How to find the next sequence of integers?

Question

I was thinking of the following problem: List all sequences \$ a_1, a_2, a_3, a_4, a_5, a_6, a_7 \$ of positive integers where \$ 1 \le a_1 < a_2 < \ldots < a_7 \le 39 \$. For any sequence with this property, my code computes the next sequence in lexicographic order. Starting with the initial sequence \$ [1,2,3,4,5,6,7] \$, calling next_row(a) repeatedly produces all such sequences.

Is there more elegant way to code the function that returns the next row than this?

def next_row(a):
    if a[-1]<39:
             a[-1]+=1
             return a
    elif a[-2]<38:
            a[-2]+=1
            a[-1]=a[-2]+1
            return a
    elif a[-3]<38:
            a[-3]+=1
            a[-2]=a[-3]+1
            a[-1]=a[-2]+1
            return a
    elif a[-4]<37:
            a[-4]+=1
            a[-3]=a[-4]+1
            a[-2]=a[-3]+1
            a[-1]=a[-2]+1
            return a
    elif a[-5]<36:
            a[-5]+=1
            a[-4]=a[-5]+1
            a[-3]=a[-4]+1
            a[-2]=a[-3]+1
            a[-1]=a[-2]+1
            return a
    elif a[-6]<35:
            a[-6]+=1
            a[-5]=a[-6]+1
            a[-4]=a[-5]+1
            a[-3]=a[-4]+1
            a[-2]=a[-3]+1
            a[-1]=a[-2]+1
            return a
    elif a[-7]<34:
            a[-7]+=1
            a[-6]=a[-7]+1
            a[-5]=a[-6]+1
            a[-4]=a[-5]+1
            a[-3]=a[-4]+1
            a[-2]=a[-3]+1
            a[-1]=a[-2]+1
            return a

Answer 1

There is a lot of repetition in your code, which makes it unnecessarily long, difficult to read, difficult to maintain, and error-prone. And in fact you have an apparent copy-paste error, can you spot it?

if a[-1]<39:
    ...
elif a[-2]<38:
    ...
elif a[-3]<38:
    ...
elif a[-4]<37:
    ...

The outer if/elif chain is to find the rightmost position in the list which can be increased, and that can be done with a loop:

for i in range(len(a)):
    if a[-1-i] < 39 - i:
       ...

The inner repeated assignments can be replaced by a loop as well. Then your function would look like this:

def next_row(a):
    for i in range(len(a)):
        if a[-1-i] < 39 - i:
            a[-1-i] += 1
            for j in range(0, i):
                a[-1-j] = a[-1-i] + i - j
            return a

Here i and j are positions from the end of the array as in your code, other choices are possible.

The new function is shorter, more general (it works with lists of arbitrary lists, not only lists of length 7), and can easily be generalized to arbitrary upper bounds (just replace the constant 39 by a function parameter).

If the intention is to enumerate all 7-element subsets of the first 39 integers then you can also use existing functions from the itertools module, in this case itertools.combinations():

import itertools

rows = itertools.combinations(range(1, 40), 7)

This creates an iterator which you can access one-by-one:

next_row = next(rows)
print(next_row) 
# (1, 2, 3, 4, 5, 6, 7)

next_row = next(rows)
print(next_row)
# (1, 2, 3, 4, 5, 6, 8)

...

or enumerate:

for row in rows:
    print(row)

As a general remark, I suggest to check your code against the PEP8 coding style, for example at PEP8 online. It will report mainly missing whitespace around operators, and wrong indentation.