Vec partitions

vec_partitions
Produces all possible partitions of a vector
Other toolboxes required	none
Function category	Helper functions

vec_partitions is a function that returns all partitions a given vector into a specified number of parts. Note that, for this function, the order of the parts does matter, but the order of the elements within the parts does not matter (for example, when partitioning the vector [1,2,3,4], the partition {[1],[2,3,4]} is the same as {[1],[3,2,4]}, but distinct from {[2,3,4],[1]}).

Syntax

• PT = vec_partitions(V,P)
• PT = vec_partitions(V,P,SZ)

Argument descriptions

• V: A vector to be partitioned.
• P: A positive integer: the number of parts to partition P into.
• SZ (optional, default [1,1,...,1]): A vector such that only the partitions with the property that the j-th part contains at least SZ(j) elements (for all j) are returned.

Examples

The following code generates all 14 partitions of the vector [1,2,3,4] into two parts. It then displays two of those partitions (the first and sixth).

>> pt = vec_partitions(1:4,2);
>> length(pt)
 
ans =
 
    14
 
>> celldisp(pt{1})
 
ans{1} =
 
     1
 
 
 
ans{2} =
 
     2     3     4
 
 
>> celldisp(pt{6})
 
ans{1} =
 
     1     3
 
 
 
ans{2} =
 
     2     4

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  VEC_PARTITIONS    Produces all possible partitions of a vector

2. %

3. %   PT = vec_partitions(V,P,SZ) computes all partitions of the vector V

4. %   into P parts. PT is a cell, each of whose entries is a 1-by-P cell

5. %   containing the P different parts.

6. %

7. %   This function has one optional argument:

8. %     SZ (default [1,1,...,1])

9. %   

10. %   PT = vec_partitions(V,P,SZ) computes all partitions of the vector V

11. %   into P parts, with the restriction that, for all j, the j-th part must

12. %   have at least SZ(j) elements.

13. %

14. %   URL: http://www.qetlab.com/vec_partitions

16. %   requires: opt_args.m

17. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

18. %   package: QETLAB

19. %   last updated: October 18, 2014

21. function pt = vec_partitions(v,p,varargin)

23. % set optional argument defaults: sz = [1,1,...,1]

24. [sz] = opt_args({ ones(1,p) },varargin{:});

26. if(p > 1) % if p > 1 then we still have work to do

27.     pt = cell(0);

28.     ub = length(v) - sum(sz(2:end));

29.     for jj = sz(1):ub

30.         tpt = nchoosek(v,jj);

31.         for kk = 1:size(tpt,1) % recurse through the other parties

32.             tpt2 = vec_partitions(setdiff(v,tpt(kk,:)),p-1,sz(2:end));

33.             for ll = 1:length(tpt2)

34.                 pt{end+1} = {tpt(kk,:), tpt2{ll}{:}};

35.             end

36.         end

37.     end

38. else % we are on the last party: only one option left

39.     pt{1}{1} = v;

40. end