The Dyck words are the elements that form the Dyck language and can be defined as any sequence of parenthesis correctly balanced, for example, these are the Dyck words having 3 pairs of parenthesis:
((())) (()()) (())() ()(()) ()()()
Also, the number of Dyck words of length 2N is equal to the Nth Catalan number, registered in OEIS as A000108, and the Dyck words can be converted to many geometric representations of the Catalan numbers.
The following program in Perl generates all the Dyck words of length 2N using the characters "0" and "1", receiving N as argument:
#!/usr/bin/perl
# dyck.pl - prints all dyck words of length 2N given N as argument
my $s=('0'x$ARGV[0]).('1'x$ARGV[0]); my $t;
do{
print $s, "\n";
}while($s=~s/(.*)01(1.*)/$1.'10'.join('',($t=$2)=~m'0'g).join('',$t=~m'1'g)/e);
On each iteration, the last instruction generates the next word transforming the current word by finding the last "011", replacing "01" with "10" and then reordering the other "1" and its following characters (if any) placing first the 0's and then the 1's.
Code explanation: The variables $1 and $2 store the begining and the end of the word found by the regular expression (.*)01(1.*) and the variable $t is used to save the content of $2 because $1 and $2 are rewritten when the other regular expression operators =~ are executed for collect and separate the 0's and the 1's of the end.
For example:
> perl dyck.pl 4
00001111
00010111
00011011
00011101
00100111
00101011
00101101
00110011
00110101
01000111
01001011
01001101
01010011
01010101
Enjoy it!
