rg123

# List representing the given puzzle:
# Number of letters in each row from top to bottom...
positions = [1,2,3,4,3,2,3,4,3,2,1]

def get_num_paths(positions):
    # Start with a list consisting of a 1 for every letter in the
    # top row.  These are totals paths so far at top row...
    totals = [1 for x in xrange(positions[0])]
    # Now for every other row...
    for i in range(1, len(positions)):
        L = len(totals)
        # If current row is wider than previous row by 1...
        if positions[i] == L + 1:
            new_totals = [totals[0]]  # New leftmost total is previous
                                      # leftmost total.
            if L > 1:
                # Calculate new totals in the middle by adding together
                # previous adjacent totals and add them to the new list...
                new_totals.extend([sum(totals[x : x + 2])
                                   for x in xrange(L - 1)])
            new_totals.append(totals[-1])  # New rightmost total is previous
                                           # rightmost total.
            totals = new_totals
        # Else if current row is narrower than previous row by 1...
        elif positions[i] == L - 1:
            # All new totals can be handled in this case by adding together
            # previous adjacent totals...
            new_totals = [sum(totals[x : x + 2])
                          for x in xrange(L - 1)]
            totals = new_totals
        # Else it isn't a puzzle this function solves...
        else:
            return "Invalid structure."
    return sum(totals)  # Sum of final path totals is the result.

print get_num_paths(positions)