,6,5,4,2]=>2
[1,4,2,3,5,6]=>4
[1,4,2,3,6,5]=>3
[1,4,2,5,3,6]=>3
[1,4,2,5,6,3]=>2
[1,4,2,6,3,5]=>2
[1,4,2,6,5,3]=>2
[1,4,3,2,5,6]=>4
[1,4,3,2,6,5]=>3
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>2
[1,4,3,6,2,5]=>2
[1,4,3,6,5,2]=>2
[1,4,5,2,3,6]=>3
[1,4,5,2,6,3]=>2
[1,4,5,3,2,6]=>3
[1,4,5,3,6,2]=>2
[1,4,5,6,2,3]=>2
[1,4,5,6,3,2]=>2
[1,4,6,2,3,5]=>2
[1,4,6,2,5,3]=>2
[1,4,6,3,2,5]=>2
[1,4,6,3,5,2]=>2
[1,4,6,5,2,3]=>2
[1,4,6,5,3,2]=>2
[1,5,2,3,4,6]=>3
[1,5,2,3,6,4]=>2
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>2
[1,5,2,6,3,4]=>2
[1,5,2,6,4,3]=>2
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>2
[1,5,3,4,2,6]=>3
[1,5,3,4,6,2]=>2
[1,5,3,6,2,4]=>2
[1,5,3,6,4,2]=>2
[1,5,4,2,3,6]=>3
[1,5,4,2,6,3]=>2
[1,5,4,3,2,6]=>3
[1,5,4,3,6,2]=>2
[1,5,4,6,2,3]=>2
[1,5,4,6,3,2]=>2
[1,5,6,2,3,4]=>2
[1,5,6,2,4,3]=>2
[1,5,6,3,2,4]=>2
[1,5,6,3,4,2]=>2
[1,5,6,4,2,3]=>2
[1,5,6,4,3,2]=>2
[1,6,2,3,4,5]=>2
[1,6,2,3,5,4]=>2
[1,6,2,4,3,5]=>2
[1,6,2,4,5,3]=>2
[1,6,2,5,3,4]=>2
[1,6,2,5,4,3]=>2
[1,6,3,2,4,5]=>2
[1,6,3,2,5,4]=>2
[1,6,3,4,2,5]=>2
[1,6,3,4,5,2]=>2
[1,6,3,5,2,4]=>2
[1,6,3,5,4,2]=>2
[1,6,4,2,3,5]=>2
[1,6,4,2,5,3]=>2
[1,6,4,3,2,5]=>2
[1,6,4,3,5,2]=>2
[1,6,4,5,2,3]=>2
[1,6,4,5,3,2]=>2
[1,6,5,2,3,4]=>2
[1,6,5,2,4,3]=>2
[1,6,5,3,2,4]=>2
[1,6,5,3,4,2]=>2
[1,6,5,4,2,3]=>2
[1,6,5,4,3,2]=>2
[2,1,3,4,5,6]=>5
[2,1,3,4,6,5]=>4
[2,1,3,5,4,6]=>4
[2,1,3,5,6,4]=>3
[2,1,3,6,4,5]=>3
[2,1,3,6,5,4]=>3
[2,1,4,3,5,6]=>4
[2,1,4,3,6,5]=>3
[2,1,4,5,3,6]=>3
[2,1,4,5,6,3]=>2
[2,1,4,6,3,5]=>2
[2,1,4,6,5,3]=>2
[2,1,5,3,4,6]=>3
[2,1,5,3,6,4]=>2
[2,1,5,4,3,6]=>3
[2,1,5,4,6,3]=>2
[2,1,5,6,3,4]=>2
[2,1,5,6,4,3]=>2
[2,1,6,3,4,5]=>2
[2,1,6,3,5,4]=>2
[2,1,6,4,3,5]=>2
[2,1,6,4,5,3]=>2
[2,1,6,5,3,4]=>2
[2,1,6,5,4,3]=>2
[2,3,1,4,5,6]=>4
[2,3,1,4,6,5]=>3
[2,3,1,5,4,6]=>3
[2,3,1,5,6,4]=>2
[2,3,1,6,4,5]=>2
[2,3,1,6,5,4]=>2
[2,3,4,1,5,6]=>3
[2,3,4,1,6,5]=>2
[2,3,4,5,1,6]=>2
[2,3,4,5,6,1]=>1
[2,3,4,6,1,5]=>1
[2,3,4,6,5,1]=>1
[2,3,5,1,4,6]=>2
[2,3,5,1,6,4]=>1
[2,3,5,4,1,6]=>2
[2,3,5,4,6,1]=>1
[2,3,5,6,1,4]=>1
[2,3,5,6,4,1]=>1
[2,3,6,1,4,5]=>1
[2,3,6,1,5,4]=>1
[2,3,6,4,1,5]=>1
[2,3,6,4,5,1]=>1
[2,3,6,5,1,4]=>1
[2,3,6,5,4,1]=>1
[2,4,1,3,5,6]=>3
[2,4,1,3,6,5]=>2
[2,4,1,5,3,6]=>2
[2,4,1,5,6,3]=>1
[2,4,1,6,3,5]=>1
[2,4,1,6,5,3]=>1
[2,4,3,1,5,6]=>3
[2,4,3,1,6,5]=>2
[2,4,3,5,1,6]=>2
[2,4,3,5,6,1]=>1
[2,4,3,6,1,5]=>1
[2,4,3,6,5,1]=>1
[2,4,5,1,3,6]=>2
[2,4,5,1,6,3]=>1
[2,4,5,3,1,6]=>2
[2,4,5,3,6,1]=>1
[2,4,5,6,1,3]=>1
[2,4,5,6,3,1]=>1
[2,4,6,1,3,5]=>1
[2,4,6,1,5,3]=>1
[2,4,6,3,1,5]=>1
[2,4,6,3,5,1]=>1
[2,4,6,5,1,3]=>1
[2,4,6,5,3,1]=>1
[2,5,1,3,4,6]=>2
[2,5,1,3,6,4]=>1
[2,5,1,4,3,6]=>2
[2,5,1,4,6,3]=>1
[2,5,1,6,3,4]=>1
[2,5,1,6,4,3]=>1
[2,5,3,1,4,6]=>2
[2,5,3,1,6,4]=>1
[2,5,3,4,1,6]=>2
[2,5,3,4,6,1]=>1
[2,5,3,6,1,4]=>1
[2,5,3,6,4,1]=>1
[2,5,4,1,3,6]=>2
[2,5,4,1,6,3]=>1
[2,5,4,3,1,6]=>2
[2,5,4,3,6,1]=>1
[2,5,4,6,1,3]=>1
[2,5,4,6,3,1]=>1
[2,5,6,1,3,4]=>1
[2,5,6,1,4,3]=>1
[2,5,6,3,1,4]=>1
[2,5,6,3,4,1]=>1
[2,5,6,4,1,3]=>1
[2,5,6,4,3,1]=>1
[2,6,1,3,4,5]=>1
[2,6,1,3,5,4]=>1
[2,6,1,4,3,5]=>1
[2,6,1,4,5,3]=>1
[2,6,1,5,3,4]=>1
[2,6,1,5,4,3]=>1
[2,6,3,1,4,5]=>1
[2,6,3,1,5,4]=>1
[2,6,3,4,1,5]=>1
[2,6,3,4,5,1]=>1
[2,6,3,5,1,4]=>1
[2,6,3,5,4,1]=>1
[2,6,4,1,3,5]=>1
[2,6,4,1,5,3]=>1
[2,6,4,3,1,5]=>1
[2,6,4,3,5,1]=>1
[2,6,4,5,1,3]=>1
[2,6,4,5,3,1]=>1
[2,6,5,1,3,4]=>1
[2,6,5,1,4,3]=>1
[2,6,5,3,1,4]=>1
[2,6,5,3,4,1]=>1
[2,6,5,4,1,3]=>1
[2,6,5,4,3,1]=>1
[3,1,2,4,5,6]=>4
[3,1,2,4,6,5]=>3
[3,1,2,5,4,6]=>3
[3,1,2,5,6,4]=>2
[3,1,2,6,4,5]=>2
[3,1,2,6,5,4]=>2
[3,1,4,2,5,6]=>3
[3,1,4,2,6,5]=>2
[3,1,4,5,2,6]=>2
[3,1,4,5,6,2]=>1
[3,1,4,6,2,5]=>1
[3,1,4,6,5,2]=>1
[3,1,5,2,4,6]=>2
[3,1,5,2,6,4]=>1
[3,1,5,4,2,6]=>2
[3,1,5,4,6,2]=>1
[3,1,5,6,2,4]=>1
[3,1,5,6,4,2]=>1
[3,1,6,2,4,5]=>1
[3,1,6,2,5,4]=>1
[3,1,6,4,2,5]=>1
[3,1,6,4,5,2]=>1
[3,1,6,5,2,4]=>1
[3,1,6,5,4,2]=>1
[3,2,1,4,5,6]=>4
[3,2,1,4,6,5]=>3
[3,2,1,5,4,6]=>3
[3,2,1,5,6,4]=>2
[3,2,1,6,4,5]=>2
[3,2,1,6,5,4]=>2
[3,2,4,1,5,6]=>3
[3,2,4,1,6,5]=>2
[3,2,4,5,1,6]=>2
[3,2,4,5,6,1]=>1
[3,2,4,6,1,5]=>1
[3,2,4,6,5,1]=>1
[3,2,5,1,4,6]=>2
[3,2,5,1,6,4]=>1
[3,2,5,4,1,6]=>2
[3,2,5,4,6,1]=>1
[3,2,5,6,1,4]=>1
[3,2,5,6,4,1]=>1
[3,2,6,1,4,5]=>1
[3,2,6,1,5,4]=>1
[3,2,6,4,1,5]=>1
[3,2,6,4,5,1]=>1
[3,2,6,5,1,4]=>1
[3,2,6,5,4,1]=>1
[3,4,1,2,5,6]=>3
[3,4,1,2,6,5]=>2
[3,4,1,5,2,6]=>2
[3,4,1,5,6,2]=>1
[3,4,1,6,2,5]=>1
[3,4,1,6,5,2]=>1
[3,4,2,1,5,6]=>3
[3,4,2,1,6,5]=>2
[3,4,2,5,1,6]=>2
[3,4,2,5,6,1]=>1
[3,4,2,6,1,5]=>1
[3,4,2,6,5,1]=>1
[3,4,5,1,2,6]=>2
[3,4,5,1,6,2]=>1
[3,4,5,2,1,6]=>2
[3,4,5,2,6,1]=>1
[3,4,5,6,1,2]=>1
[3,4,5,6,2,1]=>1
[3,4,6,1,2,5]=>1
[3,4,6,1,5,2]=>1
[3,4,6,2,1,5]=>1
[3,4,6,2,5,1]=>1
[3,4,6,5,1,2]=>1
[3,4,6,5,2,1]=>1
[3,5,1,2,4,6]=>2
[3,5,1,2,6,4]=>1
[3,5,1,4,2,6]=>2
[3,5,1,4,6,2]=>1
[3,5,1,6,2,4]=>1
[3,5,1,6,4,2]=>1
[3,5,2,1,4,6]=>2
[3,5,2,1,6,4]=>1
[3,5,2,4,1,6]=>2
[3,5,2,4,6,1]=>1
[3,5,2,6,1,4]=>1
[3,5,2,6,4,1]=>1
[3,5,4,1,2,6]=>2
[3,5,4,1,6,2]=>1
[3,5,4,2,1,6]=>2
[3,5,4,2,6,1]=>1
[3,5,4,6,1,2]=>1
[3,5,4,6,2,1]=>1
[3,5,6,1,2,4]=>1
[3,5,6,1,4,2]=>1
[3,5,6,2,1,4]=>1
[3,5,6,2,4,1]=>1
[3,5,6,4,1,2]=>1
[3,5,6,4,2,1]=>1
[3,6,1,2,4,5]=>1
[3,6,1,2,5,4]=>1
[3,6,1,4,2,5]=>1
[3,6,1,4,5,2]=>1
[3,6,1,5,2,4]=>1
[3,6,1,5,4,2]=>1
[3,6,2,1,4,5]=>1
[3,6,2,1,5,4]=>1
[3,6,2,4,1,5]=>1
[3,6,2,4,5,1]=>1
[3,6,2,5,1,4]=>1
[3,6,2,5,4,1]=>1
[3,6,4,1,2,5]=>1
[3,6,4,1,5,2]=>1
[3,6,4,2,1,5]=>1
[3,6,4,2,5,1]=>1
[3,6,4,5,1,2]=>1
[3,6,4,5,2,1]=>1
[3,6,5,1,2,4]=>1
[3,6,5,1,4,2]=>1
[3,6,5,2,1,4]=>1
[3,6,5,2,4,1]=>1
[3,6,5,4,1,2]=>1
[3,6,5,4,2,1]=>1
[4,1,2,3,5,6]=>3
[4,1,2,3,6,5]=>2
[4,1,2,5,3,6]=>2
[4,1,2,5,6,3]=>1
[4,1,2,6,3,5]=>1
[4,1,2,6,5,3]=>1
[4,1,3,2,5,6]=>3
[4,1,3,2,6,5]=>2
[4,1,3,5,2,6]=>2
[4,1,3,5,6,2]=>1
[4,1,3,6,2,5]=>1
[4,1,3,6,5,2]=>1
[4,1,5,2,3,6]=>2
[4,1,5,2,6,3]=>1
[4,1,5,3,2,6]=>2
[4,1,5,3,6,2]=>1
[4,1,5,6,2,3]=>1
[4,1,5,6,3,2]=>1
[4,1,6,2,3,5]=>1
[4,1,6,2,5,3]=>1
[4,1,6,3,2,5]=>1
[4,1,6,3,5,2]=>1
[4,1,6,5,2,3]=>1
[4,1,6,5,3,2]=>1
[4,2,1,3,5,6]=>3
[4,2,1,3,6,5]=>2
[4,2,1,5,3,6]=>2
[4,2,1,5,6,3]=>1
[4,2,1,6,3,5]=>1
[4,2,1,6,5,3]=>1
[4,2,3,1,5,6]=>3
[4,2,3,1,6,5]=>2
[4,2,3,5,1,6]=>2
[4,2,3,5,6,1]=>1
[4,2,3,6,1,5]=>1
[4,2,3,6,5,1]=>1
[4,2,5,1,3,6]=>2
[4,2,5,1,6,3]=>1
[4,2,5,3,1,6]=>2
[4,2,5,3,6,1]=>1
[4,2,5,6,1,3]=>1
[4,2,5,6,3,1]=>1
[4,2,6,1,3,5]=>1
[4,2,6,1,5,3]=>1
[4,2,6,3,1,5]=>1
[4,2,6,3,5,1]=>1
[4,2,6,5,1,3]=>1
[4,2,6,5,3,1]=>1
[4,3,1,2,5,6]=>3
[4,3,1,2,6,5]=>2
[4,3,1,5,2,6]=>2
[4,3,1,5,6,2]=>1
[4,3,1,6,2,5]=>1
[4,3,1,6,5,2]=>1
[4,3,2,1,5,6]=>3
[4,3,2,1,6,5]=>2
[4,3,2,5,1,6]=>2
[4,3,2,5,6,1]=>1
[4,3,2,6,1,5]=>1
[4,3,2,6,5,1]=>1
[4,3,5,1,2,6]=>2
[4,3,5,1,6,2]=>1
[4,3,5,2,1,6]=>2
[4,3,5,2,6,1]=>1
[4,3,5,6,1,2]=>1
[4,3,5,6,2,1]=>1
[4,3,6,1,2,5]=>1
[4,3,6,1,5,2]=>1
[4,3,6,2,1,5]=>1
[4,3,6,2,5,1]=>1
[4,3,6,5,1,2]=>1
[4,3,6,5,2,1]=>1
[4,5,1,2,3,6]=>2
[4,5,1,2,6,3]=>1
[4,5,1,3,2,6]=>2
[4,5,1,3,6,2]=>1
[4,5,1,6,2,3]=>1
[4,5,1,6,3,2]=>1
[4,5,2,1,3,6]=>2
[4,5,2,1,6,3]=>1
[4,5,2,3,1,6]=>2
[4,5,2,3,6,1]=>1
[4,5,2,6,1,3]=>1
[4,5,2,6,3,1]=>1
[4,5,3,1,2,6]=>2
[4,5,3,1,6,2]=>1
[4,5,3,2,1,6]=>2
[4,5,3,2,6,1]=>1
[4,5,3,6,1,2]=>1
[4,5,3,6,2,1]=>1
[4,5,6,1,2,3]=>1
[4,5,6,1,3,2]=>1
[4,5,6,2,1,3]=>1
[4,5,6,2,3,1]=>1
[4,5,6,3,1,2]=>1
[4,5,6,3,2,1]=>1
[4,6,1,2,3,5]=>1
[4,6,1,2,5,3]=>1
[4,6,1,3,2,5]=>1
[4,6,1,3,5,2]=>1
[4,6,1,5,2,3]=>1
[4,6,1,5,3,2]=>1
[4,6,2,1,3,5]=>1
[4,6,2,1,5,3]=>1
[4,6,2,3,1,5]=>1
[4,6,2,3,5,1]=>1
[4,6,2,5,1,3]=>1
[4,6,2,5,3,1]=>1
[4,6,3,1,2,5]=>1
[4,6,3,1,5,2]=>1
[4,6,3,2,1,5]=>1
[4,6,3,2,5,1]=>1
[4,6,3,5,1,2]=>1
[4,6,3,5,2,1]=>1
[4,6,5,1,2,3]=>1
[4,6,5,1,3,2]=>1
[4,6,5,2,1,3]=>1
[4,6,5,2,3,1]=>1
[4,6,5,3,1,2]=>1
[4,6,5,3,2,1]=>1
[5,1,2,3,4,6]=>2
[5,1,2,3,6,4]=>1
[5,1,2,4,3,6]=>2
[5,1,2,4,6,3]=>1
[5,1,2,6,3,4]=>1
[5,1,2,6,4,3]=>1
[5,1,3,2,4,6]=>2
[5,1,3,2,6,4]=>1
[5,1,3,4,2,6]=>2
[5,1,3,4,6,2]=>1
[5,1,3,6,2,4]=>1
[5,1,3,6,4,2]=>1
[5,1,4,2,3,6]=>2
[5,1,4,2,6,3]=>1
[5,1,4,3,2,6]=>2
[5,1,4,3,6,2]=>1
[5,1,4,6,2,3]=>1
[5,1,4,6,3,2]=>1
[5,1,6,2,3,4]=>1
[5,1,6,2,4,3]=>1
[5,1,6,3,2,4]=>1
[5,1,6,3,4,2]=>1
[5,1,6,4,2,3]=>1
[5,1,6,4,3,2]=>1
[5,2,1,3,4,6]=>2
[5,2,1,3,6,4]=>1
[5,2,1,4,3,6]=>2
[5,2,1,4,6,3]=>1
[5,2,1,6,3,4]=>1
[5,2,1,6,4,3]=>1
[5,2,3,1,4,6]=>2
[5,2,3,1,6,4]=>1
[5,2,3,4,1,6]=>2
[5,2,3,4,6,1]=>1
[5,2,3,6,1,4]=>1
[5,2,3,6,4,1]=>1
[5,2,4,1,3,6]=>2
[5,2,4,1,6,3]=>1
[5,2,4,3,1,6]=>2
[5,2,4,3,6,1]=>1
[5,2,4,6,1,3]=>1
[5,2,4,6,3,1]=>1
[5,2,6,1,3,4]=>1
[5,2,6,1,4,3]=>1
[5,2,6,3,1,4]=>1
[5,2,6,3,4,1]=>1
[5,2,6,4,1,3]=>1
[5,2,6,4,3,1]=>1
[5,3,1,2,4,6]=>2
[5,3,1,2,6,4]=>1
[5,3,1,4,2,6]=>2
[5,3,1,4,6,2]=>1
[5,3,1,6,2,4]=>1
[5,3,1,6,4,2]=>1
[5,3,2,1,4,6]=>2
[5,3,2,1,6,4]=>1
[5,3,2,4,1,6]=>2
[5,3,2,4,6,1]=>1
[5,3,2,6,1,4]=>1
[5,3,2,6,4,1]=>1
[5,3,4,1,2,6]=>2
[5,3,4,1,6,2]=>1
[5,3,4,2,1,6]=>2
[5,3,4,2,6,1]=>1
[5,3,4,6,1,2]=>1
[5,3,4,6,2,1]=>1
[5,3,6,1,2,4]=>1
[5,3,6,1,4,2]=>1
[5,3,6,2,1,4]=>1
[5,3,6,2,4,1]=>1
[5,3,6,4,1,2]=>1
[5,3,6,4,2,1]=>1
[5,4,1,2,3,6]=>2
[5,4,1,2,6,3]=>1
[5,4,1,3,2,6]=>2
[5,4,1,3,6,2]=>1
[5,4,1,6,2,3]=>1
[5,4,1,6,3,2]=>1
[5,4,2,1,3,6]=>2
[5,4,2,1,6,3]=>1
[5,4,2,3,1,6]=>2
[5,4,2,3,6,1]=>1
[5,4,2,6,1,3]=>1
[5,4,2,6,3,1]=>1
[5,4,3,1,2,6]=>2
[5,4,3,1,6,2]=>1
[5,4,3,2,1,6]=>2
[5,4,3,2,6,1]=>1
[5,4,3,6,1,2]=>1
[5,4,3,6,2,1]=>1
[5,4,6,1,2,3]=>1
[5,4,6,1,3,2]=>1
[5,4,6,2,1,3]=>1
[5,4,6,2,3,1]=>1
[5,4,6,3,1,2]=>1
[5,4,6,3,2,1]=>1
[5,6,1,2,3,4]=>1
[5,6,1,2,4,3]=>1
[5,6,1,3,2,4]=>1
[5,6,1,3,4,2]=>1
[5,6,1,4,2,3]=>1
[5,6,1,4,3,2]=>1
[5,6,2,1,3,4]=>1
[5,6,2,1,4,3]=>1
[5,6,2,3,1,4]=>1
[5,6,2,3,4,1]=>1
[5,6,2,4,1,3]=>1
[5,6,2,4,3,1]=>1
[5,6,3,1,2,4]=>1
[5,6,3,1,4,2]=>1
[5,6,3,2,1,4]=>1
[5,6,3,2,4,1]=>1
[5,6,3,4,1,2]=>1
[5,6,3,4,2,1]=>1
[5,6,4,1,2,3]=>1
[5,6,4,1,3,2]=>1
[5,6,4,2,1,3]=>1
[5,6,4,2,3,1]=>1
[5,6,4,3,1,2]=>1
[5,6,4,3,2,1]=>1
[6,1,2,3,4,5]=>1
[6,1,2,3,5,4]=>1
[6,1,2,4,3,5]=>1
[6,1,2,4,5,3]=>1
[6,1,2,5,3,4]=>1
[6,1,2,5,4,3]=>1
[6,1,3,2,4,5]=>1
[6,1,3,2,5,4]=>1
[6,1,3,4,2,5]=>1
[6,1,3,4,5,2]=>1
[6,1,3,5,2,4]=>1
[6,1,3,5,4,2]=>1
[6,1,4,2,3,5]=>1
[6,1,4,2,5,3]=>1
[6,1,4,3,2,5]=>1
[6,1,4,3,5,2]=>1
[6,1,4,5,2,3]=>1
[6,1,4,5,3,2]=>1
[6,1,5,2,3,4]=>1
[6,1,5,2,4,3]=>1
[6,1,5,3,2,4]=>1
[6,1,5,3,4,2]=>1
[6,1,5,4,2,3]=>1
[6,1,5,4,3,2]=>1
[6,2,1,3,4,5]=>1
[6,2,1,3,5,4]=>1
[6,2,1,4,3,5]=>1
[6,2,1,4,5,3]=>1
[6,2,1,5,3,4]=>1
[6,2,1,5,4,3]=>1
[6,2,3,1,4,5]=>1
[6,2,3,1,5,4]=>1
[6,2,3,4,1,5]=>1
[6,2,3,4,5,1]=>1
[6,2,3,5,1,4]=>1
[6,2,3,5,4,1]=>1
[6,2,4,1,3,5]=>1
[6,2,4,1,5,3]=>1
[6,2,4,3,1,5]=>1
[6,2,4,3,5,1]=>1
[6,2,4,5,1,3]=>1
[6,2,4,5,3,1]=>1
[6,2,5,1,3,4]=>1
[6,2,5,1,4,3]=>1
[6,2,5,3,1,4]=>1
[6,2,5,3,4,1]=>1
[6,2,5,4,1,3]=>1
[6,2,5,4,3,1]=>1
[6,3,1,2,4,5]=>1
[6,3,1,2,5,4]=>1
[6,3,1,4,2,5]=>1
[6,3,1,4,5,2]=>1
[6,3,1,5,2,4]=>1
[6,3,1,5,4,2]=>1
[6,3,2,1,4,5]=>1
[6,3,2,1,5,4]=>1
[6,3,2,4,1,5]=>1
[6,3,2,4,5,1]=>1
[6,3,2,5,1,4]=>1
[6,3,2,5,4,1]=>1
[6,3,4,1,2,5]=>1
[6,3,4,1,5,2]=>1
[6,3,4,2,1,5]=>1
[6,3,4,2,5,1]=>1
[6,3,4,5,1,2]=>1
[6,3,4,5,2,1]=>1
[6,3,5,1,2,4]=>1
[6,3,5,1,4,2]=>1
[6,3,5,2,1,4]=>1
[6,3,5,2,4,1]=>1
[6,3,5,4,1,2]=>1
[6,3,5,4,2,1]=>1
[6,4,1,2,3,5]=>1
[6,4,1,2,5,3]=>1
[6,4,1,3,2,5]=>1
[6,4,1,3,5,2]=>1
[6,4,1,5,2,3]=>1
[6,4,1,5,3,2]=>1
[6,4,2,1,3,5]=>1
[6,4,2,1,5,3]=>1
[6,4,2,3,1,5]=>1
[6,4,2,3,5,1]=>1
[6,4,2,5,1,3]=>1
[6,4,2,5,3,1]=>1
[6,4,3,1,2,5]=>1
[6,4,3,1,5,2]=>1
[6,4,3,2,1,5]=>1
[6,4,3,2,5,1]=>1
[6,4,3,5,1,2]=>1
[6,4,3,5,2,1]=>1
[6,4,5,1,2,3]=>1
[6,4,5,1,3,2]=>1
[6,4,5,2,1,3]=>1
[6,4,5,2,3,1]=>1
[6,4,5,3,1,2]=>1
[6,4,5,3,2,1]=>1
[6,5,1,2,3,4]=>1
[6,5,1,2,4,3]=>1
[6,5,1,3,2,4]=>1
[6,5,1,3,4,2]=>1
[6,5,1,4,2,3]=>1
[6,5,1,4,3,2]=>1
[6,5,2,1,3,4]=>1
[6,5,2,1,4,3]=>1
[6,5,2,3,1,4]=>1
[6,5,2,3,4,1]=>1
[6,5,2,4,1,3]=>1
[6,5,2,4,3,1]=>1
[6,5,3,1,2,4]=>1
[6,5,3,1,4,2]=>1
[6,5,3,2,1,4]=>1
[6,5,3,2,4,1]=>1
[6,5,3,4,1,2]=>1
[6,5,3,4,2,1]=>1
[6,5,4,1,2,3]=>1
[6,5,4,1,3,2]=>1
[6,5,4,2,1,3]=>1
[6,5,4,2,3,1]=>1
[6,5,4,3,1,2]=>1
[6,5,4,3,2,1]=>1
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Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
References
[1] Callan, D. Counting stabilized-interval-free permutations MathSciNet:2049703 arXiv:math/0310157
[2] Adin, R. M., Bagno, E., Roichman, Y. Block decomposition of permutations and Schur-positivity arXiv:1611.06979
[3] Stanley, R. P. The Descent Set and Connectivity Set of a Permutation arXiv:math/0507224
[2] Adin, R. M., Bagno, E., Roichman, Y. Block decomposition of permutations and Schur-positivity arXiv:1611.06979
[3] Stanley, R. P. The Descent Set and Connectivity Set of a Permutation arXiv:math/0507224
Code
def statistic(pi):
return len([i for i in [1..pi.size()] if sorted(pi[j] for j in range(i)) == [1..i]])
Created
Mar 26, 2013 at 06:07 by Christian Stump
Updated
Jul 12, 2017 at 16:47 by Christian Stump
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