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    <title>decomposeBetti -- write a Betti diagram as a positive combination of pure integral diagrams</title>
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o1 = R

o1 : PolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : I = ideal&quot;a3,abc,b4,c4,b2c2&quot;

3          4   4   2 2
o2 = ideal (a , a*b*c, b , c , b c )

o2 : Ideal of R</code></pre>
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              <pre><code class="language-macaulay2">i3 : B = betti res I

0 1 2 3
o3 = total: 1 5 8 4
         0: 1 . . .
         1: . . . .
         2: . 2 . .
         3: . 3 2 .
         4: . . 4 2
         5: . . 2 2

o3 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i4 : decomposeBetti(B)

1 /       0  1  2  3\     1 /       0 1  2 3\     3 /       0 1  2 3\    11 /       0 1 2 3\     1 /       0  1  2  3\
o4 = (--)|total: 8 35 42 15| + (--)|total: 2 7 14 9| + (--)|total: 1 7 14 8| + (--)|total: 1 6 8 3| + (--)|total: 3 14 32 21|
      21 |    0: 8  .  .  .|    21 |    0: 2 .  . .|    28 |    0: 1 .  . .|    48 |    0: 1 . . .|    16 |    0: 3  .  .  .|
         |    1: .  .  .  .|       |    1: . .  . .|       |    1: . .  . .|       |    1: . . . .|       |    1: .  .  .  .|
         |    2: . 35  .  .|       |    2: . 7  . .|       |    2: . .  . .|       |    2: . . . .|       |    2: .  .  .  .|
         |    3: .  . 42  .|       |    3: . .  . .|       |    3: . 7  . .|       |    3: . 6 . .|       |    3: . 14  .  .|
         \    4: .  .  . 15/       \    4: . . 14 9/       \    4: . . 14 8/       |    4: . . 8 .|       |    4: .  .  .  .|
                                                                                   \    5: . . . 3/       \    5: .  . 32 21/

o4 : Expression of class Sum</code></pre>
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We can see what the pure diagrams should be using the Herzog-Kuhl equations from Boij-Soederberg's initial paper        <table class="examples">
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              <pre><code class="language-macaulay2">i5 : decomposeBetti(B,TableEntries => HerzogKuhl)

/         0  1  2  3\       /         0  1  2  3\       /         0  1  2  3\       /         0  1  2  3\       /         0  1  2  3\
         |        1   1  1  1|       |        1   1  1  1|       |        1   1  1  1|       |        1   1  1  1|       |        1   1  1  1|
o5 = (40)|total: --- -- -- --| + (12)|total: --- -- -- --| + (18)|total: --- -- -- --| + (44)|total: --- -- -- --| + (42)|total: --- -- -- --|
         |       105 24 20 56|       |       126 36 18 28|       |       168 24 12 21|       |       192 32 24 64|       |       224 48 21 32|
         |        1          |       |        1          |       |        1          |       |        1          |       |        1          |
         |    0: ---  .  .  .|       |    0: ---  .  .  .|       |    0: ---  .  .  .|       |    0: ---  .  .  .|       |    0: ---  .  .  .|
         |       105         |       |       126         |       |       168         |       |       192         |       |       224         |
         |    1:   .  .  .  .|       |    1:   .  .  .  .|       |    1:   .  .  .  .|       |    1:   .  .  .  .|       |    1:   .  .  .  .|
         |            1      |       |            1      |       |    2:   .  .  .  .|       |    2:   .  .  .  .|       |    2:   .  .  .  .|
         |    2:   . --  .  .|       |    2:   . --  .  .|       |            1      |       |            1      |       |            1      |
         |           24      |       |           36      |       |    3:   . --  .  .|       |    3:   . --  .  .|       |    3:   . --  .  .|
         |               1   |       |    3:   .  .  .  .|       |           24      |       |           32      |       |           48      |
         |    3:   .  . --  .|       |               1  1|       |               1  1|       |               1   |       |    4:   .  .  .  .|
         |              20   |       |    4:   .  . -- --|       |    4:   .  . -- --|       |    4:   .  . --  .|       |               1  1|
         |                  1|       \              18 28/       \              12 21/       |              24   |       |    5:   .  . -- --|
         |    4:   .  .  . --|                                                               |                  1|       \              21 32/
         \                 56/                                                               |    5:   .  .  . --|
                                                                                             \                 64/

o5 : Expression of class Sum</code></pre>
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And we can also see what the realization modules from the Eisenbud-Schreyer paper will be.        <table class="examples">
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              <pre><code class="language-macaulay2">i6 : decomposeBetti(B,TableEntries => RealizationModules)

1 /        0   1   2  3\     1  /        0  1  2  3\     3  /       0  1  2  3\     11  /        0   1   2   3\     1 /        0  1   2   3\
o6 = (--)|total: 24 105 126 45| + (---)|total: 10 35 70 45| + (---)|total: 5 35 70 40| + (----)|total: 35 210 280 105| + (--)|total: 15 70 160 105|
      63 |    0: 24   .   .  .|    105 |    0: 10  .  .  .|    140 |    0: 5  .  .  .|    1680 |    0: 35   .   .   .|    80 |    0: 15  .   .   .|
         |    1:  .   .   .  .|        |    1:  .  .  .  .|        |    1: .  .  .  .|         |    1:  .   .   .   .|       |    1:  .  .   .   .|
         |    2:  . 105   .  .|        |    2:  . 35  .  .|        |    2: .  .  .  .|         |    2:  .   .   .   .|       |    2:  .  .   .   .|
         |    3:  .   . 126  .|        |    3:  .  .  .  .|        |    3: . 35  .  .|         |    3:  . 210   .   .|       |    3:  . 70   .   .|
         \    4:  .   .   . 45/        \    4:  .  . 70 45/        \    4: .  . 70 40/         |    4:  .   . 280   .|       |    4:  .  .   .   .|
                                                                                               \    5:  .   .   . 105/       \    5:  .  . 160 105/

o6 : Expression of class Sum</code></pre>
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        <h2>See also</h2>
        <ul>
          <li><span><a title="write a Betti diagram as a positive combination of pure integral diagrams" href="_decompose_lp__Betti__Tally_rp.html">decompose(BettiTally)</a></span></li>
          <li><span><a title="Find the degree sequences of pure diagrams occurring in a Boij-Soederberg decomposition of B" href="_decompose__Degrees.html">decomposeDegrees</a> -- Find the degree sequences of pure diagrams occurring in a Boij-Soederberg decomposition of B</span></li>
        </ul>
      </div>
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        <div class="waystouse">
          <h2>Ways to use <span class="tt">decomposeBetti</span>:</h2>
          <ul>
            <li><span><span class="tt">decomposeBetti(BettiTally)</span> (missing documentation)
</span></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="write a Betti diagram as a positive combination of pure integral diagrams" href="_decompose__Betti.html">decomposeBetti</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">BoijSoederberg.m2:1799:0</span>.</p>
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