Language Reference

ALLPERM Function

ALLPERM (n); ALLPERM (set, <, idx> );

The ALLPERM function generates all permutations of a set with n elements. The permutations are produced in the same order and using the same algorithm (Trotter, 1962) as the ALLPERM function in Base SAS software.

By default, the ALLPERM function returns a matrix with $n!$ rows and n columns. Each row of the returned matrix represents a single permutation. The following statements generate all permutations of the set $\{ 1, 2, 3\} $: 

n = 3;
p = allperm(n);
print p;

Figure 24.36: All Permutations of Three Items

p
1 2 3
1 3 2
3 1 2
3 2 1
2 3 1
2 1 3 




            

            


            
The first argument can be a scalar or a vector. If it is a vector, the number of elements in the vector determines the value
               of n. The ALLPERM function can compute permutations of arbitrary numeric or character matrices. For example, the following statements
               compute permutations of an unsorted character vector: 
            


a = allperm({C B A});
print a;



   

            


Figure 24.37: All Permutations of a Character Vector

               

                  

                     
                     

                        
                        
                        

                           
                           

                              
                              








a
                                       
                                    




C
                                       
                                       B
                                       
                                       A
                                       
                                    



C
                                       
                                       A
                                       
                                       B
                                       
                                    



A
                                       
                                       C
                                       
                                       B
                                       
                                    



A
                                       
                                       B
                                       
                                       C
                                       
                                    



B
                                       
                                       A
                                       
                                       C
                                       
                                    



B
                                       
                                       C
                                       
                                       A
                                       
                                    






                           
                        

                        



                     
                     
                  





            

            


            
The optional second argument, idx, can be used to control the number of rows in the output of the function. The argument must consist of consecutive integers;
               you cannot use it to randomly access permutations that are out of sequence. The second argument is often used to generate
               one or more permutations in a loop so that you do not need to allocate a huge matrix that contains all of the permutations
               at once. The following statements illustrate this usage: 
            


perm = 1:n;
do i=1 to fact(n);
   perm = allperm(perm, i);
   /* do something with the i_th permutation */
end;



If you want the permutations in lexicographic order, generate the permutations and then use the SORT subroutine
               , as follows: 
            


p = allperm(n);
call sort(p, 1:n);