I am not able to write a helper function which does the same job as map and lambda
(define (permutations lst)
(cond [(empty? lst) '()]
[(empty? (rest lst)) (list lst)]
[else???]))
map is just an abstractiom. Imagine you make a typical list eater:
;; add 2 to each element
(define (add2 lst)
(if (null? lst)
'()
(cons (+ 2 (car lst))
(add2 (cdr lst)))))
(add2 '(1 2 3)) ; ==> (3 4 5)
;; reverse each sublist in a list
(define (revel lst)
(if (null? lst)
'()
(cons (reverse (car lst))
(revel (cdr lst)))))
(revel '((1 2) (3 4))) ; ==> ((2 1) (4 3))
Do you notice how similar those two are? It's almost 90% the same code. The common code is the core of map:
(define (map1 f lst)
(if (null? lst)
'()
(cons (f (car lst))
(map1 f (cdr lst)))))
Ad now we can implement the two using map1
(define (add2 lst)
(map1 (lambda (v) (+ v 2)) lst))
(define (revel lst)
(map1 reverse lst))
Of course map is more than map1 since it also supports more list arguments, but implementing a full map is beyond the scope of this post.
So here is one approach to doing this. I assume you are allowed to write helper functions.
The first step is to realise that at some point you are going to need to take a list of elements and 'thread' something through it, producing all the possible new lists with that element inserted somewhere. So given (1 2) and 3 you want to make ((3 1 2) (1 3 2) (1 2 3)).
And for reasons which will become clear I'm going to do this in such a way that the function adds its answers to some preexisting ones. And it's going to have, itself, a little helper function.
(define (thread-list e l into)
;; Thread e through l accumulating into into
(thread-list-loop e '() l into))
(define (thread-list-loop e before after into)
;; Do the work of threading e through a list:
;; - before is the chunk of list before e;
;; - after is the chunk after e;
;; - into is what we're collecting into
(if (null? after)
;; We now just need before with e at the end
(cons (append before (list e)) into)
;; add the current thing to results and loop
(thread-list-loop
e
(append before (list (first after)))
(rest after)
(cons (append before (list e) after) into))))
So let's test this:
> (thread-list 1 '(2 3) '())
'((2 3 1) (2 1 3) (1 2 3))
Now we want a function which will take an element, and a bunch of lists, and thread the element through each of the lists, accumulating the result:
(define (thread-lists e lists into)
;; thread e through each element of lists, accumulating into into
(if (null? lists)
into
(thread-lists
e
(rest lists)
(thread-list e (first lists) into))))
And checking this:
> (thread-lists 1 '((2 3) (3 2)) '())
'((3 2 1) (3 1 2) (1 3 2) (2 3 1) (2 1 3) (1 2 3))
And I'm going to stop here, but look at what the above call did: given ((2 3) (3 2)), which is the permutations of (2 3), and 1, it returned the permutations of (1 2 3)! So you should now be able to write another function, permutations, which makes use of thread-lists to create permutations.
Note I don't know if this approach is anything like efficient and I would expect it not to be. It is, I think, fairly clear.