46 Combinatorics

This chapter describes the functions that deal with combinatorics. We mainly concentrate on two areas. One is about selections, that is the ways one can select elements from a set. The other is about partitions, that is the ways one can partition a set into the union of pairwise disjoint subsets.

First this package contains various functions that are related to the number of selections from a set (see Factorial, Binomial) or to the number of partitions of a set (see Bell, Stirling1, Stirling2). Those numbers satisfy literally thousands of identities, which we do no mention in this document, for a thorough treatment see GKP90.

Then this package contains functions to compute the selections from a set (see Combinations), ordered selections, i.e., selections where the order in which you select the elements is important (see Arrangements), selections with repetitions, i.e., you are allowed to select the same element more than once (see UnorderedTuples) and ordered selections with repetitions (see Tuples).

As special cases of ordered combinations there are functions to compute all permutations (see PermutationsList), all fixpointfree permutations (see Derangements) of a list.

This package also contains functions to compute partitions of a set (see PartitionsSet), partitions of an integer into the sum of positive integers (see Partitions, RestrictedPartitions) and ordered partitions of an integer into the sum of positive integers (see OrderedPartitions).

Moreover, it provides three functions to compute Fibonacci numbers (see Fibonacci), Lucas sequences (see Lucas), or Bernoulli numbers (see Bernoulli).

Finally, there is a function to compute the number of permutations that fit a given 1-0 matrix (see Permanent).

All these functions are in the file "LIBNAME/combinat.g".

Subsections

  1. Factorial
  2. Binomial
  3. Bell
  4. Stirling1
  5. Stirling2
  6. Combinations
  7. Arrangements
  8. UnorderedTuples
  9. Tuples
  10. PermutationsList
  11. Derangements
  12. PartitionsSet
  13. Partitions
  14. OrderedPartitions
  15. RestrictedPartitions
  16. SignPartition
  17. AssociatedPartition
  18. PowerPartition
  19. PartitionTuples
  20. Fibonacci
  21. Lucas
  22. Bernoulli
  23. Permanent

46.1 Factorial

Factorial( n )

Factorial returns the factorial n! of the positive integer n, which is defined as the product 1 * 2 * 3 * .. * n.

n! is the number of permutations of a set of n elements. 1/n! is the coefficient of x^n in the formal series e^x, which is the generating function for factorial.

    gap> List( [0..10], Factorial );
    [ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
    gap> Factorial( 30 );
    265252859812191058636308480000000 

PermutationsList (see PermutationsList) computes the set of all permutations of a list.

46.2 Binomial

Binomial( n, k )

Binomial returns the binomial coefficient {n choose k} of integers n and k, which is defined as n! / (k! (n-k)!) (see Factorial). We define {0 choose 0} = 1, {n choose k} = 0 if k<0 or n, and {n choose k} = (-1)^k {-n+k-1 choose k} if n < 0, which is consistent with {n choose k} = {n-1 choose k} + {n-1 choose k-1}.

{n choose k} is the number of combinations with k elements, i.e., the number of subsets with k elements, of a set with n elements. {n choose k} is the coefficient of the term x^k of the polynomial (x + 1)^n, which is the generating function for {n choose *}, hence the name.

    gap> List( [0..4], k->Binomial( 4, k ) );
    [ 1, 4, 6, 4, 1 ]    # Knuth calls this the trademark of Binomial
    gap> List( [0..6], n->List( [0..6], k->Binomial( n, k ) ) );;
    gap> PrintArray( last );
    [ [   1,   0,   0,   0,   0,   0,   0 ],    # the lower triangle is
      [   1,   1,   0,   0,   0,   0,   0 ],    # called Pascal\'s triangle
      [   1,   2,   1,   0,   0,   0,   0 ],
      [   1,   3,   3,   1,   0,   0,   0 ],
      [   1,   4,   6,   4,   1,   0,   0 ],
      [   1,   5,  10,  10,   5,   1,   0 ],
      [   1,   6,  15,  20,  15,   6,   1 ] ]
    gap> Binomial( 50, 10 );
    10272278170 

NrCombinations (see Combinations) is the generalization of Binomial for multisets. Combinations (see Combinations) computes the set of all combinations of a multiset.

46.3 Bell

Bell( n )

Bell returns the Bell number B(n). The Bell numbers are defined by B(0)=1 and the recurrence B(n+1) = sum_{k=0}^{n}{{n choose k}B(k)}.

B(n) is the number of ways to partition a set of n elements into pairwise disjoint nonempty subsets (see PartitionsSet). This implies of course that B(n) = sum_{k=0}^{n}{S_2(n,k)} (see Stirling2). B(n)/n! is the coefficient of x^n in the formal series e^{e^x-1}, which is the generating function for B(n).

    gap> List( [0..6], n -> Bell( n ) );
    [ 1, 1, 2, 5, 15, 52, 203 ]
    gap> Bell( 14 );
    190899322 

46.4 Stirling1

Stirling1( n, k )

Stirling1 returns the Stirling number of the first kind S_1(n,k) of the integers n and k. Stirling numbers of the first kind are defined by S_1(0,0) = 1, S_1(n,0) = S_1(0,k) = 0 if n, k <> 0 and the recurrence S_1(n,k) = (n-1) S_1(n-1,k) + S_1(n-1,k-1).

S_1(n,k) is the number of permutations of n points with k cycles. Stirling numbers of the first kind appear as coefficients in the series n! {x choose n} = sum_{k=0}^{n}{S_1(n,k) x^k} which is the generating function for Stirling numbers of the first kind. Note the similarity to x^n = sum_{k=0}^{n}{S_2(n,k) k! {x choose k}} (see Stirling2). Also the definition of S_1 implies S_1(n,k) = S_2(-k,-n) if n,k<0. There are many formulae relating Stirling numbers of the first kind to Stirling numbers of the second kind, Bell numbers, and Binomial numbers.

    gap> List( [0..4], k->Stirling1( 4, k ) );
    [ 0, 6, 11, 6, 1 ]    # Knuth calls this the trademark of $S_1$
    gap> List( [0..6], n->List( [0..6], k->Stirling1( n, k ) ) );;
    gap> PrintArray( last );
    [ [    1,    0,    0,    0,    0,    0,    0 ],    # Note the similarity
      [    0,    1,    0,    0,    0,    0,    0 ],    # with Pascal\'s
      [    0,    1,    1,    0,    0,    0,    0 ],    # triangle for the
      [    0,    2,    3,    1,    0,    0,    0 ],    # Binomial numbers
      [    0,    6,   11,    6,    1,    0,    0 ],
      [    0,   24,   50,   35,   10,    1,    0 ],
      [    0,  120,  274,  225,   85,   15,    1 ] ]
    gap> Stirling1(50,10);
    101623020926367490059043797119309944043405505380503665627365376 

46.5 Stirling2

Stirling2( n, k )

Stirling2 returns the Stirling number of the second kind S_2(n,k) of the integers n and k. Stirling numbers of the second kind are defined by S_2(0,0) = 1, S_2(n,0) = S_2(0,k) = 0 if n, k <> 0 and the recurrence S_2(n,k) = k S_2(n-1,k) + S_2(n-1,k-1).

S_2(n,k) is the number of ways to partition a set of n elements into k pairwise disjoint nonempty subsets (see PartitionsSet). Stirling numbers of the second kind appear as coefficients in the expansion of x^n = sum_{k=0}^{n}{S_2(n,k) k! {x choose k}}. Note the similarity to n! {x choose n} = sum_{k=0}^{n}{S_1(n,k) x^k} (see Stirling1). Also the definition of S_2 implies S_2(n,k) = S_1(-k,-n) if n,k<0. There are many formulae relating Stirling numbers of the second kind to Stirling numbers of the first kind, Bell numbers, and Binomial numbers.

    gap> List( [0..4], k->Stirling2( 4, k ) );
    [ 0, 1, 7, 6, 1 ]    # Knuth calls this the trademark of $S_2$
    gap> List( [0..6], n->List( [0..6], k->Stirling2( n, k ) ) );;
    gap> PrintArray( last );
    [ [   1,   0,   0,   0,   0,   0,   0 ],    # Note the similarity with
      [   0,   1,   0,   0,   0,   0,   0 ],    # Pascal\'s triangle for
      [   0,   1,   1,   0,   0,   0,   0 ],    # the Binomial numbers
      [   0,   1,   3,   1,   0,   0,   0 ],
      [   0,   1,   7,   6,   1,   0,   0 ],
      [   0,   1,  15,  25,  10,   1,   0 ],
      [   0,   1,  31,  90,  65,  15,   1 ] ]
    gap> Stirling2( 50, 10 );
    26154716515862881292012777396577993781727011 

46.6 Combinations

Combinations( mset )
Combinations( mset, k )

NrCombinations( mset )
NrCombinations( mset, k )

In the first form Combinations returns the set of all combinations of the multiset mset. In the second form Combinations returns the set of all combinations of the multiset mset with k elements.

In the first form NrCombinations returns the number of combinations of the multiset mset. In the second form NrCombinations returns the number of combinations of the multiset mset with k elements.

A combination of mset is an unordered selection without repetitions and is represented by a sorted sublist of mset. If mset is a proper set, there are {|mset| choose k} (see Binomial) combinations with k elements, and the set of all combinations is just the powerset of mset, which contains all subsets of mset and has cardinality 2^{|mset|}.

    gap> Combinations( [1,2,2,3] );
    [ [  ], [ 1 ], [ 1, 2 ], [ 1, 2, 2 ], [ 1, 2, 2, 3 ], [ 1, 2, 3 ],
      [ 1, 3 ], [ 2 ], [ 2, 2 ], [ 2, 2, 3 ], [ 2, 3 ], [ 3 ] ]
    gap> NrCombinations( [1..52], 5 );
    2598960    # number of different hands in a game of poker 

The function Arrangements (see Arrangements) computes ordered selections without repetitions, UnorderedTuples (see UnorderedTuples) computes unordered selections with repetitions and Tuples (see Tuples) computes ordered selections with repetitions.

46.7 Arrangements

Arrangements( mset )
Arrangements( mset, k )

NrArrangements( mset )
NrArrangements( mset, k )

In the first form Arrangements returns the set of arrangements of the multiset mset. In the second form Arrangements returns the set of all arrangements with k elements of the multiset mset.

In the first form NrArrangements returns the number of arrangements of the multiset mset. In the second form NrArrangements returns the number of arrangements with k elements of the multiset mset.

An arrangement of mset is an ordered selection without repetitions and is represented by a list that contains only elements from mset, but maybe in a different order. If mset is a proper set there are |mset|! / (|mset|-k)! (see Factorial) arrangements with k elements.

As an example of arrangements of a multiset, think of the game Scrabble. Suppose you have the six characters of the word settle and you have to make a four letter word. Then the possibilities are given by

    gap> Arrangements( ["s","e","t","t","l","e"], 4 );
    [ [ "e", "e", "l", "s" ], [ "e", "e", "l", "t" ],
      [ "e", "e", "s", "l" ], [ "e", "e", "s", "t" ],
      # 96 more possibilities
      [ "t", "t", "s", "e" ], [ "t", "t", "s", "l" ] ] 

Can you find the five proper English words, where lets does not count? Note that the fact that the list returned by Arrangements is a proper set means in this example that the possibilities are listed in the same order as they appear in the dictionary.

    gap> NrArrangements( ["s","e","t","t","l","e"] );
    523 

The function Combinations (see Combinations) computes unordered selections without repetitions, UnorderedTuples (see UnorderedTuples) computes unordered selections with repetitions and Tuples (see Tuples) computes ordered selections with repetitions.

46.8 UnorderedTuples

UnorderedTuples( set, k )

NrUnorderedTuples( set, k )

UnorderedTuples returns the set of all unordered tuples of length k of the set set.

NrUnorderedTuples returns the number of unordered tuples of length k of the set set.

An unordered tuple of length k of set is a unordered selection with repetitions of set and is represented by a sorted list of length k containing elements from set. There are {|set|+k-1 choose k} (see Binomial) such unordered tuples.

Note that the fact that UnOrderedTuples returns a set implies that the last index runs fastest. That means the first tuple contains the smallest element from set k times, the second tuple contains the smallest element of set at all positions except at the last positions, where it contains the second smallest element from set and so on.

As an example for unordered tuples think of a poker-like game played with 5 dice. Then each possible hand corresponds to an unordered five-tuple from the set [1..6]

    gap> NrUnorderedTuples( [1..6], 5 );
    252
    gap> UnorderedTuples( [1..6], 5 );
    [ [ 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 2 ], [ 1, 1, 1, 1, 3 ],
      [ 1, 1, 1, 1, 4 ], [ 1, 1, 1, 1, 5 ], [ 1, 1, 1, 1, 6 ],
      # 99 more tuples
      [ 1, 3, 4, 5, 6 ], [ 1, 3, 4, 6, 6 ], [ 1, 3, 5, 5, 5 ],
      # 99 more tuples
      [ 3, 3, 4, 4, 5 ], [ 3, 3, 4, 4, 6 ], [ 3, 3, 4, 5, 5 ],
      # 39 more tuples
      [ 5, 5, 6, 6, 6 ], [ 5, 6, 6, 6, 6 ], [ 6, 6, 6, 6, 6 ] ] 

The function Combinations (see Combinations) computes unordered selections without repetitions, Arrangements (see Arrangements) computes ordered selections without repetitions and Tuples (see Tuples) computes ordered selections with repetitions.

46.9 Tuples

Tuples( set, k )

NrTuples( set, k )

Tuples returns the set of all ordered tuples of length k of the set set.

NrTuples returns the number of all ordered tuples of length k of the set set.

An ordered tuple of length k of set is an ordered selection with repetition and is represented by a list of length k containing elements of set. There are |set|^k such ordered tuples.

Note that the fact that Tuples returns a set implies that the last index runs fastest. That means the first tuple contains the smallest element from set k times, the second tuple contains the smallest element of set at all positions except at the last positions, where it contains the second smallest element from set and so on.

    gap> Tuples( [1,2,3], 2 );
    [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], 
      [ 3, 1 ], [ 3, 2 ], [ 3, 3 ] ]
    gap> NrTuples( [1..10], 5 );
    100000 

Tuples(set,k) can also be viewed as the k-fold cartesian product of set (see Cartesian).

The function Combinations (see Combinations) computes unordered selections without repetitions, Arrangements (see Arrangements) computes ordered selections without repetitions, and finally the function UnorderedTuples (see UnorderedTuples) computes unordered selections with repetitions.

46.10 PermutationsList

PermutationsList( mset )

NrPermutationsList( mset )

PermutationsList returns the set of permutations of the multiset mset.

NrPermutationsList returns the number of permutations of the multiset mset.

A permutation is represented by a list that contains exactly the same elements as mset, but possibly in different order. If mset is a proper set there are |mset| ! (see Factorial) such permutations. Otherwise if the first elements appears k_1 times, the second element appears k_2 times and so on, the number of permutations is |mset|! / (k_1! k_2! ..), which is sometimes called multinomial coefficient.

    gap> PermutationsList( [1,2,3] );
    [ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 2, 1, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ],
      [ 3, 2, 1 ] ]
    gap> PermutationsList( [1,1,2,2] );
    [ [ 1, 1, 2, 2 ], [ 1, 2, 1, 2 ], [ 1, 2, 2, 1 ], [ 2, 1, 1, 2 ],
      [ 2, 1, 2, 1 ], [ 2, 2, 1, 1 ] ]
    gap> NrPermutationsList( [1,2,2,3,3,3,4,4,4,4] );
    12600 

The function Arrangements (see Arrangements) is the generalization of PermutationsList that allows you to specify the size of the permutations. Derangements (see Derangements) computes permutations that have no fixpoints.

46.11 Derangements

Derangements( list )

NrDerangements( list )

Derangements returns the set of all derangements of the list list.

NrDerangements returns the number of derangements of the list list.

A derangement is a fixpointfree permutation of list and is represented by a list that contains exactly the same elements as list, but in such an order that the derangement has at no position the same element as list. If the list list contains no element twice there are exactly |list|! (1/2! - 1/3! + 1/4! - .. (-1)^n/n!) derangements.

Note that the ratio NrPermutationsList([1..n])/NrDerangements([1..n]), which is n! / (n! (1/2! - 1/3! + 1/4! - .. (-1)^n/n!)) is an approximation for the base of the natural logarithm e = 2.7182818285, which is correct to about n digits.

As an example of derangements suppose that you have to send four different letters to four different people. Then a derangement corresponds to a way to send those letters such that no letter reaches the intended person.

    gap> Derangements( [1,2,3,4] );
    [ [ 2, 1, 4, 3 ], [ 2, 3, 4, 1 ], [ 2, 4, 1, 3 ], [ 3, 1, 4, 2 ],
      [ 3, 4, 1, 2 ], [ 3, 4, 2, 1 ], [ 4, 1, 2, 3 ], [ 4, 3, 1, 2 ],
      [ 4, 3, 2, 1 ] ]
    gap> NrDerangements( [1..10] );
    1334961
    gap> Int( 10^7*NrPermutationsList([1..10])/last );
    27182816
    gap> Derangements( [1,1,2,2,3,3] );
    [ [ 2, 2, 3, 3, 1, 1 ], [ 2, 3, 1, 3, 1, 2 ], [ 2, 3, 1, 3, 2, 1 ],
      [ 2, 3, 3, 1, 1, 2 ], [ 2, 3, 3, 1, 2, 1 ], [ 3, 2, 1, 3, 1, 2 ],
      [ 3, 2, 1, 3, 2, 1 ], [ 3, 2, 3, 1, 1, 2 ], [ 3, 2, 3, 1, 2, 1 ],
      [ 3, 3, 1, 1, 2, 2 ] ]
    gap> NrDerangements( [1,2,2,3,3,3,4,4,4,4] );
    338 

The function PermutationsList (see PermutationsList) computes all permutations of a list.

46.12 PartitionsSet

PartitionsSet( set )
PartitionsSet( set, k )

NrPartitionsSet( set )
NrPartitionsSet( set, k )

In the first form PartitionsSet returns the set of all unordered partitions of the set set. In the second form PartitionsSet returns the set of all unordered partitions of the set set into k pairwise disjoint nonempty sets.

In the first form NrPartitionsSet returns the number of unordered partitions of the set set. In the second form NrPartitionsSet returns the number of unordered partitions of the set set into k pairwise disjoint nonempty sets.

An unordered partition of set is a set of pairwise disjoint nonempty sets with union set and is represented by a sorted list of such sets. There are B( |set| ) (see Bell) partitions of the set set and S_2( |set|, k ) (see Stirling2) partitions with k elements.

    gap> PartitionsSet( [1,2,3] );
    [ [ [ 1 ], [ 2 ], [ 3 ] ], [ [ 1 ], [ 2, 3 ] ], [ [ 1, 2 ], [ 3 ] ],
      [ [ 1, 2, 3 ] ], [ [ 1, 3 ], [ 2 ] ] ]
    gap> PartitionsSet( [1,2,3,4], 2 );
    [ [ [ 1 ], [ 2, 3, 4 ] ], [ [ 1, 2 ], [ 3, 4 ] ],
      [ [ 1, 2, 3 ], [ 4 ] ], [ [ 1, 2, 4 ], [ 3 ] ],
      [ [ 1, 3 ], [ 2, 4 ] ], [ [ 1, 3, 4 ], [ 2 ] ],
      [ [ 1, 4 ], [ 2, 3 ] ] ]
    gap> NrPartitionsSet( [1..6] );
    203
    gap> NrPartitionsSet( [1..10], 3 );
    9330 

Note that PartitionsSet does currently not support multisets and that there is currently no ordered counterpart.

46.13 Partitions

Partitions( n )
Partitions( n, k )

NrPartitions( n )
NrPartitions( n, k )

In the first form Partitions returns the set of all (unordered) partitions of the positive integer n. In the second form Partitions returns the set of all (unordered) partitions of the positive integer n into sums with k summands.

In the first form NrPartitions returns the number of (unordered) partitions of the positive integer n. In the second form NrPartitions returns the number of (unordered) partitions of the positive integer n into sums with k summands.

An unordered partition is an unordered sum n = p_1+p_2 +..+ p_k of positive integers and is represented by the list p = [p_1,p_2,..,p_k], in nonincreasing order, i.e., p_1>=p_2>=..>=p_k. We write pvdash n. There are approximately E^{pi sqrt{2/3 n}} / {4 sqrt{3} n} such partitions.

It is possible to associate with every partition of the integer n a conjugacy class of permutations in the symmetric group on n points and vice versa. Therefore p(n) := NrPartitions(n) is the number of conjugacy classes of the symmetric group on n points.

Ramanujan found the identities p(5i+4) = 0 mod 5, p(7i+5) = 0 mod 7 and p(11i+6) = 0 mod 11 and many other fascinating things about the number of partitions.

Do not call Partitions with an n much larger than 40, in which case there are 37338 partitions, since the list will simply become too large.

    gap> Partitions( 7 );
    [ [ 1, 1, 1, 1, 1, 1, 1 ], [ 2, 1, 1, 1, 1, 1 ], [ 2, 2, 1, 1, 1 ],
      [ 2, 2, 2, 1 ], [ 3, 1, 1, 1, 1 ], [ 3, 2, 1, 1 ], [ 3, 2, 2 ],
      [ 3, 3, 1 ], [ 4, 1, 1, 1 ], [ 4, 2, 1 ], [ 4, 3 ], [ 5, 1, 1 ],
      [ 5, 2 ], [ 6, 1 ], [ 7 ] ]
    gap> Partitions( 8, 3 );
    [ [ 3, 3, 2 ], [ 4, 2, 2 ], [ 4, 3, 1 ], [ 5, 2, 1 ], [ 6, 1, 1 ] ]
    gap> NrPartitions( 7 );
    15
    gap> NrPartitions( 100 );
    190569292 

The function OrderedPartitions (see OrderedPartitions) is the ordered counterpart of Partitions.

46.14 OrderedPartitions

OrderedPartitions( n )
OrderedPartitions( n, k )

NrOrderedPartitions( n )
NrOrderedPartitions( n, k )

In the first form OrderedPartitions returns the set of all ordered partitions of the positive integer n. In the second form OrderedPartitions returns the set of all ordered partitions of the positive integer n into sums with k summands.

In the first form NrOrderedPartitions returns the number of ordered partitions of the positive integer n. In the second form NrOrderedPartitions returns the number of ordered partitions of the positive integer n into sums with k summands.

An ordered partition is an ordered sum n = p_1 + p_2 + .. + p_k of positive integers and is represented by the list [ p_1, p_2, .., p_k ]. There are totally 2^{n-1} ordered partitions and {n-1 choose k-1} (see Binomial) partitions with k summands.

Do not call OrderedPartitions with an n larger than 15, the list will simply become too large.

    gap> OrderedPartitions( 5 );
    [ [ 1, 1, 1, 1, 1 ], [ 1, 1, 1, 2 ], [ 1, 1, 2, 1 ], [ 1, 1, 3 ],
      [ 1, 2, 1, 1 ], [ 1, 2, 2 ], [ 1, 3, 1 ], [ 1, 4 ], [ 2, 1, 1, 1 ],
      [ 2, 1, 2 ], [ 2, 2, 1 ], [ 2, 3 ], [ 3, 1, 1 ], [ 3, 2 ], 
      [ 4, 1 ], [ 5 ] ]
    gap> OrderedPartitions( 6, 3 );
    [ [ 1, 1, 4 ], [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 4, 1 ], [ 2, 1, 3 ],
      [ 2, 2, 2 ], [ 2, 3, 1 ], [ 3, 1, 2 ], [ 3, 2, 1 ], [ 4, 1, 1 ] ]
    gap> NrOrderedPartitions(20);
    524288 

The function Partitions (see Partitions) is the unordered counterpart of OrderedPartitions.

46.15 RestrictedPartitions

RestrictedPartitions( n, set )
RestrictedPartitions( n, set, k )

NrRestrictedPartitions( n, set )
NrRestrictedPartitions( n, set, k )

In the first form RestrictedPartitions returns the set of all restricted partitions of the positive integer n with the summands of the partition coming from the set set. In the second form RestrictedPartitions returns the set of all partitions of the positive integer n into sums with k summands with the summands of the partition coming from the set set.

In the first form NrRestrictedPartitions returns the number of restricted partitions of the positive integer n with the summands coming from the set set. In the second form NrRestrictedPartitions returns the number of restricted partitions of the positive integer n into sums with k summands with the summands of the partition coming from the set set.

A restricted partition is like an ordinary partition (see Partitions) an unordered sum n = p_1+p_2 +..+ p_k of positive integers and is represented by the list p = [p_1,p_2,..,p_k], in nonincreasing order. The difference is that here the p_i must be elements from the set set, while for ordinary partitions they may be elements from [1..n].

    gap> RestrictedPartitions( 8, [1,3,5,7] );
    [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 3, 1, 1, 1, 1, 1 ], [ 3, 3, 1, 1 ],
      [ 5, 1, 1, 1 ], [ 5, 3 ], [ 7, 1 ] ]
    gap> NrRestrictedPartitions( 50, [1,5,10,25,50] );
    50 

The last example tells us that there are 50 ways to return 50 cent change using 1, 5, 10 cent coins, quarters and halfdollars.

46.16 SignPartition

SignPartition( pi )

returns the sign of a permutation with cycle structure pi.

    gap> SignPartition([6,5,4,3,2,1]);
    -1

This function actually describes a homomorphism of the symmetric group S_n into the cyclic group of order 2, whose kernel is exactly the alternating group A_n (see SignPerm). Partitions of sign 1 are called even partitions while partitions of sign -1 are called odd.

46.17 AssociatedPartition

AssociatedPartition( pi )

returns the associated partition of the partition pi.

    gap> AssociatedPartition([4,2,1]);
    [ 3, 2, 1, 1 ]
    gap> AssociatedPartition([6]);
    [ 1, 1, 1, 1, 1, 1 ]

The associated partition of a partition pi is defined to be the partition belonging to the transposed of the Young diagram of pi.

46.18 PowerPartition

PowerPartition( pi, k )

returns the partition corresponding to the k-th power of a permutation with cycle structure pi.

    gap> PowerPartition([6,5,4,3,2,1], 3);
    [ 5, 4, 2, 2, 2, 2, 1, 1, 1, 1 ]

Each part l of pi is replaced by d = gcd(l, k) parts l/d. So if pi is a partition of n then <pi>^{<k>} also is a partition of n. PowerPartition describes the powermap of symmetric groups.

46.19 PartitionTuples

PartitionTuples( n, r )

returns the list of all r--tuples of partitions that together partition n.

    gap> PartitionTuples(3, 2);
    [ [ [ 1, 1, 1 ], [  ] ], [ [ 1, 1 ], [ 1 ] ], [ [ 1 ], [ 1, 1 ] ],
      [ [  ], [ 1, 1, 1 ] ], [ [ 2, 1 ], [  ] ], [ [ 1 ], [ 2 ] ],
      [ [ 2 ], [ 1 ] ], [ [  ], [ 2, 1 ] ], [ [ 3 ], [  ] ],
      [ [  ], [ 3 ] ] ] 

r--tuples of partitions describe the classes and the characters of wreath products of groups with r conjugacy classes with the symmetric group S_n.

46.20 Fibonacci

Fibonacci( n )

Fibonacci returns the nth number of the Fibonacci sequence. The Fibonacci sequence F_n is defined by the initial conditions F_1=F_2=1 and the recurrence relation F_{n+2} = F_{n+1} + F_{n}. For negative n we define F_n = (-1)^{n+1} F_{-n}, which is consistent with the recurrence relation.

Using generating functions one can prove that F_n = phi^n - 1/phi^n, where phi is (sqrt{5} + 1)/2, i.e., one root of x^2 - x - 1 = 0. Fibonacci numbers have the property Gcd( F_m, F_n ) = F_{Gcd(m,n)}. But a pair of Fibonacci numbers requires more division steps in Euclid's algorithm (see Gcd) than any other pair of integers of the same size. Fibonnaci(k) is the special case Lucas(1,-1,k)[1] (see Lucas).

    gap> Fibonacci( 10 );
    55
    gap> Fibonacci( 35 );
    9227465
    gap> Fibonacci( -10 );
    -55 

46.21 Lucas

Lucas( P, Q, k )

Lucas returns the k-th values of the Lucas sequence with parameters P and Q, which must be integers, as a list of three integers.

Let alpha, beta be the two roots of x^2 - P x + Q then we define
Lucas( P, Q, k )[1] = U_k = (alpha^k - beta^k) / (alpha - beta) and
Lucas( P, Q, k )[2] = V_k = (alpha^k + beta^k) and as a convenience
Lucas( P, Q, k )[3] = Q^k.

The following recurrence relations are easily derived from the definition
U_0 = 0, U_1 = 1, U_k = P U_{k-1} - Q U_{k-2} and
V_0 = 2, V_1 = P, V_k = P V_{k-1} - Q V_{k-2}.
Those relations are actually used to define Lucas if alpha = beta.

Also the more complex relations used in Lucas can be easily derived
U_{2k} = U_k V_k, U_{2k+1} = (P U_{2k} + V_{2k}) / 2 and
V_{2k} = V_k^2 - 2 Q^k, V_{2k+1} = ((P^2-4Q) U_{2k} + P V_{2k}) / 2.

Fibonnaci(k) (see Fibonacci) is simply Lucas(1,-1,k)[1]. In an abuse of notation, the sequence Lucas(1,-1,k)[2] is sometimes called the Lucas sequence.

    gap> List( [0..10], i->Lucas(1,-2,i)[1] );
    [ 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341 ]    # $2^k - (-1)^k)/3$
    gap> List( [0..10], i->Lucas(1,-2,i)[2] );
    [ 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025 ]    # $2^k + (-1)^k$
    gap> List( [0..10], i->Lucas(1,-1,i)[1] );
    [ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ]    # Fibonacci sequence
    gap> List( [0..10], i->Lucas(2,1,i)[1] );
    [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]    # the roots are equal 

46.22 Bernoulli

Bernoulli( n )

Bernoulli returns the n-th Bernoulli number B_n, which is defined by B_0 = 1 and B_n = -sum_{k=0}^{n-1}{{n+1 choose k} B_k}/(n+1).

B_n/n! is the coefficient of x^n in the power series of x/{e^x-1}. Except for B_1=-1/2 the Bernoulli numbers for odd indices m are zero.

    gap> Bernoulli( 4 );
    -1/30
    gap> Bernoulli( 10 );
    5/66
    gap> Bernoulli( 12 );
    -691/2730    # there is no simple pattern in Bernoulli numbers
    gap> Bernoulli( 50 );
    495057205241079648212477525/66    # and they grow fairly fast 

46.23 Permanent

Permanent( mat )

Permanent returns the permanent of the matrix mat. The permanent is defined by sum_{p in Symm(n)}{prod_{i=1}^{n}{mat[i][i^p]}}.

Note the similarity of the definition of the permanent to the definition of the determinant. In fact the only difference is the missing sign of the permutation. However the permanent is quite unlike the determinant, for example it is not multilinear or alternating. It has however important combinatorical properties.

    gap> Permanent( [[0,1,1,1],
    >                [1,0,1,1],
    >                [1,1,0,1],
    >                [1,1,1,0]] );
    9    # inefficient way to compute 'NrDerangements([1..4])'
    gap> Permanent( [[1,1,0,1,0,0,0],
    >                [0,1,1,0,1,0,0],
    >                [0,0,1,1,0,1,0],
    >                [0,0,0,1,1,0,1],
    >                [1,0,0,0,1,1,0],
    >                [0,1,0,0,0,1,1],
    >                [1,0,1,0,0,0,1]] );
    24    # 24 permutations fit the projective plane of order 2 

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GAP 3.4.4
April 1997