Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
0=>[2]=>[1,1,0,0]=>0
1=>[1,1]=>[1,0,1,0]=>0
00=>[3]=>[1,1,1,0,0,0]=>0
01=>[2,1]=>[1,1,0,0,1,0]=>0
10=>[1,2]=>[1,0,1,1,0,0]=>1
11=>[1,1,1]=>[1,0,1,0,1,0]=>0
000=>[4]=>[1,1,1,1,0,0,0,0]=>0
001=>[3,1]=>[1,1,1,0,0,0,1,0]=>0
010=>[2,2]=>[1,1,0,0,1,1,0,0]=>1
011=>[2,1,1]=>[1,1,0,0,1,0,1,0]=>0
100=>[1,3]=>[1,0,1,1,1,0,0,0]=>1
101=>[1,2,1]=>[1,0,1,1,0,0,1,0]=>1
110=>[1,1,2]=>[1,0,1,0,1,1,0,0]=>1
111=>[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>0
0000=>[5]=>[1,1,1,1,1,0,0,0,0,0]=>0
0001=>[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>0
0010=>[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>1
0011=>[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>0
0100=>[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>1
0101=>[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>1
0110=>[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>1
0111=>[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>0
1000=>[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>1
1001=>[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>1
1010=>[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>2
1011=>[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>1
1100=>[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>1
1101=>[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>1
1110=>[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>1
1111=>[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>0
00000=>[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>0
00001=>[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>0
00010=>[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>1
00011=>[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>0
00100=>[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>1
00101=>[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>1
00110=>[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>1
00111=>[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>0
01000=>[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>1
01001=>[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>1
01010=>[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>2
01011=>[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>1
01100=>[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>1
01101=>[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>1
01110=>[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>1
01111=>[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>0
10000=>[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>1
10001=>[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>1
10010=>[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>2
10011=>[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>1
10100=>[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>2
10101=>[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>2
10110=>[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>2
10111=>[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>1
11000=>[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>1
11001=>[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>1
11010=>[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>2
11011=>[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>1
11100=>[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>1
11101=>[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>1
11110=>[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>1
11111=>[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>0
000000=>[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>0
000001=>[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>0
000010=>[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>1
000011=>[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>0
000100=>[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>1
000101=>[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>1
000110=>[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>1
000111=>[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>0
001000=>[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>1
001001=>[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>1
001010=>[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>2
001011=>[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>1
001100=>[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>1
001101=>[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>1
001110=>[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>1
001111=>[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>0
010000=>[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>1
010001=>[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>1
010010=>[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>2
010011=>[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>1
010100=>[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>2
010101=>[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>2
010110=>[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>2
010111=>[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>1
011000=>[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>1
011001=>[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>1
011010=>[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>2
011011=>[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>1
011100=>[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>1
011101=>[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>1
011110=>[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>1
011111=>[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>0
100000=>[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>1
100001=>[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>1
100010=>[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>2
100011=>[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>1
100100=>[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
100101=>[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>2
100110=>[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>2
100111=>[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>1
101000=>[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>2
101001=>[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>2
101010=>[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>3
101011=>[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>2
101100=>[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>2
101101=>[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>2
101110=>[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>2
101111=>[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>1
110000=>[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>1
110001=>[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>1
110010=>[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>2
110011=>[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>1
110100=>[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>2
110101=>[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>2
110110=>[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>2
110111=>[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>1
111000=>[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>1
111001=>[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>1
111010=>[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>2
111011=>[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>1
111100=>[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>1
111101=>[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>1
111110=>[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>1
111111=>[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>0
0100000=>[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>1
0100001=>[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>1
0100010=>[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>2
0100011=>[2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>1
0100100=>[2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
0100101=>[2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>2
0100110=>[2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>2
0100111=>[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>1
0101000=>[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>2
0101001=>[2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>2
0101010=>[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>3
0101011=>[2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>2
0101100=>[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>2
0101101=>[2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>2
0101110=>[2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>2
0101111=>[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>1
0110000=>[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>1
0110001=>[2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>1
0110010=>[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>2
0110011=>[2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>1
0110100=>[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>2
0110101=>[2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>2
0110110=>[2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>2
0110111=>[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>1
0111000=>[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>1
0111001=>[2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>1
0111010=>[2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>2
0111011=>[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>1
0111100=>[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>1
0111101=>[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>1
0111110=>[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>1
0111111=>[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>0
1000000=>[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>1
1000001=>[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>1
1000010=>[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>2
1000011=>[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>1
1000100=>[1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>2
1000101=>[1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>2
1000110=>[1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>2
1000111=>[1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>1
1001000=>[1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>2
1001001=>[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>2
1001010=>[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>3
1001011=>[1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>2
1001100=>[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>2
1001101=>[1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>2
1001110=>[1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>2
1001111=>[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>1
1010000=>[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>2
1010001=>[1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>2
1010010=>[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>3
1010011=>[1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>2
1010100=>[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>3
1010101=>[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>3
1010110=>[1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>3
1010111=>[1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>2
1011000=>[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>2
1011001=>[1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>2
1011010=>[1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>3
1011011=>[1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>2
1011100=>[1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>2
1011101=>[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>2
1011110=>[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>2
1011111=>[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>1
1100000=>[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>1
1100001=>[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>1
1100010=>[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>2
1100011=>[1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>1
1100100=>[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
1100101=>[1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>2
1100110=>[1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>2
1100111=>[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>1
1101000=>[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>2
1101001=>[1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>2
1101010=>[1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>3
1101011=>[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>2
1101100=>[1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>2
1101101=>[1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>2
1101110=>[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>2
1101111=>[1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>1
1110000=>[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>1
1110001=>[1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>1
1110010=>[1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>2
1110011=>[1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>1
1110100=>[1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>2
1110101=>[1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>2
1110110=>[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>2
1110111=>[1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>1
1111000=>[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>1
1111001=>[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>1
1111010=>[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>2
1111011=>[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>1
1111100=>[1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>1
1111101=>[1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>1
1111110=>[1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>1
1111111=>[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>0
=>[1]=>[1,0]=>0
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Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
bounce path
Description
The bounce path determined by an integer composition.
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