If py EulerianCycle.py euleriancycle.txt writes to the standard output stream (which I assume it does, since otherwise you wouldn't be able to pipe it to cat) then cat is entirely superfluous here - you can redirect standard output directly, specifying either absolute or relative path to your output file:
py EulerianCycle.py euleriancycle.txt > outputs/euleriancycleout.txt
(note: the directory outputs/ must already exist).
Neither of your other commands works the way you might imagine.
in py EulerianCycle euleriancycle.txt > cat euleriancycleout.txt, the shell creates a file named cat in the current directory, and redirects the output of py EulerianCycle to it, passing both euleriancycle.txt and euleriancycleout.txt to it as input arguments.
in py EulerianCycle.py euleriancycle.txt | cat >cd outputs/euleriancycleout.txt, the shell creates a file named cd in the current directory, cat reads outputs/euleriancycleout.txt and writes it to file cd, ignoring standard input from the pipe (cat only reads standard input when it is given no input files, or an explicit -).
Perhaps what you were aiming for here was to pipe the output to a subshell like:
py EulerianCycle.py euleriancycle.txt | (cd outputs; cat > euleriancycleout.txt)
or
py EulerianCycle.py euleriancycle.txt | (cd outputs && cat > euleriancycleout.txt)
Here, cat reads the subshell's standard input - which is provided by the pipe - after changing to the target directory. The second version only creates euleriancycleout.txt if the cd command succeeds; the first creates it in the current directory if the cd fails.
