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<head><title>fVector -- the f-vector of a simplicial complex</title>
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<div><h1>fVector -- the f-vector of a simplicial complex</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>f = fVector D</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>D</tt>, <span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span></span></li>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, such that <tt>f#i</tt> is the number of faces in <tt>D</tt> of dimension <tt>i</tt>, where <tt>-1 <= i <= dim D</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : loadPackage "SimplicialComplexes";</pre>
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The pentagonal bipyramid has 7 vertices, 15 edges and 10 triangles.<table class="examples"><tr><td><pre>i2 : R = ZZ[a..g];</pre>
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<tr><td><pre>i3 : bipyramid = simplicialComplex monomialIdeal(
a*g, b*d, b*e, c*e, c*f, d*f)
o3 = | efg bfg deg cdg bcg aef abf ade acd abc |
o3 : SimplicialComplex</pre>
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<tr><td><pre>i4 : f = fVector bipyramid
o4 = HashTable{-1 => 1}
0 => 7
1 => 15
2 => 10
o4 : HashTable</pre>
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<tr><td><pre>i5 : f#0
o5 = 7</pre>
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<tr><td><pre>i6 : f#1
o6 = 15</pre>
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<tr><td><pre>i7 : f#2
o7 = 10</pre>
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Every simplicial complex other than the void complex has a unique face of dimension -1.<table class="examples"><tr><td><pre>i8 : void = simplicialComplex monomialIdeal 1_R
o8 = 0
o8 : SimplicialComplex</pre>
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<tr><td><pre>i9 : fVector void
o9 = HashTable{-1 => 0}
o9 : HashTable</pre>
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For a larger examp;le we consider the polarization of an artinian monomial ideal from section 3.2 in Miller-Sturmfels, Combinatorial Commutative Algebra.<table class="examples"><tr><td><pre>i10 : S = ZZ[x_1..x_4, y_1..y_4, z_1..z_4];</pre>
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<tr><td><pre>i11 : I = monomialIdeal(x_1*x_2*x_3*x_4,
y_1*y_2*y_3*y_4,
z_1*z_2*z_3*z_4,
x_1*x_2*x_3*y_1*y_2*z_1,
x_1*y_1*y_2*y_3*z_1*z_2,
x_1*x_2*y_1*z_1*z_2*z_3);
o11 : MonomialIdeal of S</pre>
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<tr><td><pre>i12 : D = simplicialComplex I;</pre>
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<tr><td><pre>i13 : fVector D
o13 = HashTable{-1 => 1 }
0 => 12
1 => 66
2 => 220
3 => 492
4 => 768
5 => 837
6 => 609
7 => 264
8 => 51
o13 : HashTable</pre>
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<p/>
The f-vector is computed using the Hilbert series of the Stanley-Reisner ideal. For example, see Hosten and Smith's chapter Monomial Ideals, in Computations in Algebraic Geometry with Macaulay2, Springer 2001.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="simplicial complexes">SimplicialComplexes</a> -- simplicial complexes</span></li>
<li><span><a href="_faces.html" title="the i-faces of a simplicial complex ">faces</a> -- the i-faces of a simplicial complex </span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>fVector</tt> :</h2>
<ul><li>fVector(SimplicialComplex)</li>
</ul>
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