Sets of all disjoint pairs - c

Question

Given a set {1,2,3,4,5...n} of n elements, we need to find all sets of disjoint pairs.
For example, if n=4, the output would be
{(1,2),(3,4)}, {(1,3),(2,4)}, {(1,4),(2,3)}
I am not even able to figure out how to start. I am hoping someone can give me a suggestion about which algorithm to use, and possibly some implementation details as well.

Answer 1

Edit:
Delphi code for recursive generation of (n-1)!! sets (1*3*5*7...n-1) from n=2*k elements
var
A: TArray<Integer>;
procedure Swap(i, j: integer);
var
t : integer;
begin
t := A[i];
A[i] := A[j];
A[j] := t;
end;
procedure MakePairs(Start: Integer; Pairs: string);
var
i: Integer;
begin
if Start >= Length(A) then
Writeln(Pairs)
else
for i := Start + 1 to High(A) do begin
Swap(Start + 1, i); //store used element in the array beginning
MakePairs(Start + 2, Pairs + Format('(%d,%d)', [A[Start], A[Start + 1]]));
Swap(Start + 1, i); //get it back
end;
end;
begin
A := TArray<Integer>.Create(1,2,3,4,5,6);
//be sure that array length is even!!!
MakePairs(0, '');
Writeln(PairCount);
Output:
(1,2)(3,4)(5,6)
(1,2)(3,5)(4,6)
(1,2)(3,6)(5,4)
(1,3)(2,4)(5,6)
(1,3)(2,5)(4,6)
(1,3)(2,6)(5,4)
(1,4)(3,2)(5,6)
(1,4)(3,5)(2,6)
(1,4)(3,6)(5,2)
(1,5)(3,4)(2,6)
(1,5)(3,2)(4,6)
(1,5)(3,6)(2,4)
(1,6)(3,4)(5,2)
(1,6)(3,5)(4,2)
(1,6)(3,2)(5,4)
15
Addition
Variant that works with odd-length array too (weird ordering)
procedure MakePairs(Start: Integer; Pairs: string);
var
i: Integer;
OddFlag: Integer;
begin
if Start >= Length(A) then
Memo1.Lines.Add(Pairs)
else begin
Oddflag := (High(A) - Start) and 1;
for i := Start + OddFlag to High(A) do begin
Swap(Start + OddFlag, i);
if OddFlag = 1 then
MakePairs(Start + 2, Pairs + Format('(%d,%d)', [A[Start], A[Start + 1]]))
else
MakePairs(Start + 1, Pairs);
Swap(Start + OddFlag, i);
end;
end;
end;
for (1,2,3,4,5):
(2,3)(4,5)
(2,4)(3,5)
(2,5)(4,3)
(1,3)(4,5)
(1,4)(3,5)
(1,5)(4,3)
(2,1)(4,5)
(2,4)(1,5)
(2,5)(4,1)
(2,3)(1,5)
(2,1)(3,5)
(2,5)(1,3)
(2,3)(4,1)
(2,4)(3,1)
(2,1)(4,3)
15
Not relevant now:
If every pair should occur just once (it is not clear from your example with n=4), then you can use round-robin tournament algorithm
n=4 case example here

Answer 2

You have to see the pattern here.
For {1, 2, 3, 4}.
Take the first element and make pairs with all the elements on the right.
(1, 2), (1, 3), (1, 4)
Take the second element and make pairs with all the elements on the right.
(2, 3), (2, 4)
Take the third element and make pairs with all the elements on the right.
(3, 4)
...and so on
Notice the pattern here.
You would need an outer loop to iterate over the elements and select each element one by one.
And another inner loop to iterate over the elements on the right of the selected element and make a pair with each one of them.