Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
St001732: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
[1]=>[1,0]=>1
[1,1]=>[1,0,1,0]=>1
[2]=>[1,1,0,0]=>1
[1,1,1]=>[1,0,1,0,1,0]=>1
[1,2]=>[1,0,1,1,0,0]=>2
[2,1]=>[1,1,0,0,1,0]=>1
[3]=>[1,1,1,0,0,0]=>1
[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>1
[1,1,2]=>[1,0,1,0,1,1,0,0]=>2
[1,2,1]=>[1,0,1,1,0,0,1,0]=>2
[1,3]=>[1,0,1,1,1,0,0,0]=>2
[2,1,1]=>[1,1,0,0,1,0,1,0]=>1
[2,2]=>[1,1,0,0,1,1,0,0]=>1
[3,1]=>[1,1,1,0,0,0,1,0]=>1
[4]=>[1,1,1,1,0,0,0,0]=>1
[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>1
[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>2
[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>2
[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>2
[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>2
[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>2
[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>2
[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>2
[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>1
[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>1
[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>1
[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>2
[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>1
[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>1
[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>1
[5]=>[1,1,1,1,1,0,0,0,0,0]=>1
[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>1
[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>2
[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>2
[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>2
[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>2
[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>2
[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>2
[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>2
[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>2
[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>2
[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>2
[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>3
[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>2
[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>2
[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>2
[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>2
[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>1
[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>1
[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>1
[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>2
[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>1
[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>1
[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>2
[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>2
[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>1
[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>1
[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>1
[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>1
[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>1
[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>1
[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>1
[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>1
[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>1
[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>2
[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>2
[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>2
[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>2
[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>2
[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>2
[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>2
[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>2
[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>2
[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>2
[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>3
[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>2
[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>2
[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>2
[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>2
[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>2
[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>2
[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>2
[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>3
[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>2
[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>2
[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>3
[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>3
[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>2
[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>2
[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>2
[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>2
[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>2
[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>2
[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>2
[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>1
[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>1
[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>1
[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>2
[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>1
[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>1
[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>2
[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>2
[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>1
[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>1
[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>1
[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>2
[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>2
[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>2
[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>2
[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>2
[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>1
[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>1
[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>1
[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>1
[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>1
[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>1
[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>1
[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>2
[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>1
[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>1
[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>1
[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>1
[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>1
[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>1
[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>1
[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>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
[1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>2
[1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>2
[1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>2
[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>2
[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>2
[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>2
[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>2
[1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>2
[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>2
[1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>2
[1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>3
[1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>2
[1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>2
[1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>2
[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>2
[1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>2
[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>2
[1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>2
[1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>3
[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>2
[1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>2
[1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>3
[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>3
[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>2
[1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>2
[1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>2
[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
[1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>2
[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>2
[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>2
[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>2
[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>2
[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>2
[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>2
[1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>3
[1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>2
[1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>2
[1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>3
[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>3
[1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>2
[1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>2
[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>2
[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>3
[1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>3
[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>3
[1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>3
[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>3
[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>2
[1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>2
[1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>2
[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>2
[1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>2
[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>2
[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>2
[1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>3
[1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>2
[1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>2
[1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>2
[1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>2
[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>2
[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>2
[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>2
[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>2
[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>1
[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>1
[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>1
[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>2
[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>1
[2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>1
[2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>2
[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>2
[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>1
[2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>1
[2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>1
[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>2
[2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>2
[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>2
[2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>2
[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>2
[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>1
[2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>1
[2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>1
[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>2
[2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>1
[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>1
[2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>2
[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>2
[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>2
[2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>2
[2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>2
[2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
[2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>2
[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>2
[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>2
[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>2
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Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.
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