Perfect matchings

From QETLAB
Jump to navigation Jump to search
perfect_matchings
Gives all perfect matchings of N objects

Other toolboxes required none
Related functions BrauerStates
one_factorization
Function category Helper functions
This is a helper function that only exists to aid other functions in QETLAB. If you are an end-user of QETLAB, you likely will never have a reason to use this function.

perfect_matchings is a function that returns all perfect matchings of a given list of objects. That is, it returns all ways of grouping an even number of objects into pairs.

Syntax

  • PM = perfect_matchings(N)

Argument descriptions

  • N: Either an even integer, indicating that you would like all perfect matchings of the integers $1, 2, \ldots, N$, or a vector containing an even number of distinct entries, indicating that you would like all perfect matchings of those entries.

Examples

Perfect matchings of four objects

The following code generates all perfect matchings of the numbers $1,2,3,4$:

>> perfect_matchings(4)

ans =

     1     2     3     4
     1     3     2     4
     1     4     3     2

The perfect matchings are read "naively" left-to-right. For example, the first row of the output above indicates that one valid perfect matching is $\{\{1,2\},\{3,4\}\}$. The second row says that another perfect matching is $\{\{1,3\},\{2,4\}\}$. Finally, the third row says that the third (and last) perfect matching is $\{\{1,4\},\{2,3\}\}$.

Notes

If $N = 2k$ then there are exactly $(2k)!/(k!\cdot 2^k)$ perfect matchings of $N$ objects. If $N$ is odd, there are no perfect matchings (and thus PM will have zero rows).

Source code

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