%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A combinat.tex GAP documentation Martin Schoenert %% %A @(#)$Id: combinat.tex,v 3.13 1993/05/04 11:43:16 fceller Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes the functions that mainly deal with combinatorics. %% %H $Log: combinat.tex,v $ %H Revision 3.13 1993/05/04 11:43:16 fceller %H fixed a spelling error %H %H Revision 3.12 1993/02/19 11:30:42 gap %H removed overfull hboxes %H %H Revision 3.11 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.10 1993/02/12 12:01:43 felsch %H examples adjusted to line length 72 %H %H Revision 3.9 1993/02/09 13:56:41 felsch %H examples fixed %H %H Revision 3.8 1992/04/27 11:55:51 martin %H replaced '\size{}' with '\|\|' %H %H Revision 3.7 1992/03/27 16:17:37 martin %H fixed the citation %H %H Revision 3.6 1992/03/13 14:57:45 goetz %H added some functions from 'charsymm.tex'. %H %H Revision 3.5 1991/12/27 16:07:04 martin %H revised everything for GAP 3.1 manual %H %H Revision 3.4 1991/07/22 14:52:20 martin %H added 'RestrictedPartitions' %H %H Revision 3.3 1991/07/21 12:01:00 martin %H changed 'Partitions' to return partitions in standard form %H %H Revision 3.2 1991/07/01 13:00:00 martin %H added 'Bell' %H %H Revision 3.1 1991/06/28 11:22:00 martin %H fixed some minor typos %H %H Revision 3.0 1991/04/11 11:28:53 martin %H Initial revision under RCS %H %% \Chapter{Combinatorics}% \index{selections}\index{partitions} 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 \cite{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 and the number of permutations that fit a given 1-0 matrix (see "Permanent"). 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"). Finally this package also contains some miscellaneous functions (see "Fibonacci", "Lucas" and "Bernoulli"). All these functions are in the file '\"LIBNAME/combinat.g\"'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Factorial} 'Factorial( )' 'Factorial' returns the *factorial* $n!$ of the positive integer , 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Binomial}% \index{coefficient!binomial}\index{number!binomial} 'Binomial( , )' 'Binomial' returns the *binomial coefficient* ${n \choose k}$ of integers and , 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\ 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Bell}% \index{number!Bell} 'Bell( )' '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 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Stirling1}% \index{Stirling number of the first kind}% \index{number!Stirling, of the first kind} 'Stirling1( , )' 'Stirling1' returns the *Stirling number of the first kind* $S_1(n,k)$ of the integers and . 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 points with 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Stirling2}% \index{Stirling number of the second kind}% \index{number!Stirling, of the second kind} 'Stirling2( , )' 'Stirling2' returns the *Stirling number of the second kind* $S_2(n,k)$ of the integers and . 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 elements into 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Combinations}% \index{NrCombinations}\index{powerset}\index{subsets} 'Combinations( )' \\ 'Combinations( , )' 'NrCombinations( )' \\ 'NrCombinations( , )' In the first form 'Combinations' returns the set of all combinations of the multiset . In the second form 'Combinations' returns the set of all combinations of the multiset with elements. In the first form 'NrCombinations' returns the number of combinations of the multiset . In the second form 'NrCombinations' returns the number of combinations of the multiset with elements. A *combination* of is an unordered selection without repetitions and is represented by a sorted sublist of . If is a proper set, there are ${\|mset\| \choose k}$ (see "Binomial") combinations with elements, and the set of all combinations is just the *powerset* of , which contains all *subsets* of 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Arrangements} 'Arrangements( )' \\ 'Arrangements( , )' 'NrArrangements( )' \\ 'NrArrangements( , )' In the first form 'Arrangements' returns the set of arrangements of the multiset . In the second form 'Arrangements' returns the set of all arrangements with elements of the multiset . In the first form 'NrArrangements' returns the number of arrangements of the multiset . In the second form 'NrArrangements' returns the number of arrangements with elements of the multiset . An *arrangement* of is an ordered selection without repetitions and is represented by a list that contains only elements from , but maybe in a different order. If is a proper set there are $\|mset\|! / (\|mset\|-k)!$ (see "Factorial") arrangements with 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{UnorderedTuples}% \index{NrUnorderedTuples} 'UnorderedTuples( , )' 'NrUnorderedTuples( , )' 'UnorderedTuples' returns the set of all unordered tuples of length of the set . 'NrUnorderedTuples' returns the number of unordered tuples of length of the set . An *unordered tuple* of length of is a unordered selection with repetitions of and is represented by a sorted list of length containing elements from . 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 times, the second tuple contains the smallest element of at all positions except at the last positions, where it contains the second smallest element from and so on. As an example for unordered tuples think of a poker-like game played with 5 dices. 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Tuples}% \index{NrTuples} 'Tuples( , )' 'NrTuples( , )' 'Tuples' returns the set of all ordered tuples of length of the set . 'NrTuples' returns the number of all ordered tuples of length of the set . An *ordered tuple* of length of is an ordered selection with repetition and is represented by a list of length containing elements of . 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 times, the second tuple contains the smallest element of at all positions except at the last positions, where it contains the second smallest element from 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(,)' can also be viewed as the -fold cartesian product of (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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PermutationsList}% \index{NrPermutationsList}\index{permutations!list} 'PermutationsList( )' 'NrPermutationsList( )' 'PermutationsList' returns the set of permutations of the multiset . 'NrPermutationsList' returns the number of permutations of the multiset . A *permutation* is represented by a list that contains exactly the same elements as , but possibly in different order. If 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Derangements}% \index{NrDerangements}\index{permutations!fixpointfree} 'Derangements( )' 'NrDerangements( )' 'Derangements' returns the set of all derangements of the list . 'NrDerangements' returns the number of derangements of the list . A *derangement* is a fixpointfree permutation of and is represented by a list that contains exactly the same elements as , but in such an order that the derangement has at no position the same element as . If the 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Permanent} 'Permanent( )' 'Permanent' returns the *permanent* of the matrix . 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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PartitionsSet}% \index{NrPartitionsSet}\index{partitions!of a set} 'PartitionsSet( )' \\ 'PartitionsSet( , )' 'NrPartitionsSet( )' \\ 'NrPartitionsSet( , )' In the first form 'PartitionsSet' returns the set of all unordered partitions of the set . In the second form 'PartitionsSet' returns the set of all unordered partitions of the set into pairwise disjoint nonempty sets. In the first form 'NrPartitionsSet' returns the number of unordered partitions of the set . In the second form 'NrPartitionsSet' returns the number of unordered partitions of the set into pairwise disjoint nonempty sets. An *unordered partition* of is a set of pairwise disjoint nonempty sets with union and is represented by a sorted list of such sets. There are $B( \|set\| )$ (see "Bell") partitions of the set and $S_2( \|set\|, k )$ (see "Stirling2") partitions with 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Partitions}% \index{NrPartitions}\index{partitions!of an integer} 'Partitions( )' \\ 'Partitions( , )' 'NrPartitions( )' \\ 'NrPartitions( , )' In the first form 'Partitions' returns the set of all (unordered) partitions of the positive integer . In the second form 'Partitions' returns the set of all (unordered) partitions of the positive integer into sums with summands. In the first form 'NrPartitions' returns the number of (unordered) partitions of the positive integer . In the second form 'NrPartitions' returns the number of (unordered) partitions of the positive integer into sums with 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 $p\vdash n$. 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 a conjugacy class of permutations in the symmetric group on points and vice versa. Therefore $p(n) \:= NrPartitions(n)$ is the number of conjugacy classes of the symmetric group on 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 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'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{OrderedPartitions}% \index{NrOrderedPartitions}% \index{partitions!ordered, of an integer}% \index{partitions!improper, of an integer}% 'OrderedPartitions( )' \\ 'OrderedPartitions( , )' 'NrOrderedPartitions( )' \\ 'NrOrderedPartitions( , )' In the first form 'OrderedPartitions' returns the set of all ordered partitions of the positive integer . In the second form 'OrderedPartitions' returns the set of all ordered partitions of the positive integer into sums with summands. In the first form 'NrOrderedPartitions' returns the number of ordered partitions of the positive integer . In the second form 'NrOrderedPartitions' returns the number of ordered partitions of the positive integer into sums with 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 summands. Do not call 'OrderedPartitions' with an 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'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{RestrictedPartitions}% \index{NrRestrictedPartitions}% \index{partitions!restricted, of an integer} 'RestrictedPartitions( , )' \\ 'RestrictedPartitions( , , )' 'NrRestrictedPartitions( , )' \\ 'NrRestrictedPartitions( , , )' In the first form 'RestrictedPartitions' returns the set of all restricted partitions of the positive integer with the summands of the partition coming from the set . In the second form 'RestrictedPartitions' returns the set of all partitions of the positive integer into sums with summands with the summands of the partition coming from the set . In the first form 'NrRestrictedPartitions' returns the number of restricted partitions of the positive integer with the summands coming from the set . In the second form 'NrRestrictedPartitions' returns the number of restricted partitions of the positive integer into sums with summands with the summands of the partition coming from the 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 , 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{SignPartition} 'SignPartition( )' returns the sign of a permutation with cycle structure . | 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*. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AssociatedPartition} 'AssociatedPartition( )' returns the associated partition of the partition . | gap> AssociatedPartition([4,2,1]); [ 3, 2, 1, 1 ] gap> AssociatedPartition([6]); [ 1, 1, 1, 1, 1, 1 ]| The *associated partition* of a partition is defined to be the partition belonging to the transposed of the Young diagram of . %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PowerPartition}\index{symmetric group!powermap} 'PowerPartition( , )' returns the partition corresponding to the -th power of a permutation with cycle structure . | gap> PowerPartition([6,5,4,3,2,1], 3); [ 5, 4, 2, 2, 2, 2, 1, 1, 1, 1 ]| Each part $l$ of is replaced by $d = \gcd(l, k)$ parts $l/d$. So if is a partition of $n$ then $^{}$ also is a partition of $n$. 'PowerPartition' describes the powermap of symmetric groups. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PartitionTuples} 'PartitionTuples( , )' returns the list of all --tuples of partitions that together partition . | 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 ] ] ] | --tuples of partitions describe the classes and the characters of wreath products of groups with conjugacy classes with the symmetric group $S_n$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Fibonacci}% \index{sequence!fibonacci} 'Fibonacci( )' 'Fibonacci' returns the th 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()' is the special case 'Lucas(1,-1,)[1]' (see "Lucas"). | gap> Fibonacci( 10 ); 55 gap> Fibonacci( 35 ); 9227465 gap> Fibonacci( -10 ); -55 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Lucas}% \index{sequence!lucas} 'Lucas(

, , )' 'Lucas' returns the -th values of the *Lucas sequence* with parameters

and , 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()' (see "Fibonacci") is simply 'Lucas(1,-1,)[1]'. In an abuse of notation, the sequence 'Lucas(1,-1,)[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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Bernoulli}% \index{sequence!bernoulli} 'Bernoulli( )' 'Bernoulli' returns the -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 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%