CL-PERMUTATION

A library for operating on permutations and permutation groups.

Creating Permutations

Permutations are represented by the structure PERM, which is read-only/immutable at the API boundary. A permutation of size N is essentially a sequence of numbers from 1 to N. One-based permutations were chosen because that is the dominating convention in mathematics. All we lose, essentially, is direct compatibility with array indexing, and one fixnum worth of space. (Internally, the permutations are stored in an array of size N+1, where the zeroth element is always zero).

A permutation can be created via MAKE-PERM:

PERM> (make-perm 1 2 3)
#<PERM 1 2 3>

The permutation will be checked for validity.

PERM> (make-perm 1 2 5)
Given permutation must contain the numbers 1 to 3
   [Condition of type SIMPLE-ERROR]

One can also create permutations with LIST-TO-PERM, which converts a list to a permutation. The companion function PERM-TO-LIST does the opposite operation, but it's not recommended to use list representations of permutations.

One can also create permutations with VECTOR-TO-PERM, which is analogous to LIST-TO-PERM, except it works for vectors. The reverse is PERM-TO-VECTOR.

Lastly, there is an experimental reader macro for permutations, which are created at read time. To enable the syntax, use

(enable-perm-reader)

and then one may type

#[3 1 2 4 5]

for permutations instead.

Permutation Operations

There is a slew of permutation operations:

• perm-identity: construct an identity perm
• perm-identity-p: check if a perm is an identity perm
• random-perm: construct a random perm with specified parity
• perm-ref: zero-based reference
• perm-eval: one-based (standard) reference
• perm-eval*: one-based (standard) reference with out-of-bounds handling
• perm-inverse-eval: one-based (standard) reference of inverse
• perm-inverse-eval*: one-based (standard) reference of inverse with out-of-bounds handling
• perm=: check for equality
• perm=*: check for equality of different sized perms
• perm-size: the size of the permutation (number of mapped elements)
• perm-length: number of inversions
• perm-even-p: check for evenness/oddness
• perm-odd-p: ''
• perm-sign: ''
• perm-compose: compose two permutations
• perm-expt: compose a perm with itself a number of times
• perm-order: order of a permutation
• perm-transpose-indexes: swap two indexes, keeping the entries fixed
• perm-transpose-entries: swap two entries, keeping the indexes fixed
• perm-inverse: invert a permutation
• perm-fixpoints: compute the fixed points of a permutation
• permute: permute an array according to a permutation
• commutesp: check if two permutations commute
• perm<: check lexicographic order

Permutation Generation

There are ways of efficiently generating all permutations of a given length incrementally. Instead of generating all permutations at once in memory -- which takes O(n*n!) space -- we generate permutations on the fly.

The first way is to iterate over the permutations using a DOLIST-style macro called DOPERMS.

PERM> (let ((i 1))
        (doperms (p 3)
          (format t "~D: ~A~%" i p)
          (incf i)))
1: #<PERM 1 2 3>
2: #<PERM 1 3 2>
3: #<PERM 3 1 2>
4: #<PERM 3 2 1>
5: #<PERM 2 3 1>
6: #<PERM 2 1 3>

The other way is to produce a generator object (a closure, in fact) which generates the permutations. Simply FUNCALL the object to receive the next permutation. When they're all exhausted, the closure will return NIL.

PERM> (defparameter S3 (make-perm-generator 3))
S3
PERM> (defparameter S2 (make-perm-generator 2))
S2
PERM> (list (funcall S2) (funcall S3))
(#<PERM 1 2> #<PERM 1 2 3>)
PERM> (list (funcall S2) (funcall S3))
(#<PERM 2 1> #<PERM 1 3 2>)
PERM> (list (funcall S2) (funcall S3))
(NIL #<PERM 3 1 2>)
PERM> (list (funcall S2) (funcall S3))
(NIL #<PERM 3 2 1>)
PERM> (list (funcall S2) (funcall S3))
(NIL #<PERM 2 3 1>)

Cycle Operations

There's also a number of operations for cycles. Cycles are represented by the CYCLE structure. We can convert to and from cycle representation using TO-CYCLES and FROM-CYCLES. Cycles created by TO-CYCLES are automatically canonicalized with CANONICALIZE-CYCLES. Canonicalization is defined as:

• Cycles contain their least element positionally first.
• Cycles are listed in descending order of their first element.
• No null cycles exist.
• The sum of the cycle lengths of a decomposition of a permutation of size N is N.

Cycles that have not been canonicalized are printed with an asterisk '*'. We can observe this by explicitly disabling cycle canonicalization:

PERM> (make-cycle 3 1)
#<CYCLE (1 3)>                ; no asterisk
PERM> (let ((*canonicalize-cycle-on-creation* nil))
        (make-cycle 3 1))
#<CYCLE (3 1)*>               ; asterisk

An example use of TO-CYCLES is as follows:

PERM> (let ((r (random-perm 10)))
        (values r (to-cycles r)))
#<PERM 7 4 8 5 2 10 3 9 1 6>
(#<CYCLE (6 10)> #<CYCLE (2 4 5)> #<CYCLE (1 7 3 8 9)>)

FROM-CYCLES allows the specification of the permutation's length. For example:

PERM> (from-cycles (list (make-cycle 1 3 2)))
#<PERM 3 1 2>
PERM> (from-cycles (list (make-cycle 1 3 2)) 5)
#<PERM 3 1 2 4 5>

Lastly, there is a (mostly useless) function CYCLES-TO-ONE-LINE which converts cycles to one-line notation. That is, the cycles

(1 2 3)(4 5)

gets converted to the permutation

12345.

For example,

PERM> (cycles-to-one-line (list (make-cycle 1 2 3)
                                (make-cycle 4 5)))
#<PERM 1 2 3 4 5>

If one takes a permutation which has been canonically decomposed into cycles, then interestingly, there exists a bijection between one-line notation and the cycle decomposition.

Combinatorial Specifications

A "combinatorial specification" describes a space of combinatorial objects. They have a nice property that they all can be mapped to and from integers sharply. See the section "Ranking and Unranking Combinatorial Specifications".

Currently, only objects of linear structure exist. All of them are represented as subclasses of COMBINATORIAL-SPEC. They are as follows.

RADIX-SPEC: Base-B Non-Negative Integers

These are a representation of a base-B non-negative integer, for a base B > 1. They are handled by the RADIX-SPEC class. Within CL-PERMUTATION, the digits are written left-to-right to correspond with natural vector index ordering. A RADIX-SPEC can be made with MAKE-RADIX-SPEC. Here is the enumeration of all two-digit trinary numbers:

PERM> (print-objects-of-spec (make-radix-spec 3 2))
0 ==> #(0 0) ==> 0
1 ==> #(1 0) ==> 1
2 ==> #(2 0) ==> 2
3 ==> #(0 1) ==> 3
4 ==> #(1 1) ==> 4
5 ==> #(2 1) ==> 5
6 ==> #(0 2) ==> 6
7 ==> #(1 2) ==> 7
8 ==> #(2 2) ==> 8

MIXED-RADIX-SPEC: Non-Negative Mixed-Radix Integers

A mixed-radix integer is a generalization of a base-B integer. The digits in a mixed-radix numeral correspond to different bases. Mixed-radix specifications can be made with VECTOR-TO-MIXED-RADIX-SPEC. For example, the following are all numerals of radix (2, 3, 1):

PERM> (print-objects-of-spec (vector-to-mixed-radix-spec #(2 3 1)))
0 ==> #(0 0 0) ==> 0
1 ==> #(1 0 0) ==> 1
2 ==> #(0 1 0) ==> 2
3 ==> #(1 1 0) ==> 3
4 ==> #(0 2 0) ==> 4
5 ==> #(1 2 0) ==> 5

Notice again we use vector index ordering.

PERM-SPEC: Permutations

The space of permutations of length N (also known as S_N) can be represented. These are represented by the PERM-SPEC class.

PERM> (print-objects-of-spec (make-perm-spec 3))
0 ==> #(0 1 2) ==> 0
1 ==> #(0 2 1) ==> 1
2 ==> #(1 0 2) ==> 2
3 ==> #(1 2 0) ==> 3
4 ==> #(2 0 1) ==> 4
5 ==> #(2 1 0) ==> 5

Currently, actual PERM objects are not generated (see below about ranking/unranking). However, one can easily convert between the two.

COMBINATION-SPEC: Combinations

Combinations represent the selection of M objects from a collection of N objects. These are represented by a vector containing M 1's and N 0's. The class that manages this is a COMBINATION-SPEC. For example, all combinations of 2 objects of a total of 4 can be listed by the following:

PERM> (print-objects-of-spec (make-combination-spec 4 2))
0 ==> #(0 0 1 1) ==> 0
1 ==> #(0 1 0 1) ==> 1
2 ==> #(1 0 0 1) ==> 2
3 ==> #(0 1 1 0) ==> 3
4 ==> #(1 0 1 0) ==> 4
5 ==> #(1 1 0 0) ==> 5

WORD-SPEC: Words

A word is similar to a permutation except that it may have repeated, indistinct elements. These are represented by a WORD-SPEC. It can be created by supplying a sample word to the function VECTOR-TO-WORD-SPEC. For example, all words of the form 1123 can be listed as follows:

PERM> (print-objects-of-spec (vector-to-word-spec #(1 1 2 3)))
0 ==> #(1 1 2 3) ==> 0
1 ==> #(1 1 3 2) ==> 1
2 ==> #(1 2 1 3) ==> 2
3 ==> #(1 2 3 1) ==> 3
4 ==> #(1 3 1 2) ==> 4
5 ==> #(1 3 2 1) ==> 5
6 ==> #(2 1 1 3) ==> 6
7 ==> #(2 1 3 1) ==> 7
8 ==> #(2 3 1 1) ==> 8
9 ==> #(3 1 1 2) ==> 9
10 ==> #(3 1 2 1) ==> 10
11 ==> #(3 2 1 1) ==> 11

Ranking and Unranking Combinatorial Specifications

Each combinatorial specification represents a finite space of N > 0 objects. N is called the "cardinality" of the specification and can be computed with the CARDINALITY method.

> (cardinality (make-perm-spec 3))
6
> (cardinality (vector-to-word-spec #(1 1 2 3)))
12

The cardinality is computed only once for a combinatorial specification and is then cached for fast access.

Obviously, every object in a particular finite combinatorial space can be bijected to and from integers below the cardinality of that space. CL-PERMUTATION provides fast and efficient mechanisms for computing one such bijection for each combinatorial specification. Mapping from an object to an integer is called "ranking" and mapping from an integer back to an object is called "unranking".

When a lexicographic ordering makes sense, there will be 1-to-1 correspondence with the ordering on integers. In other words for objects X1 and X2 and their ranks R1 and R2, X1 lex< X2 iff R1 < R2.

The method UNRANK takes a combinatorial specification and an integer, and maps it to the corresponding object representation (usually a vector). It takes an optional keyword argument :SET which acts as a destination of the unranked object (for efficiency purposes).

The method RANK takes a combinatorial specification and an object produced by UNRANK (again, usually a sensible vector) and returns the integer (the "rank") of that object. PRINT-OBJECTS-OF-SPEC, as used above, prints the rank of every object in a combinatorial space.

One can map over all objects and ranks by using MAP-SPEC, which takes a binary function (rank and object) as well as a combinatorial specification, and applies that function to each object and their rank.

Permutation Groups

There is initial support for permutation groups at the moment. Permutation groups are represented by the structure PERM-GROUP.

We can create a permutation group from its generators via GENERATE-PERM-GROUP. A shorthand syntax is provided which, instead of taking a list of PERM objects, takes a list of lists representing perms. This shorthand is GROUP-FROM. For example, the following two are the same group:

PERM> (generate-perm-group (list (make-perm 1 3 2 4)
                                 (make-perm 3 2 4 1)))
#<PERM-GROUP of 2 generators>
PERM> (group-from '((1 3 2 4)
                    (3 2 4 1)))
#<PERM-GROUP of 2 generators>

We can generate a permutation group from a list of cycles as well. The above is equivalent to

PERM> (group-from-cycles (list (list (make-cycle 2 3))
                               (list (make-cycle 1 3 4)))
                         4) 
#<PERM-GROUP of 2 generators>

Once we have generated a group, we can do some operations on it.

For example, let's define the group for 3x3 Rubik's cubes. A cube has six moves: we can turn the front, back, left, right, top, and bottom. Label each sticker with a number like so:

                     +--------------+
                     |              |
                     |  1    2    3 |
                     |              |
                     |  4   up    5 |
                     |              |
                     |  6    7    8 |
                     |              |
      +--------------+--------------+--------------+--------------+
      |              |              |              |              |
      |  9   10   11 | 17   18   19 | 25   26   27 | 33   34   35 |
      |              |              |              |              |
      | 12  left  13 | 20 front  21 | 28 right  29 | 36  back  37 |
      |              |              |              |              |
      | 14   15   16 | 22   23   24 | 30   31   32 | 38   39   40 |
      |              |              |              |              |
      +--------------+--------------+--------------+--------------+
                     |              |
                     | 41   42   43 |
                     |              |
                     | 44  down  45 |
                     |              |
                     | 46   47   48 |
                     |              |
                     +--------------+

Each turn corresponds to a permutation of stickers. I'll do the hard work of specifying them:

(defparameter rubik-3x3
  (group-from
   '((3 5 8 2 7 1 4 6 33 34 35 12 13 14 15 16 9 10 11 20 21 22 23 24 17 
      18 19 28 29 30 31 32 25 26 27 36 37 38 39 40 41 42 43 44 45 46 47 48)
     (17 2 3 20 5 22 7 8 11 13 16 10 15 9 12 14 41 18 19 44 21 46 23 24 
      25 26 27 28 29 30 31 32 33 34 6 36 4 38 39 1 40 42 43 37 45 35 47 48) 
     (1 2 3 4 5 25 28 30 9 10 8 12 7 14 15 6 19 21 24 18 23 17 20 22 43 
      26 27 42 29 41 31 32 33 34 35 36 37 38 39 40 11 13 16 44 45 46 47 48) 
     (1 2 38 4 36 6 7 33 9 10 11 12 13 14 15 16 17 18 3 20 5 22 23 8 27 
      29 32 26 31 25 28 30 48 34 35 45 37 43 39 40 41 42 19 44 21 46 47 24) 
     (14 12 9 4 5 6 7 8 46 10 11 47 13 48 15 16 17 18 19 20 21 22 23 24
      25 26 1 28 2 30 31 3 35 37 40 34 39 33 36 38 41 42 43 44 45 32 29 27) 
     (1 2 3 4 5 6 7 8 9 10 11 12 13 22 23 24 17 18 19 20 21 30 31 32 25
      26 27 28 29 38 39 40 33 34 35 36 37 14 15 16 43 45 48 42 47 41 44 46))))

Now we have our group:

PERM> rubik-3x3
#<PERM-GROUP of 6 generators>

Let's query the group's order:

PERM> (group-order rubik-3x3)
43252003274489856000

A lot of positions! Let's generate a random cube:

PERM> (random-group-element rubik-3x3)
#<PERM 1 20 24 39 12 40 29 41 9 47 46 21 45 11 34 8 14 36 22 31 44 25 10 48
       16 37 43 15 26 32 7 33 30 13 35 5 28 27 23 17 19 4 38 2 18 6 42 3>

And as cycles...

PERM> (to-cycles *)
(#<CYCLE (35)>
 #<CYCLE (30 32 33)>
 #<CYCLE (27 43 38)>
 #<CYCLE (9)>
 #<CYCLE (8 41 19 22 25 16)>
 #<CYCLE (6 40 17 14 11 46)>
 #<CYCLE (4 39 23 10 47 42)>
 #<CYCLE (3 24 48)>
 #<CYCLE (2 20 31 7 29 26 37 28 15 34 13 45 18 36 5 12 21 44)>
 #<CYCLE (1)>)

Let's check if flipping an edge piece is valid:

PERM> (group-element-p (from-cycles (list (make-cycle 7 18)) 48) rubik-3x3)
NIL

No it's not. How about four edge pieces?

PERM> (group-element-p (from-cycles (list (make-cycle 7 18)
                                          (make-cycle 2 34)
                                          (make-cycle 4 10)
                                          (make-cycle 5 26))
                                    48)
                       rubik-3x3)
T

As can be seen, the few operations we have are powerful in studying finite groups.