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<head><title>partition -- partition a set or list by values of a function</title>
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<div><h1>partition -- partition a set or list by values of a function</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>partition(f,L)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Function.html">function</a></span></span></li>
<li><span><tt>L</tt>, <span>a <a href="___Visible__List.html">visible list</a></span>, or <span>a <a href="___Tally.html">tally</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Hash__Table.html">hash table</a></span>, whose keys are the possible values of the function <tt>f</tt> on the elements of the list, set or tally <tt>L</tt>, and the corresponding value of the hash table is the list, set or tally of elements of <tt>L</tt> which take that value under <tt>f</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : L = {1,3,6,5,3,1,2,8,8,8}
o1 = {1, 3, 6, 5, 3, 1, 2, 8, 8, 8}
o1 : List</pre>
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<tr><td><pre>i2 : partition(odd, L)
o2 = HashTable{false => {6, 2, 8, 8, 8}}
true => {1, 3, 5, 3, 1}
o2 : HashTable</pre>
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<tr><td><pre>i3 : partition(odd, set L)
o3 = HashTable{false => set {2, 6, 8}}
true => set {1, 3, 5}
o3 : HashTable</pre>
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<tr><td><pre>i4 : partition(odd, tally L)
o4 = HashTable{false => Tally{2 => 1}}
6 => 1
8 => 3
true => Tally{1 => 2}
3 => 2
5 => 1
o4 : HashTable</pre>
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The following example parititions the generators of an ideal by degree.<table class="examples"><tr><td><pre>i5 : R = QQ[a..f]
o5 = R
o5 : PolynomialRing</pre>
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<tr><td><pre>i6 : I = ideal"ab,ade,ac3,d4,b3,adf,f4,e10"
3 4 3 4 10
o6 = ideal (a*b, a*d*e, a*c , d , b , a*d*f, f , e )
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : partition(f -> first degree f, flatten entries gens I)
o7 = HashTable{2 => {a*b} }
3
3 => {a*d*e, b , a*d*f}
3 4 4
4 => {a*c , d , f }
10
10 => {e }
o7 : HashTable</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_tally.html" title="tally the elements of a list or sequence">tally</a> -- tally the elements of a list or sequence</span></li>
<li><span><a href="_partitions.html" title="list the partitions of an integer">partitions</a> -- list the partitions of an integer</span></li>
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<div class="waystouse"><h2>Ways to use <tt>partition</tt> :</h2>
<ul><li>partition(Function,Tally)</li>
<li>partition(Function,VisibleList)</li>
</ul>
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