One Hat Cyber Team

One Hat Cyber Team

Your IP : 216.73.216.135

Server IP : 194.44.31.54

Server : Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64

Server Software : Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k

PHP Version : 5.6.40

Buat File | Buat Folder

Dir : ~/usr/share/doc/Macaulay2/Permanents/html/

Edit File: index.html

<!DOCTYPE html>
<html lang="en">
  <head>
    <title>Permanents -- Computes the permanent of a square matrix.</title>
    <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
    <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css">

<link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css">
    <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script>
    <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script>
    <script>
      var macros = {
          "\\break": "\\\\",
          "\\ZZ": "\\mathbb{Z}",
          "\\NN": "\\mathbb{N}",
          "\\QQ": "\\mathbb{Q}",
          "\\RR": "\\mathbb{R}",
          "\\CC": "\\mathbb{C}",
          "\\PP": "\\mathbb{P}"
      }, delimiters = [
          { left: "$$",  right: "$$",  display: true},
          { left: "\\[", right: "\\]", display: true},
          { left: "$",   right: "$",   display: false},
          { left: "\\(", right: "\\)", display: false}
      ], ignoredTags = [
	  "kbd", "var", "samp", "script", "noscript",
	  "style", "textarea", "pre", "code", "option" ];
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true });
      });
    </script>
    <style>.katex { font-size: 1em; }</style>
    <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script>
    <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script>
    <script src="../../../../Macaulay2/Style/prism.js"></script>
    <script>var current_version = '1.25.06';</script>
    <script src="../../../../Macaulay2/Style/version-select.js"></script>
    <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif">
  </head>
  <body>
    <div id="buttons">
      <div>
<a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Computes the permanent of a square matrix." href="index.html">Permanents</a> :: <a title="Computes the permanent of a square matrix." href="index.html">Permanents</a></span>      </div>
      <div class="right">
<form method="get" action="https://www.google.com/search">
  <input placeholder="Search" type="text" name="q" value="">
  <input type="hidden" name="q" value="site:macaulay2.com/doc">
</form>
<a href="_glynn.html">next</a> | previous | <a href="_glynn.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a>      </div>
    </div>
    <hr>
    <div>
      <h1>Permanents -- Computes the permanent of a square matrix.</h1>
      <div>
        <h2>Description</h2>
        <div>
          <p><em>Permanents</em> is a package of functions for computing the permanent of a square matrix.</p>
          <p>Computing the permanent is believed to be computationally intractable. In Valiant's theory of algebraic complexity the permanent polynomial is complete for the class VNP. Even computing the permanent of 0-1 matrices is #P-complete. See Valiant, Leslie G. (1979), &quot;The Complexity of Computing the Permanent,&quot; <em>Theoretical Computer Science (Elsevier)</em> 8 (2): 189-201.</p>
          <p>The permanent of a $n\times n$ matrix $(x_{i,j})$ is defined in analogy to the determinant as $\sum_{\sigma \in S_n} \prod x_{i,\sigma(i)}$. There are two other formulas for the permanent polynomial, Ryser's formula and Glynn's formula, both of which have the asymptotically smaller formula size of $O(2^n n^2)$. This can be improved further to {$O(2^n n)$} arithmetic operations with the use of Gray codes, and we do so in this package. The connection between the two formulae and the possibility of others is discussed in Glynn, David G. (2013), &quot;Permanent Formulae from the Veronesean.&quot; <em>Designs Codes and Cryptography</em>, 68(1-3) pp. 39-47.</p>
          <p>It is conjectured that the permanent polynomial does not have a polynomial size formula. By Valiant's theory, a possible strategy for proving this is to show that the permanent of the $n\times n$ generic matrix $N$ cannot be the determinant of a $p(n) \times p(n)$ matrix $M$ with entries affine linear entries of the variables of $M$ where $p(n)$ is a polynomial. The best lower bound is quadratic, i.e. the permanent of the $nxn$ generic matrix is not the affine projection of the determinant of a $n^2/2xn^2/2$ matrix. See T. Mignon, N. Ressayre. &quot;A Quadratic Bound for the Determinant and Permanent Problem.&quot; (2004).</p>
          <p>The best known upper bound is $2^n-1$ due to Grenet. More specifically, Grenet constructs a $2^n-1x2^n-1$ matrix $M$ with entries $0,1,-1$ and individual variables of the $nxn$ generic matrix $N$, such that the determinant of $M$ is equal to the permanent of $N$. See B. Grenet, &quot;An Upper Bound for the Permanent versus Determinant Problem&quot; (2012).</p>
        </div>
      </div>
      <div>
        <h2>Caveat</h2>
        <div>
          <p>Computationally intensive</p>
        </div>
      </div>
      <div>
        <h2>See also</h2>
        <ul>
          <li><span><a title="ideal generated by square permanents of a matrix" href="../../Macaulay2Doc/html/_permanents.html">permanents</a> -- ideal generated by square permanents of a matrix</span></li>
          <li><span><a title="determinant of a matrix" href="../../Macaulay2Doc/html/_determinant.html">determinant</a> -- determinant of a matrix</span></li>
        </ul>
      </div>
      <div>
        <div>
          <div>
            <h2>Author</h2>
            <ul>
              <li><a href="http://www.math.cornell.edu/~takhmejanov">Tair Akhmejanov</a><span> &lt;<a href="mailto:ta328%40cornell.edu">ta328@cornell.edu</a>></span></li>
            </ul>
          </div>
          <div>
            <h2>Version</h2>
            <p>This documentation describes version <b>0.9</b> of Permanents, released <b>July 11, 2014</b>.</p>
          </div>
          <div>
            <h2>Citation</h2>
            <p>If you have used this package in your research, please cite it as follows:</p>
            <table class="examples">
              <tr>
                <td>
                  <pre><code class="language-bib">@misc{PermanentsSource,
  title = {{Permanents: permanents of a matrix. Version~0.9}},
  author = {Tair Akhmejanov},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}
</code></pre>
                </td>
              </tr>
            </table>
          </div>
          <div>
            <h2>Exports</h2>
            <div class="exports">
              <ul>
                <li>Functions and commands                  <ul>
                    <li><span><a title="compute permanent using Glynn's formula" href="_glynn.html">glynn</a> -- compute permanent using Glynn's formula</span></li>
                    <li><span><a title="Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix" href="_grenet.html">grenet</a> -- Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix</span></li>
                    <li><span><a title="Return ideal generated by pminors of a specified size" href="_pminors.html">pminors</a> -- Return ideal generated by pminors of a specified size</span></li>
                    <li><span><a title="compute permanent using Ryser's formula" href="_ryser.html">ryser</a> -- compute permanent using Ryser's formula</span></li>
                  </ul>
                </li>
                <li>Methods                  <ul>
                    <li><span><kbd>glynn(Matrix)</kbd> -- see <span><a title="compute permanent using Glynn's formula" href="_glynn.html">glynn</a> -- compute permanent using Glynn's formula</span></span></li>
                    <li><span><kbd>grenet(Matrix)</kbd> -- see <span><a title="Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix" href="_grenet.html">grenet</a> -- Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix</span></span></li>
                    <li><span><kbd>pminors(ZZ,Matrix)</kbd> -- see <span><a title="Return ideal generated by pminors of a specified size" href="_pminors.html">pminors</a> -- Return ideal generated by pminors of a specified size</span></span></li>
                    <li><span><kbd>ryser(Matrix)</kbd> -- see <span><a title="compute permanent using Ryser's formula" href="_ryser.html">ryser</a> -- compute permanent using Ryser's formula</span></span></li>
                  </ul>
                </li>
              </ul>
            </div>
          </div>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="Computes the permanent of a square matrix." href="index.html">Permanents</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Permanents.m2</span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Permanents.m2:400:0</span>.</p>
        </div>
      </div>
    </div>
  </body>

</html>