<p>This chapter describes functions that deal with combinatorics. We mainly concentrate on two areas. One is about <em>selections</em>, that is the ways one can select elements from a set. The other is about <em>partitions</em>, that is the ways one can partition a set into the union of pairwise disjoint subsets.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Factorial</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <em>factorial</em> <span class="SimpleMath">n!</span> of the positive integer <var class="Arg">n</var>, which is defined as the product <span class="SimpleMath">1 ⋅ 2 ⋅ 3 ⋯ n</span>.</p>
<p><span class="SimpleMath">n!</span> is the number of permutations of a set of <span class="SimpleMath">n</span> elements. <span class="SimpleMath">1 / n!</span> is the coefficient of <span class="SimpleMath">x^n</span> in the formal series <span class="SimpleMath">exp(x)</span>, which is the generating function for factorial.</p>
<p><code class="func">PermutationsList</code> (<a href="chap16.html#X7B0143FB83F359B7"><spanclass="RefLink">16.2-12</span></a>) computes the set of all permutations of a list.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Binomial</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <em>binomial coefficient</em> <span class="SimpleMath">{n choose k}</span> of integers <var class="Arg">n</var> and <var class="Arg">k</var>. This is defined by the conditions <span class="SimpleMath">{n choose k} = 0</span> for <span class="SimpleMath">k < 0</span>, <span class="SimpleMath">{0 choose k} = 0</span> for <span class="SimpleMath">k ≠ 0</span>, <span class="SimpleMath">{0 choose 0} = 1</span> and the relation <span class="SimpleMath">{n choose k} = {n-1 choose k} + {n-1 choose k-1}</span> for all <span class="SimpleMath">n</span> and <span class="SimpleMath">k</span>.</p>
<p>There are many ways of describing this function. For example, if <span class="SimpleMath">n ≥ 0</span> and <span class="SimpleMath">0 ≤ k ≤ n</span>, then <span class="SimpleMath">{n choose k} = n! / (k! (n-k)!)</span> and for <span class="SimpleMath">n < 0</span> and <span class="SimpleMath">k ≥ 0</span> we have <span class="SimpleMath">{n choose k} = (-1)^k {-n+k-1 choose k}</span>.</p>
<p>If <span class="SimpleMath">n ≥ 0</span> then <span class="SimpleMath">{n choose k}</span> is the number of subsets with <span class="SimpleMath">k</span> elements of a set with <span class="SimpleMath">n</span> elements. Also, <span class="SimpleMath">{n choose k}</span> is the coefficient of <span class="SimpleMath">x^k</span> in the polynomial <span class="SimpleMath">(x + 1)^n</span>, which is the generating function for <span class="SimpleMath">{n choose .}</span>, hence the name.</p>
<p><code class="func">NrCombinations</code> (<a href="chap16.html#X82A6E98C85714FD0"><span class="RefLink">16.2-3</span></a>) is the generalization of <code class="func">Binomial</code> for multisets. <code class="func">Combinations</code> (<a href="chap16.html#X8770F16D794C0ADB"><span class="RefLink">16.2-1</span></a>) computes the set of all combinations of a multiset.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Bell</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <em>Bell number</em> <span class="SimpleMath">B(n)</span>. The Bell numbers are defined by <span class="SimpleMath">B(0) = 1</span> and the recurrence <span class="SimpleMath">B(n+1) = ∑_{k = 0}^n {n choose k} B(k)</span>.</p>
<p><span class="SimpleMath">B(n)</span> is the number of ways to partition a set of <var class="Arg">n</var> elements into pairwise disjoint nonempty subsets (see <code class="func">PartitionsSet</code> (<a href="chap16.html#X7A13D8DC8204525F"><span class="RefLink">16.2-16</span></a>)). This implies of course that <span class="SimpleMath">B(n) = ∑_{k = 0}^n S_2(n,k)</span> (see <code class="func">Stirling2</code> (<a href="chap16.html#X7C93E14D7BC360F0"><span class="RefLink">16.1-6</span></a>)). <span class="SimpleMath">B(n)/n!</span> is the coefficient of <span class="SimpleMath">x^n</span> in the formal series <span class="SimpleMath">exp( exp(x)-1 )</span>, which is the generating function for <span class="SimpleMath">B(n)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Bernoulli</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <var class="Arg">n</var>-th <em>Bernoulli number</em> <span class="SimpleMath">B_n</span>, which is defined by <span class="SimpleMath">B_0 = 1</span> and <span class="SimpleMath">B_n = -∑_{k = 0}^{n-1} {n+1 choose k} B_k/(n+1)</span>.</p>
<p><span class="SimpleMath">B_n / n!</span> is the coefficient of <span class="SimpleMath">x^n</span> in the power series of <span class="SimpleMath">x / (exp(x)-1)</span>. Except for <span class="SimpleMath">B_1 = -1/2</span> the Bernoulli numbers for odd indices are zero.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Bernoulli( 4 );</span>
-1/30
<span class="GAPprompt">gap></span> <span class="GAPinput">Bernoulli( 10 );</span> 5/66
<span class="GAPprompt">gap></span> <span class="GAPinput">Bernoulli( 12 ); # there is no simple pattern in Bernoulli numbers</span>
-691/2730
<span class="GAPprompt">gap></span> <span class="GAPinput">Bernoulli( 50 ); # and they grow fairly fast</span> 495057205241079648212477525/66
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Stirling1</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <em>Stirling number of the first kind</em> <span class="SimpleMath">S_1(n,k)</span> of the integers <var class="Arg">n</var> and <var class="Arg">k</var>. Stirling numbers of the first kind are defined by <span class="SimpleMath">S_1(0,0) = 1</span>, <span class="SimpleMath">S_1(n,0) = S_1(0,k) = 0</span> if <span class="SimpleMath">n, k ne 0</span> and the recurrence <span class="SimpleMath">S_1(n,k) = (n-1) S_1(n-1,k) + S_1(n-1,k-1)</span>.</p>
<p><span class="SimpleMath">S_1(n,k)</span> is the number of permutations of <var class="Arg">n</var> points with <var class="Arg">k</var> cycles. Stirling numbers of the first kind appear as coefficients in the series <span class="SimpleMath">n! {x choose n} = ∑_{k = 0}^n S_1(n,k) x^k</span> which is the generating function for Stirling numbers of the first kind. Note the similarity to <span class="SimpleMath">x^n = ∑_{k = 0}^n S_2(n,k) k! {x choose k}</span> (see <code class="func">Stirling2</code> (<a href="chap16.html#X7C93E14D7BC360F0"><span class="RefLink">16.1-6</span></a>)). Also the definition of <span class="SimpleMath">S_1</span> implies <span class="SimpleMath">S_1(n,k) = S_2(-k,-n)</span> if <span class="SimpleMath">n, k < 0</span>. There are many formulae relating Stirling numbers of the first kind to Stirling numbers of the second kind, Bell numbers, and Binomial coefficients.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Stirling2</code>( <var class="Arg">n</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <em>Stirling number of the second kind</em> <span class="SimpleMath">S_2(n,k)</span> of the integers <var class="Arg">n</var> and <var class="Arg">k</var>. Stirling numbers of the second kind are defined by <span class="SimpleMath">S_2(0,0) = 1</span>, <span class="SimpleMath">S_2(n,0) = S_2(0,k) = 0</span> if <span class="SimpleMath">n, k ne 0</span> and the recurrence <span class="SimpleMath">S_2(n,k) = k S_2(n-1,k) + S_2(n-1,k-1)</span>.</p>
<p><span class="SimpleMath">S_2(n,k)</span> is the number of ways to partition a set of <var class="Arg">n</var> elements into <var class="Arg">k</var> pairwise disjoint nonempty subsets (see <code class="func">PartitionsSet</code> (<a href="chap16.html#X7A13D8DC8204525F"><span class="RefLink">16.2-16</span></a>)). Stirling numbers of the second kind appear as coefficients in the expansion of <span class="SimpleMath">x^n = ∑_{k = 0}^n S_2(n,k) k! {x choose k}</span>. Note the similarity to <span class="SimpleMath">n! {x choose n} = ∑_{k = 0}^n S_1(n,k) x^k</span> (see <code class="func">Stirling1</code> (<a href="chap16.html#X85037456785BB33C"><span class="RefLink">16.1-5</span></a>)). Also the definition of <span class="SimpleMath">S_2</span> implies <span class="SimpleMath">S_2(n,k) = S_1(-k,-n)</span> if <span class="SimpleMath">n, k < 0</span>. There are many formulae relating Stirling numbers of the second kind to Stirling numbers of the first kind, Bell numbers, and Binomial coefficients.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Combinations</code>( <var class="Arg">mset</var>[, <var class="Arg">k</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the set of all combinations of the multiset <var class="Arg">mset</var> (a list of objects <
<p>A <em>combination</em> of <var class="Arg">mset</var> is an unordered selection without repetitions and is represented by a sorted sublist of <var class="Arg">mset</var>. If <var class="Arg">mset</var> is a proper set, there are <span class="SimpleMath">{|<var class="Arg">mset</var>| choose <var class="Arg">k</var>}</span> (see <code class="func">Binomial</code> (<a href="chap16.html#X7A9AF5F58682819D"><span class="java.lang.StringIndexOutOfBoundsException: Index 424 out of bounds for length 0
<p>To loop over combinations of a larger multiset use <code class="func">span class="ContSS"><br /><span class="nocss"> </span><a href="chap16.html1- /a>/>
<h5>16.2-2 <span class="Heading">Iterator and enumerator of combinations</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IteratorOfCombinations</code>( <var class="Arg">mset</var>[, <var class="Arg">k</var>] )</td><td class="tdrightspanclass""<br /<class=nocss">&;&;<span>a"hap16#X7C93E14D7BC360F0"16.-6Stirling2/a>/pan>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EnumeratorOfCombinationsspan
<">Iterator</code> (<#> "efLink"30.-1</span></> for combinations see code "unc></code> < hrefhref="hap16.X8770F16D794C0ADB"< class=R"1621/</a)) the multiset <class=Arg></>. non-negative integer <var class="Arg">k</var> is given as second argument then only the combinations with <var class="Arg">k</var> entries are produced, otherwise all combinations.</p>
<p><code class="func">EnumeratorOfCombinations</code> returns an <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) of the given multiset <var class="Arg">mset</var>. Currently only a variant without second argument <var class="Arg">k</var> is implementedspan =">br /< =nocss>&;nbsp;<span< =chap16.html#X78DD5C0D81057540"1622<span ="" and java.lang.StringIndexOutOfBoundsException: Range [176, 174) out of bounds for length 185
<>Theordering fromthese functionscanbe alsofrom listreturned class""/java.lang.StringIndexOutOfBoundsException: Range [153, 152) out of bounds for length 240
< class">>
<span class="GAPprompt">gap>< =>< >span=nocss&;/>< =.#java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 141
<span class="GAPprompt">gap></span> <span class="GAPinput">NrCombinations(m);</span> 49152
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 143
<span class="GAPprompt">gap></span> <span class="GAPinput">i;</span> 49152
<span class="GAPprompt">gap>/span>< class=""> : (m;;<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cm[1000];</span>
[ 1, 2, 3, 6, 7, 8, 9, <pan class=ContSS">br/<span class="nocss"> /span<a href"hap16.
<span class="GAPprompt">gap></span> <span class="GAPinput">Position(cm, [1,13,15,15]);</span> 36866
</pre></div>
<p><a id="X82A6E98C85714FD0" name="X82A6E98C85714FD0">spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 135
<>163 h5
<=""><class""width"%">tdjava.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 252
<p>returns the number of <code class="code">Combinations(<var class="Arg">mset</var>,<var class="Arg">k</var>)</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Combinations( [1,2,2,3<pan java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 142
[ [ ], [ 1 ], ["ontSS< /<class"nocss>&</java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 144
[ 1, 3 ], [ 2 ], [ 2, 2 ], [ 2, < java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 145
<span class="GAPprompt">gap></span> <span class="GAPinput">class=ContSS><br < class=">&".>2-25 java.lang.StringIndexOutOfBoundsException: Range [136, 134) out of bounds for length 145
<span class="GAPprompt">gap></span> <span class="GAPinput">NrCombinations( [1..52], 5 );</span> 2598960
</pre></div>
<pThe function codeclass=func"Arrangements</code> (<a href="chap16.html#X7837B3357C7566C8"><span class="RefLink">16.2-4</span></a>) computes ordered selections without repetitions, <span ">br><>nbsp&<span< java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 144
<divjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 250
< returns all arrangements of <var class=Arg>/./>
<p>An <em>arrangement</em> of <var class="Arg">mset</var> is an ordered selection without repetitions and is represented by a list that contains only elements from <var class="Arg">mset</var>, but maybe in a different order. If <var class="Arg"span =ContSS><br /spanclass"">nbsp;nbsp</a".#8516-Fjava.lang.StringIndexOutOfBoundsException: Range [123, 122) out of bounds for length 133
<>< "java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<h5>16.2-5 NrArrangements</h5>
<div class="func"><table class="func" width=""pan class="ContSS"</href"html#>6.41 </a></span>
<> the <">java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 122
<>As of ofgame Suppose havethesixof <class="><code> and you have to afour .Then given<pjava.lang.StringIndexOutOfBoundsException: Index 239 out of bounds for length 239
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput
[ [ "e", "e"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[e,e,s t ] [ e,e,t,""],[ """""t"s ,
... 93 more possibilities ...
[ "t", "t", "l"
</pre></div>
<p>Can you find the five proper English words, where <code class="code">"lets"</code> does not count? Note that the fact
<div class="funcpthe em> /><=">n k</ <varclass"Arg"n</ar varclass=A"k<var. This definedby ""{ }= 0</span>for span =SimpleMath"0span,spanclass"java.lang.StringIndexOutOfBoundsException: Range [310, 309) out of bounds for length 609
<p>returns the set of all unordered tuples of length <var class="Arg">k</var> of the set <var class="Arg">set</var>.</p>
<p>An <em>unordered tuple</em> of length <var class="Arg">k</var> of <var class="Arg">set</var> is an unordered selection with repetitions of <var class="Arg">set</var> and is represented by a sorted list of length <var class="Arg">k</var> containing elements from <var class="Arg">set</var>. There are <span class="SimpleMath">{|set| + k - 1 choose k}</span> (see <code class="func">Binomial</code>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Note that the fact that <code class="func">UnorderedTuples</code> returns a set implies that the last index runs fastest. That means the first tuple contains the smallest element from <var class=
<p>The function <code class="func">Combinations</code> (<a href="chap16.html#X8770F16D794C0ADB"><span class="RefLink">16.2-1</span></a>) computes unordered selections without repetitions, <code class="func">Arrangements</code> (<a href="chap16.html#X7837B3357C7566C8"><span class="RefLink">16.2-4<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><<> ""B</>. by <span">B0) = 1</span> recurrence<span class="impleMath"B(+1 =∑k 0^ {choose } (k)<span>./p>
<p>returns the set of all ordered tuples of length <var class="Arg">k</var> of the set <var class="Arg">set</var>.</p>
<p>An <em>ordered tuple</em> of length <var class="Arg">k</var> of <var class="Arg">set</var> is an ordered selection
<p>Note that the fact that <[11251552, java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
<p><a idjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>.- </5java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
<div class="func"><table class="func" width="100%p> the var class=Arg"><var- emBernoulli /> span=SimpleMath><span> which is <class=SimpleMath> 1/java.lang.StringIndexOutOfBoundsException: Range [165, 164) out of bounds for length 254
<p>This function is referred to as an example of enumerators that are defined by functions but are not constructed from
<h5>16.2-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< =>table"func"width="%">tr>tdclass""codeclass=f">‣ IteratorOfTuples</code> < class=Arg">set<var,< class=Arg>/> )td>< java.lang.StringIndexOutOfBoundsException: Range [195, 194) out of bounds for length 251
<p>For a set <var class="Arg">set</var> and a positive integer <var class="Arg">k</var>, <code class="func">IteratorOfTuples</code>/30
<p><a id="X85E18A9A87FD4CA2" name="span class="GAPprompt">gap></span> <span"APinput"Bernoulli 12) #there is no simple in Bernoullinumbers</>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PermutationsList</code>( <var class="Arg">mset</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PermutationsList</code> returns the set of permutations of the multiset <var class="Arg">mset</var>.</p>
<p>A <em>permutation</em> is represented by a list that contains exactly the same elements <var class="rg"><var>but in java.lang.StringIndexOutOfBoundsException: Range [159, 158) out of bounds for length 750
class<""width100%> "#; <code>(< =Arg"mset/var<td<td =tdright"( unction&;</></r><table>/java.lang.StringIndexOutOfBoundsException: Index 228 out of bounds for length 228
<p>returns the java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
<p>The function <code class="func">Arrangements</code> (<a href="chap16.html#X7837B3357C7566C8"><span class="RefLink">16.2-4</span></a>) is the generalization of <code
<div class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [10, 2) out of bounds for length 83
<p> thenumber of <code class="code">Derangements(<var class="Arg">list</var>)</code>.</p>
<p>Note that the ratio <code class="code">NrPermutationsList( [ 1 .. n ] ) / NrDerangements( [ 1 .. n ] )</code>, which is <span class="SimpleMath">n! / (n! (1/2! - 1/3! + 1/4! - ⋯ + (-1)^n / n!))</span> is an approximation for the base of the natural logarithm <span class="SimpleMath">e = 2.7182818285...</span>, which is correct to about <span class="SimpleMath">n</span> digits.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NrDerangements</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number <pan class="APprompt"gapgt</span> <span class="GAPinput">Derangements( [1,2,3,4] );</span>
<p>As an example of derangements suppose that you have to send four different letters to four different people. java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 65
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="
<p>returns the set of all unordered partitions of the set <var class="Arg">set</var> into <var class="Arg">k</var> pairwise disjoint nonempty sets. If <
<p>An <em>java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
<" ="%>tr<<java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 252
<p>returns the number of <code class="code">PartitionsSet(<var class="Arg">set</var>,<var java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 0
<p>Note that <code class="func">PartitionsSet</code> (<a href="chap16.html#X7A13D8DC8204525F"><span class="RefLink">16.2-16</span></a>) does currently not support multisets and[ ,2 3, 4 ] [ 12,4 ,[3]],
< ""<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 245
<>returns ()ofthe java.lang.StringIndexOutOfBoundsException: Range [78, 77) out of bounds for length 290
<p>An <em>unordered partition</em> is an unordered sum <span class="SimpleMath">n = p_1 + p_2 + ⋯ + p_k</span> of positive integers and is represented by the list <span class="SimpleMath">p = [ p_1, p_2, ..., p_k ]</span>, in nonincreasing order, i.e., <span class="SimpleMath">p_1 ≥ p_2 ≥ ... ≥ p_k</span>. We write <span class="SimpleMath">p ⊢ n</span>. There are approximately <span class="SimpleMath">exp(π sqrt{2/3 n}) / (4 sqrt{3} n)</span> such partitions, use <code class="func">NrPartitions</code> (<ahref="chap16.html#X86933C4F795C4EBD"><span class="RefLink">16.2-21</span></a>) to compute the precise number.</p>
< loop java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 247
<p>It is possible to associate with every partition of the integer <var class="Arg">n</var> a conjugacy class of permutations in the symmetric group on <var class="Arg">n<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Ramanujan found the identities <span class="SimpleMath">p(5i+4) = 0</span> mod 5, <span class="SimpleMath">p(7i+5) = 0</span> mod 7 and <span class="SimpleMath">p(11i+6) = 0</span> mod 11 and many other fascinating things about the number of partitions.</p>
<p>The function <code class="func">OrderedPartitions</code> (<a href="chap16.html#X820DF201871F2723"><span class="RefLink">16.2-22</span></a>) is the ordered counterpart of <code class="func">Partitions</code> (<a href="chap16.html#X84A6D15F8107008B"><span class="RefLink">16.2-18</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OrderedPartitions</code>( <var class="Arg">n</var>[, <var class="Arg">k</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the set of all span class="GAPprompt">gap></span class="GAPinput">SignPartition([6,5,4,3,2,1]);</span>
<p>An <em>ordered partition</em> is an ordered sum <span class="SimpleMath">n = p_1 + p_2 + ... + p_k</span> of positive integers and is represented by the list <span class="SimpleMath">[ p_1, p_2, ..., p_k ]</span>. There are totally <span class="SimpleMath">2^{n-1}</span> ordered partitions and <span class="SimpleMath">{n-1 choose k-1}</span> (see <code class="java.lang.StringIndexOutOfBoundsException: Index 369 out of bounds for length 2
<> with an <var=Arg><varmuch than <span class="SimpleMath">5/span> the list simply become large</p>
<>The function <code class="func">Partitions</code> (<a href="chap16.html#X84A6D15F8107008B"><span class="RefLink">16.2-18</span></a>) is the unordered counterpart of <span class="SimpleMath">l/d</span>. So if <var class="Arg">pi</var> is a partition of <span class="SimpleMath">n</span> then <span class="SimpleMath"><var class="Arg">pi</var>^<var class="Arg">k</var></span> also is a partition of <span class="SimpleMath">n</span>. <code class="func">PowerPartition</code> describes the power map of symmetric groups.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PartitionsGreatestLE</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></<><a id="877D997B7F66A119" name="X877D997B7F66A119"></a></p>
<p>returns the set of all (unordered) partitions of the integer <var class="Arg">n</var> having parts less or equal to the integer <var class="Arg">m</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PartitionsGreatestEQ</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>pcode =func>PartitionTuples</ode>returns thelist all varclassArg"</ar>tuples of partitions which together form a partition of <var class="Arg">n</var>.</p>
<>eturns setof all unordered partitions of the integer <var class="Arg">n</var> having greatest part equal to the integer <var class="Arg">m</var>.</p>
<p>A <em>restricted partition</em> is like an ordinary partition (see <code class="func">Partitions</code> (<a href="chap16.html#X84A6D15F8107008B"><span class="RefLink">16.2-18</span></a>)) an unordered sum <span class="SimpleMath">n = p_1 + p_2 + .<5>162 />
<p>The last example tells us that there are 451 ways to return 50 pence change using 1, 2, 5,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PowerPartition</code>( <var class="Arg">pi</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></java.lang.StringIndexOutOfBoundsException: Index 246 out of bounds for length 0
<p><code class="func"java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<p>Each part <span class="SimpleMath">l</span> of <var /><div
<div class="example">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerPartition([java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[5 4, 22 2 1 ,1,]
</pre></div>
< java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 62
<>231 java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 32
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PartitionTuples</code>( <var class="Arg">n</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PartitionTuples</code> returns the list of all <var class="Arg">r</var>-tuples of partitions which together forma partition of <var class="Arg">n</var>.</p>
<p><var class="Arg">r</var>-tuples of partitionsk<)/code seecode func"Fibonacci</>(a href=java.lang.StringIndexOutOfBoundsException: Range [123, 122) out of bounds for length 413
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BetaSet</code>( <var class="Arg">alpha</var> )</[0,1,234,, ,7,,910
<p>For a list <var class="Arg">alpha</var> that describes a partition of a nonnegative integer (see <code class="func">Partitions</code> (<a
<div class="func"><table class="func" width="100%"><tr><td class=span class""gt<span><java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 81
<p>returns the <var class="Arg">n</var>th number of the <em>Fibonacci sequence</em>. The Fibonacci sequence <span class="SimpleMath">F_n</span> is defined by the initial conditions <span class="SimpleMath">F_1 = F_2 = 1</span> and the recurrence relation <span class="SimpleMath">F_{n+2} = F_{n+1} class">&;<span span class=G>[,,,] ; # to computeNrDerangements([1..4])</span>
<p>Using generating functions one can prove that <span class="SimpleMath">F_n = ϕ^n - 1/ϕ^n</span>, where <span class="SimpleMath">ϕ</span> is <span class="java.lang.StringIndexOutOfBoundsException: Index 160 out of bounds for length 87
<div class="example"><pre>
;/java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85 55
<span class="GAPprompt">gap></span> spanclass="">gt</><span class"> 010,,,,],/> 9227465
<span class="GAPprompt">gap></span> 24
-55
/>div
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&<;hr >
<> with parameters <var class="Arg">P/varand <var class=Arg>Q/> which must be integers, as a list of three integers. If <var class="Arg">k</var> is a negative integer, then the values of the Lucas sequence may be nonintegral rational numbers, with denominator roughly <var class="Arg">Q</var>^<var class="Arg">k</var>.</p>
<p>Let <span class="SimpleMath">α, β</span> be the two roots of <span class="SimpleMath">x^2 - P x + Q</span> then we define <code class="code">Lucas( <var class="Arg">P</var>, <var class="Arg">Q</var>, <var class="Arg">k</var> )[1]</code> <span class="SimpleMath">= U_k = (α^k - β^k) / (α - β)</span> and <code class="code">Lucas( <var class="Arg">P</var>, <var class="Arg">Q</var>, <var class="Arg">k</var> )[2]</code> <span class="SimpleMath">= V_k = (α^k + β^k)</span> and as a convenience <code class="code">Lucas( <var class="Arg">P</var>, <var class="Arg">Q</var>, <var class="Arg">k</var> )[3]</code> <span class="SimpleMath">= Q^k</span>.</p>
<p>The following recurrence relations are easily derived from the definition <span class="SimpleMath">U_0 = 0, U_1 = 1, U_k = P U_{k-1} - Q U_{k-2}</span> and <span class="SimpleMath">V_0 = 2, V_1 = P, V_k = P V_{k-1} - Q V_{k-2}</span>. Those relations are actually used to define <code class="func">Lucas</code> if <span class="SimpleMath">α = β</span>.</p>
<p>Also the more complex relations used in <code class="func">Lucas</code> can be easily derived <span class="SimpleMath">U_2k = U_k V_k</span>, <span class="SimpleMath">U_{2k+1} = (P U_2k + V_2k) / 2</span> and <span class="SimpleMath">V_2k = V_k^2 - 2Q^k</span>, <span class="SimpleMath">V_{2k+1} = ((P^2-4Q) U_2k + P V_2k) / 2</span>.</p>
<p><code class="code">Fibonacci(<var class="Arg">k</var>)</code> (see <code class="func">Fibonacci</code> (<a href="chap16.html#X85AE1D70803A886C"><span class="RefLink">16.3-1</span></a>)) is simply <code class="code">Lucas(1,-1,<var class="Arg">k</var>)[1]</code>. In an abuse of notation, the sequence <code class="code">Lucas(1,-1,<var class="Arg">k</var>)[2]</code> is sometimes called the Lucas sequence.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Permanent</code>( <var class="Arg">mat</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the <em>permanent</em> of the matrix <var class="Arg">mat</var>. The permanent is defined by <span class="SimpleMath">∑_{p ∈ Sym(n)} ∏_{i = 1}^n mat[i][i^p]</span>.</p>
<p>Note the similarity of the definition of the permanent to the definition of the determinant (see <code class="func">DeterminantMat</code> (<a href="chap24.html#X8488D69A7ADDB4E2"><span class="RefLink">24.4-4</span></a>)). 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.</p>
