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16 Combinatorics
16.1 Combinatorial Numbers
16.1-1 Factorial
16.1-2 Binomial
16.1-3 Bell
16.1-4 Bernoulli
16.1-5 Stirling1
16.1-6 Stirling2
16.2 Combinations, Arrangements and Tuples
16.2-1 Combinations
16.2-2 Iterator and enumerator of combinations
16.2-3 NrCombinations
16.2-4 Arrangements
16.2-5 NrArrangements
16.2-6 UnorderedTuples
16.2-7 NrUnorderedTuples
16.2-8 Tuples
16.2-9 EnumeratorOfTuples
16.2-10 IteratorOfTuples
16.2-11 NrTuples
16.2-12 PermutationsList
16.2-13 NrPermutationsList
16.2-14 Derangements
16.2-15 NrDerangements
16.2-16 PartitionsSet
16.2-17 NrPartitionsSet
16.2-18 Partitions
16.2-19 IteratorOfPartitions
16.2-20 Iterator and enumerator of unordered set partitions
16.2-21 NrPartitions
16.2-22 OrderedPartitions
16.2-23 NrOrderedPartitions
16.2-24 PartitionsGreatestLE
16.2-25 PartitionsGreatestEQ
16.2-26 RestrictedPartitions
16.2-27 NrRestrictedPartitions
16.2-28 SignPartition
16.2-29 AssociatedPartition
16.2-30 PowerPartition
16.2-31 PartitionTuples
16.2-32 NrPartitionTuples
16.2-33 BetaSet
16.3 Fibonacci and Lucas Sequences
16.3-1 Fibonacci
16.3-2 Lucas
16.4 Permanent of a Matrix
16.4-1 Permanent

### 16 Combinatorics

This chapter describes 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.

#### 16.1 Combinatorial Numbers

##### 16.1-1 Factorial

`‣ Factorial`( n ) | ( function )

returns the factorial \(n!\) of the positive integer n, which is defined as the product \(1 \cdot 2 \cdot 3 \cdots n\).

\(n!\) is the number of permutations of a set of \(n\) elements. \(1 / n!\) is the coefficient of \(x^n\) in the formal series \(\exp(x)\), 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` (16.2-12) computes the set of all permutations of a list.

##### 16.1-2 Binomial

`‣ Binomial`( n, k ) | ( function )

returns the binomial coefficient \({{n \choose k}}\) of integers n and k. This is defined by the conditions \({{n \choose k}} = 0\) for \(k < 0\), \({{0 \choose k}} = 0\) for \(k \neq 0\), \({{0 \choose 0}} = 1\) and the relation \({{n \choose k}} = {{n-1 \choose k}} + {{n-1 \choose k-1}}\) for all \(n\) and \(k\).

There are many ways of describing this function. For example, if \(n \geq 0\) and \(0 \leq k \leq n\), then \({{n \choose k}} = n! / (k! (n-k)!)\) and for \(n < 0\) and \(k \geq 0\) we have \({{n \choose k}} = (-1)^k {{-n+k-1 \choose k}}\).

If \(n \geq 0\) then \({{n \choose k}}\) is the number of subsets with \(k\) elements of a set with \(n\) elements. Also, \({{n \choose k}}\) is the coefficient of \(x^k\) in the polynomial \((x + 1)^n\), which is the generating function for \({{n \choose .}}\), hence the name.

    gap> # Knuth calls this the trademark of Binomial:
    gap> List( [0..4], k->Binomial( 4, k ) );
    [ 1, 4, 6, 4, 1 ]
    gap> List( [0..6], n->List( [0..6], k->Binomial( n, k ) ) );;
    gap> # the lower triangle is called Pascal's triangle:
    gap> PrintArray( last );
    [ [   1,   0,   0,   0,   0,   0,   0 ],
      [   1,   1,   0,   0,   0,   0,   0 ],
      [   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` (16.2-3) is the generalization of `Binomial` for multisets. `Combinations` (16.2-1) computes the set of all combinations of a multiset.

##### 16.1-3 Bell

`‣ Bell`( n ) | ( function )

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` (16.2-16)). This implies of course that \(B(n) = \sum_{{k = 0}}^n S_2(n,k)\) (see `Stirling2` (16.1-6)). \(B(n)/n!\) is the coefficient of \(x^n\) in the formal series \(\exp( \exp(x)-1 )\), 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

##### 16.1-4 Bernoulli

`‣ Bernoulli`( n ) | ( function )

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 \(x / (\exp(x)-1)\). Except for \(B_1 = -1/2\) the Bernoulli numbers for odd indices are zero.

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

##### 16.1-5 Stirling1

`‣ Stirling1`( n, k ) | ( function )

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 \(n, k \ne 0\) 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` (16.1-6)). 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 coefficients.

    gap> # Knuth calls this the trademark of S_1:
    gap> List( [0..4], k -> Stirling1( 4, k ) );
    [ 0, 6, 11, 6, 1 ]
    gap> List( [0..6], n->List( [0..6], k->Stirling1( n, k ) ) );;
    gap> # note the similarity with Pascal's triangle for Binomial numbers
    gap> PrintArray( last );
    [ [    1,    0,    0,    0,    0,    0,    0 ],
      [    0,    1,    0,    0,    0,    0,    0 ],
      [    0,    1,    1,    0,    0,    0,    0 ],
      [    0,    2,    3,    1,    0,    0,    0 ],
      [    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

##### 16.1-6 Stirling2

`‣ Stirling2`( n, k ) | ( function )

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 \(n, k \ne 0\) 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` (16.2-16)). 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` (16.1-5)). 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 coefficients.

    gap> # Knuth calls this the trademark of S_2:
    gap> List( [0..4], k->Stirling2( 4, k ) );
    [ 0, 1, 7, 6, 1 ]
    gap> List( [0..6], n->List( [0..6], k->Stirling2( n, k ) ) );;
    gap> # note the similarity with Pascal's triangle for Binomial numbers
    gap> PrintArray( last );
    [ [   1,   0,   0,   0,   0,   0,   0 ],
      [   0,   1,   0,   0,   0,   0,   0 ],
      [   0,   1,   1,   0,   0,   0,   0 ],
      [   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

#### 16.2 Combinations, Arrangements and Tuples

##### 16.2-1 Combinations

`‣ Combinations`( mset[, k] ) | ( function )

returns the set of all combinations of the multiset mset (a list of objects which may contain the same object several times) with k elements; if k is not given it returns all combinations of mset.

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 \({{|\textit{mset}| \choose \textit{k}}}\) (see `Binomial` (16.1-2)) combinations with k elements, and the set of all combinations is just the power set of mset, which contains all subsets of mset and has cardinality \(2^{{|\textit{mset}|}}\).

To loop over combinations of a larger multiset use `IteratorOfCombinations` (16.2-2) which produces combinations one by one and may save a lot of memory. Another memory efficient representation of the list of all combinations is provided by `EnumeratorOfCombinations` (16.2-2).

##### 16.2-2 Iterator and enumerator of combinations

`‣ IteratorOfCombinations`( mset[, k] ) | ( function )

`‣ EnumeratorOfCombinations`( mset ) | ( function )

`IteratorOfCombinations` returns an `Iterator` (30.8-1) for combinations (see `Combinations` (16.2-1)) of the given multiset mset. If a non-negative integer k is given as second argument then only the combinations with k entries are produced, otherwise all combinations.

`EnumeratorOfCombinations` returns an `Enumerator` (30.3-2) of the given multiset mset. Currently only a variant without second argument k is implemented.

The ordering of combinations from these functions can be different and also different from the list returned by `Combinations` (16.2-1).

    gap> m:=[1..15];; Add(m, 15);
    gap> NrCombinations(m);
    49152
    gap> i := 0;; for c in Combinations(m) do i := i+1; od;
    gap> i;
    49152
    gap> cm := EnumeratorOfCombinations(m);;
    gap> cm[1000];
    [ 1, 2, 3, 6, 7, 8, 9, 10 ]
    gap> Position(cm, [1,13,15,15]);
    36866

##### 16.2-3 NrCombinations

`‣ NrCombinations`( mset[, k] ) | ( function )

returns the number of `Combinations(mset,k)`.

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

The function `Arrangements` (16.2-4) computes ordered selections without repetitions, `UnorderedTuples` (16.2-6) computes unordered selections with repetitions, and `Tuples` (16.2-8) computes ordered selections with repetitions.

##### 16.2-4 Arrangements

`‣ Arrangements`( mset[, k] ) | ( function )

returns the set of arrangements of the multiset mset that contain k elements. If k is not given it returns all arrangements of 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` (16.1-1)) arrangements with k elements.

##### 16.2-5 NrArrangements

`‣ NrArrangements`( mset[, k] ) | ( function )

returns the number of `Arrangements(mset,k)`.

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" ], [ "e", "e", "t", "l" ], [ "e", "e", "t", "s" ],
      ... 93 more possibilities ...
      [ "t", "t", "l", "s" ], [ "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` (16.2-4) 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` (16.2-1) computes unordered selections without repetitions, `UnorderedTuples` (16.2-6) computes unordered selections with repetitions, and `Tuples` (16.2-8) computes ordered selections with repetitions.

##### 16.2-6 UnorderedTuples

`‣ UnorderedTuples`( set, k ) | ( function )

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

An unordered tuple of length k of set is an 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` (16.1-2)) 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.

##### 16.2-7 NrUnorderedTuples

`‣ NrUnorderedTuples`( set, k ) | ( function )

returns the number of `UnorderedTuples(set,k)`.

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 \(\{ 1, 2, \ldots, 6 \}\).

    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 ], [ 1, 1, 1, 2, 2 ], [ 1, 1, 1, 2, 3 ],
      ... 100 more tuples ...
      [ 1, 3, 5, 5, 6 ], [ 1, 3, 5, 6, 6 ], [ 1, 3, 6, 6, 6 ], [ 1, 4, 4, 4, 4 ],
      ... 100 more tuples ...
      [ 3, 3, 5, 5, 5 ], [ 3, 3, 5, 5, 6 ], [ 3, 3, 5, 6, 6 ], [ 3, 3, 6, 6, 6 ],
      ... 32 more tuples ...
      [ 5, 5, 5, 6, 6 ], [ 5, 5, 6, 6, 6 ], [ 5, 6, 6, 6, 6 ], [ 6, 6, 6, 6, 6 ] ]

The function `Combinations` (16.2-1) computes unordered selections without repetitions, `Arrangements` (16.2-4) computes ordered selections without repetitions, and `Tuples` (16.2-8) computes ordered selections with repetitions.

##### 16.2-8 Tuples

`‣ Tuples`( set, k ) | ( function )

returns the set 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 \(|\textit{set}|^{\textit{k}}\) 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.

##### 16.2-9 EnumeratorOfTuples

`‣ EnumeratorOfTuples`( set, k ) | ( function )

This function is referred to as an example of enumerators that are defined by functions but are not constructed from a domain. The result is equal to that of `Tuples( set, k )`. However, the entries are not stored physically in the list but are created/identified on demand.

##### 16.2-10 IteratorOfTuples

`‣ IteratorOfTuples`( set, k ) | ( function )

For a set set and a positive integer k, `IteratorOfTuples` returns an iterator (see 30.8) of the set of all ordered tuples (see `Tuples` (16.2-8)) of length k of the set set. The tuples are returned in lexicographic order.

##### 16.2-11 NrTuples

`‣ NrTuples`( set, k ) | ( function )

returns the number of `Tuples(set,k)`.

    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` (21.20-15)).

The function `Combinations` (16.2-1) computes unordered selections without repetitions, `Arrangements` (16.2-4) computes ordered selections without repetitions, and finally the function `UnorderedTuples` (16.2-6) computes unordered selections with repetitions.

##### 16.2-12 PermutationsList

`‣ PermutationsList`( mset ) | ( function )

`PermutationsList` returns the set 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 \(|\textit{mset}| !\) (see `Factorial` (16.1-1)) 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 \(|\textit{mset}| ! / (k_1! k_2! \ldots)\), which is sometimes called multinomial coefficient.

##### 16.2-13 NrPermutationsList

`‣ NrPermutationsList`( mset ) | ( function )

returns the number of `PermutationsList(mset)`.

    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` (16.2-4) is the generalization of `PermutationsList` (16.2-12) that allows you to specify the size of the permutations. `Derangements` (16.2-14) computes permutations that have no fixed points.

##### 16.2-14 Derangements

`‣ Derangements`( list ) | ( function )

returns the set of all 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 \(|\textit{list}|! (1/2! - 1/3! + 1/4! - \cdots + (-1)^n / n!)\) derangements.

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

##### 16.2-15 NrDerangements

`‣ NrDerangements`( list ) | ( function )

returns the number of `Derangements(list)`.

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` (16.2-12) computes all permutations of a list.

##### 16.2-16 PartitionsSet

`‣ PartitionsSet`( set[, k] ) | ( function )

returns the set of all unordered partitions of the set set into k pairwise disjoint nonempty sets. If k is not given it returns all unordered partitions of set for all k.

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` (16.1-3)) partitions of the set set and \(S_2( |set|, k )\) (see `Stirling2` (16.1-6)) partitions with k elements.

##### 16.2-17 NrPartitionsSet

`‣ NrPartitionsSet`( set[, k] ) | ( function )

returns the number of `PartitionsSet(set,k)`.

    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` (16.2-16) does currently not support multisets and that there is currently no ordered counterpart.

##### 16.2-18 Partitions

`‣ Partitions`( n[, k] ) | ( function )

returns the set of all (unordered) partitions of the positive integer n into sums with k summands. If k is not given it returns all unordered partitions of n for all k.

An unordered partition is an unordered sum \(n = p_1 + p_2 + \cdots + p_k\) of positive integers and is represented by the list \(p = [ p_1, p_2, \ldots, p_k ]\), in nonincreasing order, i.e., \(p_1 \geq p_2 \geq \ldots \geq p_k\). We write \(p \vdash n\). There are approximately \(\exp(\pi \sqrt{{2/3 n}}) / (4 \sqrt{{3}} n)\) such partitions, use `NrPartitions` (16.2-21) to compute the precise number.

If you want to loop over all partitions of some larger n use the more memory efficient `IteratorOfPartitions` (16.2-19).

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.

##### 16.2-19 IteratorOfPartitions

`‣ IteratorOfPartitions`( n ) | ( function )

For a positive integer n, `IteratorOfPartitions` returns an iterator (see 30.8) of the set of partitions of n (see `Partitions` (16.2-18)). The partitions of n are returned in lexicographic order.

##### 16.2-20 Iterator and enumerator of unordered set partitions

`‣ IteratorOfPartitionsSet`( set[, k[, flag]] ) | ( function )

`‣ EnumeratorOfPartitionsSet`( set[, k[, flag]] ) | ( function )

`IteratorOfPartitionsSet` returns an iterator (see 30.8) for all unordered partitions of the set set into pairwise disjoint nonempty sets (see `PartitionsSet` (16.2-16)). If k given and flag is omitted or equal to `false`, then only partitions of size k are computed. If k is given and flag is equal to `true`, then only partitions of size at most k are computed.

`EnumeratorOfPartitionsSet` returns an enumerator (see `Enumerator` (30.3-2)) for all unordered partitions of the set set. The arguments k and flag function in the same manner as for the iterator.

The ordering of the partitions from these functions can be different and also different from the list returned by `PartitionsSet` (16.2-16).

    gap> m:=[1..9];;
    gap> Bell(9);
    21147
    gap> Length(PartitionsSet(m));
    21147
    gap> cm := EnumeratorOfPartitionsSet(m);;
    gap> cm[1000];
    [ [ 1, 2, 4, 9 ], [ 3, 6, 8 ], [ 5, 7 ] ]
    gap> Position(cm, last);
    1000

##### 16.2-21 NrPartitions

`‣ NrPartitions`( n[, k] ) | ( function )

returns the number of `Partitions(set,k)`.

    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` (16.2-22) is the ordered counterpart of `Partitions` (16.2-18).

##### 16.2-22 OrderedPartitions

`‣ OrderedPartitions`( n[, k] ) | ( function )

returns the set of all ordered partitions of the positive integer n into sums with k summands. If k is not given it returns all ordered partitions of set for all k.

An ordered partition is an ordered sum \(n = p_1 + p_2 + \ldots + p_k\) of positive integers and is represented by the list \([ p_1, p_2, \ldots, p_k ]\). There are totally \(2^{{n-1}}\) ordered partitions and \({{n-1 \choose k-1}}\) (see `Binomial` (16.1-2)) ordered partitions with k summands.

Do not call `OrderedPartitions` with an n much larger than \(15\), the list will simply become too large.

##### 16.2-23 NrOrderedPartitions

`‣ NrOrderedPartitions`( n[, k] ) | ( function )

returns the number of `OrderedPartitions(set,k)`.

    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` (16.2-18) is the unordered counterpart of `OrderedPartitions` (16.2-22).

##### 16.2-24 PartitionsGreatestLE

`‣ PartitionsGreatestLE`( n, m ) | ( function )

returns the set of all (unordered) partitions of the integer n having parts less or equal to the integer m. For \(\textit{n} = 0\) and \(\textit{m} \geq 0\), this returns the empty partition `[[]]`. Negative arguments yield the empty list.

    gap> PartitionsGreatestLE( 6, 3 );
    [ [ 1, 1, 1, 1, 1, 1 ], [ 2, 1, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 2, 2 ],
      [ 3, 1, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ] ]
    gap> List( [ 0 .. 3 ], m -> PartitionsGreatestLE( 0, m ) );
    [ [ [  ] ], [ [  ] ], [ [  ] ], [ [  ] ] ]

##### 16.2-25 PartitionsGreatestEQ

`‣ PartitionsGreatestEQ`( n, m ) | ( function )

returns the set of all (unordered) partitions of the integer n having greatest part equal to the integer m. Since partitions use positive parts, this returns the empty list if \(\textit{n} \leq 0\) or \(\textit{m} \leq 0\).

    gap> PartitionsGreatestEQ( 6, 3 );
    [ [ 3, 1, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ] ]
    gap> PartitionsGreatestEQ( 0, 1 );
    [  ]

##### 16.2-26 RestrictedPartitions

`‣ RestrictedPartitions`( n, set[, k] ) | ( function )

In the first form `RestrictedPartitions` returns the set of all restricted partitions of the positive integer n into sums with k summands with the summands of the partition coming from the set set. If k is not given all restricted partitions for all k are returned.

A restricted partition is like an ordinary partition (see `Partitions` (16.2-18)) an unordered sum \(n = p_1 + p_2 + \ldots + p_k\) of positive integers and is represented by the list \(p = [ p_1, p_2, \ldots, p_k ]\), 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 ]`.

##### 16.2-27 NrRestrictedPartitions

`‣ NrRestrictedPartitions`( n, set[, k] ) | ( function )

returns the number of `RestrictedPartitions(n,set,k)`.

    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,[1,2,5,10,20,50]);
    451

The last example tells us that there are 451 ways to return 50 pence change using 1, 2, 5, 10, 20 and 50 pence coins.

##### 16.2-28 SignPartition

`‣ SignPartition`( pi ) | ( function )

returns the sign of a permutation with cycle structure pi.

This function actually describes a homomorphism from the symmetric group \(S_n\) into the cyclic group of order 2, whose kernel is exactly the alternating group \(A_n\) (see `SignPerm` (42.4-1)). Partitions of sign 1 are called even partitions while partitions of sign \(-1\) are called odd.

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

##### 16.2-29 AssociatedPartition

`‣ AssociatedPartition`( pi ) | ( function )

`AssociatedPartition` returns the associated partition of the partition pi which is obtained by transposing the corresponding Young diagram.

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

##### 16.2-30 PowerPartition

`‣ PowerPartition`( pi, k ) | ( function )

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

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

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

##### 16.2-31 PartitionTuples

`‣ PartitionTuples`( n, r ) | ( function )

`PartitionTuples` returns the list of all r-tuples of partitions which together form a partition of n.

r-tuples of partitions describe the classes and the characters of wreath products of groups with r conjugacy classes with the symmetric group on n points, see `CharacterTableWreathSymmetric` (71.20-6) and `CharacterValueWreathSymmetric` (71.20-7).

##### 16.2-32 NrPartitionTuples

`‣ NrPartitionTuples`( n, r ) | ( function )

returns the number of `PartitionTuples( n, r )`.

    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 ] ] ]

##### 16.2-33 BetaSet

`‣ BetaSet`( alpha ) | ( function )

For a list alpha that describes a partition of a nonnegative integer (see `Partitions` (16.2-18)), `BetaSet` returns the list of integers obtained by reversing the order of alpha and then adding the sequence `[ 0, 1, 2, ... ]` of the same length, cf. [JK81, Section 2.7].

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

#### 16.3 Fibonacci and Lucas Sequences

##### 16.3-1 Fibonacci

`‣ Fibonacci`( n ) | ( function )

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` (56.7-1)) than any other pair of integers of the same size. `Fibonacci(k)` is the special case `Lucas(1,-1,k)[1]` (see `Lucas` (16.3-2)).

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

##### 16.3-2 Lucas

`‣ Lucas`( P, Q, k ) | ( function )

returns the k-th values of the Lucas sequence with parameters P and Q, which must be integers, as a list of three integers. If k is a negative integer, then the values of the Lucas sequence may be nonintegral rational numbers, with denominator roughly Q^k.

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 \(V_{2k} = V_k^2 - 2 Q^k\), \(V_{{2k+1}} = ((P^2-4Q) U_{2k} + P V_{2k}) / 2\).

`Fibonacci(k)` (see `Fibonacci` (16.3-1)) 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] );     # 2^k - (-1)^k)/3
    [ 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341 ]
    gap> List( [0..10], i -> Lucas(1,-2,i)[2] );     # 2^k + (-1)^k
    [ 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025 ]
    gap> List( [0..10], i -> Lucas(1,-1,i)[1] );     # Fibonacci sequence
    [ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ]
    gap> List( [0..10], i -> Lucas(2,1,i)[1] );      # the roots are equal
    [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]

#### 16.4 Permanent of a Matrix

##### 16.4-1 Permanent

`‣ Permanent`( mat ) | ( attribute )

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

Note the similarity of the definition of the permanent to the definition of the determinant (see `DeterminantMat` (24.4-4)). 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 combinatorial properties.

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

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