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<head><title>BettiTally -- the class of all Betti tallies</title>
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<div><h1>BettiTally -- the class of all Betti tallies</h1>
<div class="single"><h2>Description</h2>
<div>A Betti tally is a special type of <a href="___Tally.html" title="the class of all tally results">Tally</a> that is printed as a display of graded Betti numbers. The class was created so the function <a href="_betti.html" title="display degrees">betti</a> could return something that both prints nicely and from which information can be extracted. The keys are triples <tt>(i,d,h)</tt>, where <tt>i</tt> is the homological degree, <tt>d</tt> is a list of integers giving a multidegree, and <tt>h</tt> is the result of applying a weight covector to <tt>d</tt>. Only <tt>i</tt> and <tt>h</tt> are used in printing.<table class="examples"><tr><td><pre>i1 : t = new BettiTally from { (0,{0},0) => 1, (1,{1},1) => 2, (2,{3},3) => 3, (2,{4},4) => 4 }
0 1 2
o1 = total: 1 2 7
0: 1 2 .
1: . . 3
2: . . 4
o1 : BettiTally</pre>
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<tr><td><pre>i2 : peek oo
o2 = BettiTally{(0, {0}, 0) => 1}
(1, {1}, 1) => 2
(2, {3}, 3) => 3
(2, {4}, 4) => 4</pre>
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For convenience, the operations of direct sum (<a href="__pl_pl.html" title="a binary operator, usually used for direct sum">++</a>), tensor product (<a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a>), <a href="_codim.html" title="compute the codimension">codim</a>, <a href="_degree.html" title="">degree</a>, <a href="_dual.html" title="dual module or map">dual</a>, <a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a>, <a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a>, <a href="_pdim.html" title="calculate the projective dimension">pdim</a>, <a href="_poincare.html" title="assemble degrees into polynomial">poincare</a>, <a href="_regularity.html" title="compute the Castelnuovo-Mumford regularity">regularity</a>, and degree shifting (numbers in brackets or parentheses), have been implemented for Betti tallies. These operations mimic the corresponding operations on chain complexes.<table class="examples"><tr><td><pre>i3 : t(5)
0 1 2
o3 = total: 1 2 7
-5: 1 2 .
-4: . . 3
-3: . . 4
o3 : BettiTally</pre>
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<tr><td><pre>i4 : t[-5]
5 6 7
o4 = total: 1 2 7
-5: 1 2 .
-4: . . 3
-3: . . 4
o4 : BettiTally</pre>
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<tr><td><pre>i5 : t ++ oo
0 1 2 3 4 5 6 7
o5 = total: 1 2 7 . . 1 2 7
-5: . . . . . 1 2 .
-4: . . . . . . . 3
-3: . . . . . . . 4
-2: . . . . . . . .
-1: . . . . . . . .
0: 1 2 . . . . . .
1: . . 3 . . . . .
2: . . 4 . . . . .
o5 : BettiTally</pre>
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<tr><td><pre>i6 : t ** t
0 1 2 3 4
o6 = total: 1 4 18 28 49
0: 1 4 4 . .
1: . . 6 12 .
2: . . 8 16 9
3: . . . . 24
4: . . . . 16
o6 : BettiTally</pre>
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<tr><td><pre>i7 : dual t
-2 -1 0
o7 = total: 7 2 1
-2: 4 . .
-1: 3 . .
0: . 2 1
o7 : BettiTally</pre>
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<tr><td><pre>i8 : regularity t
o8 = 2</pre>
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A Betti tally can be multiplied by an integer or by a rational number, and the values can be lifted to integers, when possible.<table class="examples"><tr><td><pre>i9 : (1/2) * t
0 1 2
o9 = total: 1/2 1 7/2
0: 1/2 1 .
1: . . 3/2
2: . . 2
o9 : BettiTally</pre>
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<tr><td><pre>i10 : 2 * oo
0 1 2
o10 = total: 1 2 7
0: 1 2 .
1: . . 3
2: . . 4
o10 : BettiTally</pre>
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<tr><td><pre>i11 : lift(oo,ZZ)
0 1 2
o11 = total: 1 2 7
0: 1 2 .
1: . . 3
2: . . 4
o11 : BettiTally</pre>
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Various combinations of the degree vectors can be displayed by using <a href="_betti_lp__Betti__Tally_rp.html" title="view and set the weights of a betti display">betti(BettiTally)</a>.</div>
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<div class="waystouse"><h2>Functions and methods returning a Betti tally :</h2>
<ul><li><span><a href="_betti.html" title="display degrees">betti</a> -- display degrees</span></li>
</ul>
<h2>Methods that use a Betti tally :</h2>
<ul><li><span>BettiTally == BettiTally, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span><a href="_betti_lp__Betti__Tally_rp.html" title="view and set the weights of a betti display">betti(BettiTally)</a> -- view and set the weights of a betti display</span></li>
<li>BettiTally ** BettiTally</li>
<li>BettiTally ++ BettiTally</li>
<li>BettiTally Array</li>
<li>BettiTally ZZ</li>
<li>codim(BettiTally)</li>
<li>degree(BettiTally)</li>
<li>dual(BettiTally)</li>
<li>hilbertPolynomial(ZZ,BettiTally)</li>
<li>hilbertSeries(ZZ,BettiTally)</li>
<li>lift(BettiTally,type of ZZ)</li>
<li>pdim(BettiTally)</li>
<li>poincare(BettiTally)</li>
<li>QQ * BettiTally</li>
<li>regularity(BettiTally)</li>
<li>ZZ * BettiTally</li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Virtual__Tally.html" title="">VirtualTally</a> < <a href="___Tally.html" title="the class of all tally results">Tally</a> < <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
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