How do I iterate over all the $k$-element subsets of $\{1,2,\dots, n\}$ by switching one element at a time?

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This comes from Ch2 of Combinatorial Algorithms by Nienhuis and Wilf.

Equivalently I am asking for a Hamiltonian circuit on the Johnson graph of $k$ element subsets of a set of $n$ elements connected if their intersection has $k-1$ elements.

I am trying to understand how the equation $$A(n,k) = A(n-1,k), \overline{ A(n-1,k-1)}\otimes \{n\}$$ from Nienhuis-Wilf leads to a type of "gray code" for subsets. In fact, it is the gray code when you restruct to $k$-element sets.

Using this formation I came up a means of listing all the subsets in order:

def gc(n,k):
    if(k==1):
        return [[i+1] for i in range(n)]
    elif(n == 0):
        return []
    else:
        L = [ x+ [n] for x in gc(n-1,k-1)]
        return gc(n-1,k)+ L[::-1]

How do I find the predecessor or successor of a given subset without generating all the subsets? I came up with some buggy python code for this, which is different from what is in the textbook.

def S(n,k,a):
    if k == 1:
        return [(a[0] + 1)%n]
    elif(a[-1] == n-1):
        return P(n-1,k-1, a[:-1]) + [n-1]
    else:
        return S(n-1,k,a)

def P(n,k,a):
    if k == 1:
        return [(a[0] - 1)%n]
    elif(a[-1] == n-1):
        return S(n-1,k-1, a[:-1]) + [n-1]
    else:
        return P(n-1,k,a)

This looks pretty close to the recursion in Nienhuis-Wilf but I would like to understand where I am going wrong in my implementation.