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    <title>BettiTally -- the class of all Betti tallies</title>
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0 1 2
o1 = total: 1 2 7
         0: 1 2 .
         1: . . 3
         2: . . 4

o1 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i2 : betti(t, Weights => {2})

0 1 2
o2 = total: 1 2 7
         0: 1 . .
         1: . 2 .
         2: . . .
         3: . . .
         4: . . 3
         5: . . .
         6: . . 4

o2 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i3 : peek oo

o3 = BettiTally{(0, {0}, 0) => 1}
                (1, {1}, 2) => 2
                (2, {3}, 6) => 3
                (2, {4}, 8) => 4</code></pre>
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          <p>For convenience, the operations of direct sum (<a title="a binary operator, usually used for direct sum" href="__pl_pl.html">++</a>), tensor product (<a title="a binary operator, usually used for tensor product or Cartesian product" href="__st_st.html">**</a>), <a title="compute the codimension" href="_codim.html">codim</a>, <a href="_degree.html">degree</a>, <a title="dual module or map" href="_dual.html">dual</a>, <a title="compute the projective dimension" href="_pdim.html">pdim</a>, <a title="assemble degrees of a ring, module, or ideal into a polynomial" href="_poincare.html">poincare</a>, <a title="compute the Castelnuovo-Mumford regularity" href="_regularity.html">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.</p>
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              <pre><code class="language-macaulay2">i4 : t(5)

0 1 2
o4 = total: 1 2 7
        -5: 1 2 .
        -4: . . 3
        -3: . . 4

o4 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i5 : t[-5]

5 6 7
o5 = total: 1 2 7
        -5: 1 2 .
        -4: . . 3
        -3: . . 4

o5 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i6 : dual oo

-7 -6 -5
o6 = total:  7  2  1
         3:  4  .  .
         4:  3  .  .
         5:  .  2  1

o6 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i7 : t ++ oo

-7 -6 -5 -4 -3 -2 -1 0 1 2
o7 = total:  7  2  1  .  .  .  . 1 2 7
         0:  .  .  .  .  .  .  . 1 2 .
         1:  .  .  .  .  .  .  . . . 3
         2:  .  .  .  .  .  .  . . . 4
         3:  4  .  .  .  .  .  . . . .
         4:  3  .  .  .  .  .  . . . .
         5:  .  2  1  .  .  .  . . . .

o7 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i8 : t ** t

0 1  2  3  4
o8 = total: 1 4 18 28 49
         0: 1 4  4  .  .
         1: . .  6 12  .
         2: . .  8 16  9
         3: . .  .  . 24
         4: . .  .  . 16

o8 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i9 : pdim t

o9 = 2</code></pre>
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              <pre><code class="language-macaulay2">i10 : codim t

o10 = 0</code></pre>
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              <pre><code class="language-macaulay2">i11 : degree t

o11 = 6</code></pre>
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              <pre><code class="language-macaulay2">i12 : poincare t

3     4
o12 = 1 - 2T + 3T  + 4T

o12 : ZZ[T]</code></pre>
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              <pre><code class="language-macaulay2">i13 : regularity t

o13 = 2</code></pre>
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          <p>If the Betti tally represents the Betti numbers of a resolution of a module $M$ on a polynomial ring $R = K[x_0,...,x_n]$, then while the data does not uniquely determine $M$, it suffices to compute the <a title="compute the Hilbert polynomial" href="_hilbert__Polynomial.html">Hilbert polynomial</a> and <a title="compute the Hilbert series" href="_hilbert__Series.html">Hilbert series</a> of $M$.</p>
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              <pre><code class="language-macaulay2">i14 : n = 3

o14 = 3</code></pre>
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              <pre><code class="language-macaulay2">i15 : hilbertSeries(n, t)

3     4
      1 - 2T + 3T  + 4T
o15 = ------------------
                  3
           (1 - T)

o15 : Expression of class Divide</code></pre>
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              <pre><code class="language-macaulay2">i16 : hilbertPolynomial(n, t)

o16 = 33*P  - 23*P  + 6*P
          0       1      2

o16 : ProjectiveHilbertPolynomial</code></pre>
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          <p>A Betti tally can be multiplied by an integer or by a rational number, and the values can be lifted to integers, when possible.</p>
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              <pre><code class="language-macaulay2">i17 : (1/2) * t

0 1 2
             1   7
o17 = total: - 1 -
             2   2
             1   
          0: - 1 .
             2   
                 3
          1: . . -
                 2
          2: . . 2

o17 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i18 : 2 * oo

0 1 2
o18 = total: 1 2 7
          0: 1 2 .
          1: . . 3
          2: . . 4

o18 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i19 : lift(oo,ZZ)

0 1 2
o19 = total: 1 2 7
          0: 1 2 .
          1: . . 3
          2: . . 4

o19 : BettiTally</code></pre>
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        <h3>Menu</h3>
        <ul>
          <li><span><a title="the class of all multigraded Betti tallies" href="___Multigraded__Betti__Tally.html">MultigradedBettiTally</a> -- the class of all multigraded Betti tallies</span></li>
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        <div class="waystouse">
          <h2>Types of Betti tally:</h2>
          <ul>
            <li><span><a title="the class of all multigraded Betti tallies" href="___Multigraded__Betti__Tally.html">MultigradedBettiTally</a> -- the class of all multigraded Betti tallies</span></li>
          </ul>
          <h2>Functions and methods returning a Betti tally:</h2>
          <ul>
            <li><span><a title="display or modify a Betti diagram" href="_betti.html">betti</a> -- display or modify a Betti diagram</span></li>
            <li><kbd>BettiTally ** BettiTally</kbd></li>
            <li><kbd>BettiTally ++ BettiTally</kbd></li>
            <li><kbd>BettiTally Array</kbd></li>
            <li><kbd>BettiTally ZZ</kbd></li>
            <li><kbd>dual(BettiTally)</kbd></li>
            <li><kbd>lift(BettiTally,type of ZZ)</kbd></li>
            <li><kbd>QQ * BettiTally</kbd></li>
            <li><kbd>ZZ * BettiTally</kbd></li>
            <li><span><a title="minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module" href="_minimal__Betti.html">minimalBetti</a> -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module</span></li>
            <li><span><kbd>minimalBetti(Ideal)</kbd> -- see <span><a title="minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module" href="_minimal__Betti.html">minimalBetti</a> -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module</span></span></li>
            <li><span><kbd>minimalBetti(Module)</kbd> -- see <span><a title="minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module" href="_minimal__Betti.html">minimalBetti</a> -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module</span></span></li>
            <li><span><span class="tt">truncate(BettiTally,InfiniteNumber,InfiniteNumber)</span> (missing documentation)
</span></li>
            <li><span><span class="tt">truncate(BettiTally,InfiniteNumber,ZZ)</span> (missing documentation)
</span></li>
            <li><span><span class="tt">truncate(BettiTally,ZZ,InfiniteNumber)</span> (missing documentation)
</span></li>
            <li><span><span class="tt">truncate(BettiTally,ZZ,ZZ)</span> (missing documentation)
</span></li>
          </ul>
          <h2>Methods that use a Betti tally:</h2>
          <ul>
            <li><span><a title="view and set the weight vector of a Betti diagram" href="_betti_lp__Betti__Tally_rp.html">betti(BettiTally)</a> -- view and set the weight vector of a Betti diagram</span></li>
            <li><kbd>codim(BettiTally)</kbd></li>
            <li><kbd>degree(BettiTally)</kbd></li>
            <li><kbd>hilbertPolynomial(ZZ,BettiTally)</kbd></li>
            <li><kbd>hilbertSeries(ZZ,BettiTally)</kbd></li>
            <li><kbd>pdim(BettiTally)</kbd></li>
            <li><kbd>poincare(BettiTally)</kbd></li>
            <li><kbd>regularity(BettiTally)</kbd></li>
            <li><span><kbd>multigraded(BettiTally)</kbd> -- see <span><a title="convert a Betti tally into a multigraded Betti tally" href="_multigraded.html">multigraded</a> -- convert a Betti tally into a multigraded Betti tally</span></span></li>
            <li><span><a title="construct a chain complex with prescribed Betti table" href="../../OldChainComplexes/html/___Ring_sp%5E_sp__Betti__Tally.html">Ring ^ BettiTally</a></span></li>
          </ul>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="the class of all Betti tallies" href="___Betti__Tally.html">BettiTally</a> is <span>a <a title="the class of all mutable types" href="___Type.html">type</a></span>, with ancestor classes <a href="___Virtual__Tally.html">VirtualTally</a> &lt; <a title="the class of all hash tables" href="___Hash__Table.html">HashTable</a> &lt; <a title="the class of all things" href="___Thing.html">Thing</a>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Macaulay2Doc/functions/betti-doc.m2:249:0</span>.</p>
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Edit File: ___Betti__Tally.html

Edit File: _BettiTally.html