I've written several articles that show how to generate permutations in SAS. In the SAS DATA step, you can use the ALLPEM subroutine to generate all permutations of a DATA step array that contain a small number (18 or fewer) elements. In addition, the PLAN procedure enables you to generate permutations that are useful for experimental design.

I also wrote about how to generate random permutations in the SAS/IML language. The random-permutation algorithm also works in the DATA step, but alternatively you can call the RANPERM subroutine in the DATA step to generate a random permutation of elements in an array.

However, I wrote those articles back in 2010, and a lot has changed since then in the SAS/IML language. Beginning in SAS 9.3, the SAS/IML language supports the following functions for generating permutations:

• The ALLPERM function generates all n! permutations of a set with n elements, for n ≤ 18
• The RANPERM function generates random permutations of a set.
• The RANPERK function generates a permutation of k elements chosen from a set that contains n elements.

The RANPERM and RANPERK functions can be used for algorithms in SAS that require random sampling without replacement. (Random sampling without replacement is also supported by the SURVEYSELECT procedure. See the documentation of the METHOD=SRS option.)

Generating all permutations of N elements

For small sets, the ALLPERM function returns a vector that contains all n! permutations of the elements of the set. Each row is a permutation. For example, what are all the possible arrangements of four players at a card table? (The positions around the table are traditionally labeled North, East, South, and West.) The following SAS/IML statements display all the possible arrangements of players at the four positions:

proc iml;
call randseed(1234);
pName = "Player1":"Player4";
perms = allperm({N E S W}); /* use North, East, South, and West as labels */
print perms[c=pName];

For an application that is more statistically oriented, you can use the complete list of permutations to construct an exact permutation test (see documentation for the FREQ procedure).

Generating random permutations of N elements

Often the complete list of permutations of N elements is too large to compute. In that case, you can often use random permutations to simulate draws from the set of complete permutations. Random permutations are also important because they represent a random rearrangement (a shuffle) of elements.

/* function to generate a deck of 52 playing cards: 2C, 3C,..., KS, AS */
start MakeCardDeck();
   suits = {H D C S};   
   suits52 = rowvec( repeat(suits`,1,13) );
   vals = char(2:10,2) || {J Q K A};
   vals52 = right( repeat( vals, 1, 4) );
   return( vals52 + suits52 );
finish;
Cards = MakeCardDeck();
 
/* randomly permute columns (or rows) of a matrix */
shuffle = ranperm(Cards);
BridgeHand = shape(shuffle, 0, 4);
print BridgeHand[c=("Player1":"Player4")];

The previous SAS/IML statements create a 52-element character vector to represent to suit and face value of the cards. The array is then randomly permuted. The cards are then "dealt" to four players, who each receive 13 cards. Notice that enumerating all 8.06 x 10⁶⁷ permutations of 52 elements is infeasible.

Generating random permutations of k elements from a set of N elements

Sometimes you don't need the entire permutation, but just k elements, which you can think of as the first k elements of the permutation. If that is the case, you don't need to generate the entire permutation. Instead, use the RANPERK function. For example, the following statements return 20 cards (four five-card hands) randomly drawn from a deck (without replacement):

NumPlayers = 4;
NumCards = 5;
Deal = ranperk(Cards, NumCards*NumPlayers);
Hands = shape(Deal, 0, NumPlayers);
print Hands[c=("Player1":"Player4")];

Besides being useful, I like the name of the RANPERK function. A "perk" (slang for perquisite) is an additional benefit given to someone, like free coffee and snacks in the break room. The thought of the SAS/IML language giving me random perks makes me smile!

In the field of statistical sampling, a permutation is a "random sample drawn without replacement" from a finite set where there is an equal probability of choosing any item. The RANPERK function represents drawing k items without replacement from a box that contains N items. The RANPERM function represents drawing without replacement until the box is empty. Thus you can use these two SAS/IML functions to support random sampling without replacement for items that are drawn wih equal probability.