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<a name="Symmetries"></a>
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<h1 class="chapter"> 32. Symmetries </h1>


<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC141">32.1 Introduction to Symmetries</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC142">32.2 Functions and Variables for Symmetries</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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<p><a name="Item_003a-Introduction-to-Symmetries"></a>
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<h2 class="section"> 32.1 Introduction to Symmetries </h2>

<p><code>sym</code> is a package for working with symmetric groups of polynomials.
</p>
<p>It was written for Macsyma-Symbolics by Annick Valibouze (<a href="http://www-calfor.lip6.fr/~avb/">http://www-calfor.lip6.fr/~avb/</a>).
The algorithms are described in the following papers:
</p>
<ol>
<li>
Fonctions sym&eacute;triques et changements de bases. Annick Valibouze.
EUROCAL'87 (Leipzig, 1987), 323-332, Lecture Notes in Comput. Sci 378.
Springer, Berlin, 1989.<br>
<a href="http://www.stix.polytechnique.fr/publications/1984-1994.html">http://www.stix.polytechnique.fr/publications/1984-1994.html</a>

</li><li> R&eacute;solvantes et fonctions sym&eacute;triques. Annick Valibouze.
Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic
and Algebraic Computation, ISSAC'89 (Portland, Oregon).
ACM Press, 390-399, 1989.<br>
<a href="http://www-calfor.lip6.fr/~avb/DonneesTelechargeables/MesArticles/issac89ACMValibouze.pdf">http://www-calfor.lip6.fr/~avb/DonneesTelechargeables/MesArticles/issac89ACMValibouze.pdf</a>

</li><li> Symbolic computation with symmetric polynomials, an extension to Macsyma.
Annick Valibouze. Computers and Mathematics (MIT, USA, June 13-17, 1989),
Springer-Verlag, New York Berlin, 308-320, 1989.<br>
<a href="http://www.stix.polytechnique.fr/publications/1984-1994.html">http://www.stix.polytechnique.fr/publications/1984-1994.html</a>

</li><li> Th&eacute;orie de Galois Constructive. Annick Valibouze. M&eacute;moire d'habilitation
&agrave; diriger les recherches (HDR), Universit&eacute; P. et M. Curie (Paris VI), 1994.
</li></ol>


<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Group-theory">Group theory</a>
 &middot;
<a href="maxima_95.html#Category_003a-Polynomials">Polynomials</a>
 &middot;
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
 &middot;
<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>



<p><a name="Item_003a-Functions-and-Variables-for-Symmetries"></a>
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<h2 class="section"> 32.2 Functions and Variables for Symmetries </h2>


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<h3 class="subsection"> 32.2.1 Changing bases </h3>


<p><a name="Item_003a-comp2pui"></a>
</p><dl>
<dt><u>Function:</u> <b>comp2pui</b><i> (<var>n</var>, <var>L</var>)</i>
<a name="IDX1183"></a>
</dt>
<dd><p>implements passing from the complete symmetric functions given in the list
<var>L</var> to the elementary symmetric functions from 0 to <var>n</var>. If the
list <var>L</var> contains fewer than <var>n+1</var> elements, it will be completed with
formal values of the type <var>h1</var>, <var>h2</var>, etc. If the first element
of the list <var>L</var> exists, it specifies the size of the alphabet,
otherwise the size is set to <var>n</var>.
</p>
<pre class="example">(%i1) comp2pui (3, [4, g]);
                        2                    2
(%o1)    [4, g, 2 h2 - g , 3 h3 - g h2 + g (g  - 2 h2)]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-ele2pui"></a>
</p><dl>
<dt><u>Function:</u> <b>ele2pui</b><i> (<var>m</var>, <var>L</var>)</i>
<a name="IDX1184"></a>
</dt>
<dd><p>goes from the elementary symmetric functions to the complete functions.
Similar to <code>comp2ele</code> and <code>comp2pui</code>.
</p>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-ele2comp"></a>
</p><dl>
<dt><u>Function:</u> <b>ele2comp</b><i> (<var>m</var>, <var>L</var>)</i>
<a name="IDX1185"></a>
</dt>
<dd><p>Goes from the elementary symmetric functions to the compete functions.
Similar to <code>comp2ele</code> and <code>comp2pui</code>.
</p>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-elem"></a>
</p><dl>
<dt><u>Function:</u> <b>elem</b><i> (<var>ele</var>, <var>sym</var>, <var>lvar</var>)</i>
<a name="IDX1186"></a>
</dt>
<dd><p>decomposes the symmetric polynomial <var>sym</var>, in the variables
contained in the list <var>lvar</var>, in terms of the elementary symmetric
functions given in the list <var>ele</var>.  If the first element of
<var>ele</var> is given, it will be the size of the alphabet, otherwise the
size will be the degree of the polynomial <var>sym</var>.  If values are
missing in the list <var>ele</var>, formal values of the type <var>e1</var>,
<var>e2</var>, etc. will be added.  The polynomial <var>sym</var> may be given in
three different forms: contracted (<code>elem</code> should then be 1, its
default value), partitioned (<code>elem</code> should be 3), or extended
(i.e. the entire polynomial, and <code>elem</code> should then be 2).  The
function <code>pui</code> is used in the same way.
</p>
<p>On an alphabet of size 3 with <var>e1</var>, the first elementary symmetric
function, with value 7, the symmetric polynomial in 3 variables whose
contracted form (which here depends on only two of its variables) is
<var>x^4-2*x*y</var> decomposes as follows in elementary symmetric functions:
</p>
<pre class="example">(%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
(%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3

                                         + (- 2 (49 - e2) - 2) e2
(%i2) ratsimp (%);
                              2
(%o2)             28 e3 + 2 e2  - 198 e2 + 2401
</pre>

<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-mon2schur"></a>
</p><dl>
<dt><u>Function:</u> <b>mon2schur</b><i> (<var>L</var>)</i>
<a name="IDX1187"></a>
</dt>
<dd><p>The list <var>L</var> represents the Schur function <em>S_L</em>: we have
<em>L = [i_1, i_2, ..., i_q]</em>, with <em>i_1 &lt;= i_2 &lt;= ... &lt;= i_q</em>.
The Schur function <em>S_[i_1, i_2, ..., i_q]</em> is the minor
of the infinite matrix <em>h_[i-j]</em>, <em>i &lt;= 1, j &lt;= 1</em>,
consisting of the <em>q</em> first rows and the columns <em>1 + i_1,
2 + i_2, ..., q + i_q</em>.
</p>
<p>This Schur function can be written in terms of monomials by using
<code>treinat</code> and <code>kostka</code>.  The form returned is a symmetric
polynomial in a contracted representation in the variables
<em>x_1,x_2,...</em>
</p>
<pre class="example">(%i1) mon2schur ([1, 1, 1]);
(%o1)                       x1 x2 x3
(%i2) mon2schur ([3]);
                                  2        3
(%o2)                x1 x2 x3 + x1  x2 + x1
(%i3) mon2schur ([1, 2]);
                                      2
(%o3)                  2 x1 x2 x3 + x1  x2
</pre>

<p>which means that for 3 variables this gives:
</p>
<pre class="example">   2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1
    + x2^2 x3 + x3^2 x2
</pre>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-multi_005felem"></a>
</p><dl>
<dt><u>Function:</u> <b>multi_elem</b><i> (<var>l_elem</var>, <var>multi_pc</var>, <var>l_var</var>)</i>
<a name="IDX1188"></a>
</dt>
<dd><p>decomposes a multi-symmetric polynomial in the multi-contracted form
<var>multi_pc</var> in the groups of variables contained in the list of lists
<var>l_var</var> in terms of the elementary symmetric functions contained in
<var>l_elem</var>.
</p>
<pre class="example">(%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3,
      [[x, y], [a, b]]);
                                                  3
(%o1)         - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1
(%i2) ratsimp (%);
                         2                       3
(%o2)         - 2 f2 + f1  + e1 f1 - 3 e1 e2 + e1
</pre>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-multi_005fpui"></a>
</p><dl>
<dt><u>Function:</u> <b>multi_pui</b>
<a name="IDX1189"></a>
</dt>
<dd><p>is to the function <code>pui</code> what the function <code>multi_elem</code> is to
the function <code>elem</code>.
</p>
<pre class="example">(%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3,
      [[x, y], [a, b]]);
                                            3
                                3 p1 p2   p1
(%o1)              t2 + p1 t1 + ------- - ---
                                   2       2
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-pui"></a>
</p><dl>
<dt><u>Function:</u> <b>pui</b><i> (<var>L</var>, <var>sym</var>, <var>lvar</var>)</i>
<a name="IDX1190"></a>
</dt>
<dd><p>decomposes the symmetric polynomial <var>sym</var>, in the variables in the
list <var>lvar</var>, in terms of the power functions in the list <var>L</var>.
If the first element of <var>L</var> is given, it will be the size of the
alphabet, otherwise the size will be the degree of the polynomial
<var>sym</var>.  If values are missing in the list <var>L</var>, formal values of
the type <var>p1</var>, <var>p2</var> , etc. will be added. The polynomial
<var>sym</var> may be given in three different forms: contracted (<code>elem</code>
should then be 1, its default value), partitioned (<code>elem</code> should be
3), or extended (i.e. the entire polynomial, and <code>elem</code> should then
be 2). The function <code>pui</code> is used in the same way.
</p>
<pre class="example">(%i1) pui;
(%o1)                           1
(%i2) pui ([3, a, b], u*x*y*z, [x, y, z]);
                       2
                   a (a  - b) u   (a b - p3) u
(%o2)              ------------ - ------------
                        6              3
(%i3) ratsimp (%);
                                       3
                      (2 p3 - 3 a b + a ) u
(%o3)                 ---------------------
                                6
</pre>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>



<p><a name="Item_003a-pui2comp"></a>
</p><dl>
<dt><u>Function:</u> <b>pui2comp</b><i> (<var>n</var>, <var>lpui</var>)</i>
<a name="IDX1191"></a>
</dt>
<dd><p>renders the list of the first <var>n</var> complete functions (with the
length first) in terms of the power functions given in the list
<var>lpui</var>. If the list <var>lpui</var> is empty, the cardinal is <var>n</var>,
otherwise it is its first element (as in <code>comp2ele</code> and
<code>comp2pui</code>).
</p>
<pre class="example">(%i1) pui2comp (2, []);
                                       2
                                p2 + p1
(%o1)                   [2, p1, --------]
                                   2
(%i2) pui2comp (3, [2, a1]);
                                            2
                                 a1 (p2 + a1 )
                         2  p3 + ------------- + a1 p2
                  p2 + a1              2
(%o2)     [2, a1, --------, --------------------------]
                     2                  3
(%i3) ratsimp (%);
                            2                     3
                     p2 + a1   2 p3 + 3 a1 p2 + a1
(%o3)        [2, a1, --------, --------------------]
                        2               6
</pre>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
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<p><a name="Item_003a-pui2ele"></a>
</p><dl>
<dt><u>Function:</u> <b>pui2ele</b><i> (<var>n</var>, <var>lpui</var>)</i>
<a name="IDX1192"></a>
</dt>
<dd><p>effects the passage from power functions to the elementary symmetric functions.
If the flag <code>pui2ele</code> is <code>girard</code>, it will return the list of
elementary symmetric functions from 1 to <var>n</var>, and if the flag is
<code>close</code>, it will return the <var>n</var>-th elementary symmetric function.
</p>
<p>Other functions for changing bases: <code>comp2ele</code>.
</p>
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<p><a name="Item_003a-puireduc"></a>
</p><dl>
<dt><u>Function:</u> <b>puireduc</b><i> (<var>n</var>, <var>lpui</var>)</i>
<a name="IDX1193"></a>
</dt>
<dd><p><var>lpui</var> is a list whose first element is an integer <var>m</var>.
<code>puireduc</code> gives the first <var>n</var> power functions in terms of the
first <var>m</var>.
</p>
<pre class="example">(%i1) puireduc (3, [2]);
                                         2
                                   p1 (p1  - p2)
(%o1)          [2, p1, p2, p1 p2 - -------------]
                                         2
(%i2) ratsimp (%);
                                           3
                               3 p1 p2 - p1
(%o2)              [2, p1, p2, -------------]
                                     2
</pre>
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</dd></dl>


<p><a name="Item_003a-schur2comp"></a>
</p><dl>
<dt><u>Function:</u> <b>schur2comp</b><i> (<var>P</var>, <var>l_var</var>)</i>
<a name="IDX1194"></a>
</dt>
<dd><p><var>P</var> is a polynomial in the variables of the list <var>l_var</var>.  Each
of these variables represents a complete symmetric function.  In
<var>l_var</var> the <var>i</var>-th complete symmetric function is represented by
the concatenation of the letter <code>h</code> and the integer <var>i</var>:
<code>h<var>i</var></code>.  This function expresses <var>P</var> in terms of Schur
functions.
</p>

<pre class="example">(%i1) schur2comp (h1*h2 - h3, [h1, h2, h3]);
(%o1)                         s
                               1, 2
(%i2) schur2comp (a*h3, [h3]);
(%o2)                         s  a
                               3
</pre>
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<h3 class="subsection"> 32.2.2 Changing representations </h3>

<p><a name="Item_003a-cont2part"></a>
</p><dl>
<dt><u>Function:</u> <b>cont2part</b><i> (<var>pc</var>, <var>lvar</var>)</i>
<a name="IDX1195"></a>
</dt>
<dd><p>returns the partitioned polynomial associated to the contracted form
<var>pc</var> whose variables are in <var>lvar</var>.
</p>
<pre class="example">(%i1) pc: 2*a^3*b*x^4*y + x^5;
                           3    4      5
(%o1)                   2 a  b x  y + x
(%i2) cont2part (pc, [x, y]);
                                   3
(%o2)              [[1, 5, 0], [2 a  b, 4, 1]]
</pre>
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<dl>
<dt><u>Function:</u> <b>contract</b><i> (<var>psym</var>, <var>lvar</var>)</i>
<a name="IDX1196"></a>
</dt>
<dd><p>returns a contracted form (i.e. a monomial orbit under the action of the
symmetric group) of the polynomial <var>psym</var> in the variables contained
in the list <var>lvar</var>.  The function <code>explose</code> performs the
inverse operation.  The function <code>tcontract</code> tests the symmetry of
the polynomial.
</p>
<pre class="example">(%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]);
         3      4      3      4      3    4        3    4
(%o1) 2 a  b y z  + 2 a  b x z  + 2 a  b y  z + 2 a  b x  z

                                           3      4      3    4
                                      + 2 a  b x y  + 2 a  b x  y
(%i2) contract (psym, [x, y, z]);
                              3    4
(%o2)                      2 a  b x  y
</pre>
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<p><a name="Item_003a-explose"></a>
</p><dl>
<dt><u>Function:</u> <b>explose</b><i> (<var>pc</var>, <var>lvar</var>)</i>
<a name="IDX1197"></a>
</dt>
<dd><p>returns the symmetric polynomial associated with the contracted form
<var>pc</var>. The list <var>lvar</var> contains the variables.
</p>
<pre class="example">(%i1) explose (a*x + 1, [x, y, z]);
(%o1)                  a z + a y + a x + 1
</pre>
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<p><a name="Item_003a-part2cont"></a>
</p><dl>
<dt><u>Function:</u> <b>part2cont</b><i> (<var>ppart</var>, <var>lvar</var>)</i>
<a name="IDX1198"></a>
</dt>
<dd><p>goes from the partitioned form to the contracted form of a symmetric polynomial.
The contracted form is rendered with the variables in <var>lvar</var>.
</p>
<pre class="example">(%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]);
                              3    4
(%o1)                      2 a  b x  y
</pre>
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<p><a name="Item_003a-partpol"></a>
</p><dl>
<dt><u>Function:</u> <b>partpol</b><i> (<var>psym</var>, <var>lvar</var>)</i>
<a name="IDX1199"></a>
</dt>
<dd><p><var>psym</var> is a symmetric polynomial in the variables of the list
<var>lvar</var>. This function retturns its partitioned representation.
</p>
<pre class="example">(%i1) partpol (-a*(x + y) + 3*x*y, [x, y]);
(%o1)               [[3, 1, 1], [- a, 1, 0]]
</pre>
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<p><a name="Item_003a-tcontract"></a>
</p><dl>
<dt><u>Function:</u> <b>tcontract</b><i> (<var>pol</var>, <var>lvar</var>)</i>
<a name="IDX1200"></a>
</dt>
<dd><p>tests if the polynomial <var>pol</var> is symmetric in the variables of the
list <var>lvar</var>.  If so, it returns a contracted representation like the
function <code>contract</code>.
</p>
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<p><a name="Item_003a-tpartpol"></a>
</p><dl>
<dt><u>Function:</u> <b>tpartpol</b><i> (<var>pol</var>, <var>lvar</var>)</i>
<a name="IDX1201"></a>
</dt>
<dd><p>tests if the polynomial <var>pol</var> is symmetric in the variables of the
list <var>lvar</var>.  If so, it returns its partitioned representation like
the function <code>partpol</code>.
</p>
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<h3 class="subsection"> 32.2.3 Groups and orbits </h3>


<p><a name="Item_003a-direct"></a>
</p><dl>
<dt><u>Function:</u> <b>direct</b><i> ([<var>p_1</var>, ..., <var>p_n</var>], <var>y</var>, <var>f</var>, [<var>lvar_1</var>, ..., <var>lvar_n</var>])</i>
<a name="IDX1202"></a>
</dt>
<dd><p>calculates the direct image (see M. Giusti, D. Lazard et A. Valibouze,
ISSAC 1988, Rome) associated to the function <var>f</var>, in the lists of
variables <var>lvar_1</var>, ..., <var>lvar_n</var>, and in the polynomials
<var>p_1</var>, ..., <var>p_n</var> in a variable <var>y</var>.  The arity of the
function <var>f</var> is important for the calulation.  Thus, if the
expression for <var>f</var> does not depend on some variable, it is useless
to include this variable, and not including it will also considerably
reduce the amount of computation.
</p>
<pre class="example">(%i1) direct ([z^2  - e1* z + e2, z^2  - f1* z + f2],
              z, b*v + a*u, [[u, v], [a, b]]);
       2
(%o1) y  - e1 f1 y

                                 2            2             2   2
                  - 4 e2 f2 - (e1  - 2 e2) (f1  - 2 f2) + e1  f1
                + -----------------------------------------------
                                         2
(%i2) ratsimp (%);
              2                2                   2
(%o2)        y  - e1 f1 y + (e1  - 4 e2) f2 + e2 f1
(%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2  - f1* z + f2],
              z, b*v + a*u, [[u, v], [a, b]]));
       6            5         2                        2    2   4
(%o3) y  - 2 e1 f1 y  + ((2 e1  - 6 e2) f2 + (2 e2 + e1 ) f1 ) y

                          3                               3   3
 + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y

         2       2        4    2
 + ((9 e2  - 6 e1  e2 + e1 ) f2

                    2       2       2                   2    4
 + (- 9 e1 e3 - 6 e2  + 3 e1  e2) f1  f2 + (2 e1 e3 + e2 ) f1 )

  2          2                      2     3          2
 y  + (((9 e1  - 27 e2) e3 + 3 e1 e2  - e1  e2) f1 f2

                 2            2    3                5
 + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1  f2 - 2 e2 e3 f1 ) y

           2                   3           3     2   2    3
 + (- 27 e3  + (18 e1 e2 - 4 e1 ) e3 - 4 e2  + e1  e2 ) f2

         2      3                   3    2   2
 + (27 e3  + (e1  - 9 e1 e2) e3 + e2 ) f1  f2

                   2    4        2   6
 + (e1 e2 e3 - 9 e3 ) f1  f2 + e3  f1
</pre>
<p>Finding the polynomial whose roots are the sums <em>a+u</em> where <em>a</em>
is a root of <em>z^2 - e_1 z + e_2</em> and <em>u</em> is a root of <em>z^2 -
f_1 z + f_2</em>.
</p>
<pre class="example">(%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
                          z, a + u, [[u], [a]]));
       4                    3             2
(%o1) y  + (- 2 f1 - 2 e1) y  + (2 f2 + f1  + 3 e1 f1 + 2 e2

     2   2                              2               2
 + e1 ) y  + ((- 2 f1 - 2 e1) f2 - e1 f1  + (- 2 e2 - e1 ) f1

                  2                     2            2
 - 2 e1 e2) y + f2  + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1  + e1 e2 f1

     2
 + e2
</pre>
<p><code>direct</code> accepts two flags: <code>elementaires</code> and
<code>puissances</code> (default) which allow decomposing the symmetric
polynomials appearing in the calculation into elementary symmetric
functions, or power functions, respectively.
</p>
<p>Functions of <code>sym</code> used in this function:
</p>
<p><code>multi_orbit</code> (so <code>orbit</code>), <code>pui_direct</code>, <code>multi_elem</code>
(so <code>elem</code>), <code>multi_pui</code> (so <code>pui</code>), <code>pui2ele</code>, <code>ele2pui</code>
(if the flag <code>direct</code> is in <code>puissances</code>).
</p>
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<p><a name="Item_003a-multi_005forbit"></a>
</p><dl>
<dt><u>Function:</u> <b>multi_orbit</b><i> (<var>P</var>, [<var>lvar_1</var>, <var>lvar_2</var>,..., <var>lvar_p</var>])</i>
<a name="IDX1203"></a>
</dt>
<dd><p><var>P</var> is a polynomial in the set of variables contained in the lists
<var>lvar_1</var>, <var>lvar_2</var>, ..., <var>lvar_p</var>. This function returns the
orbit of the polynomial <var>P</var> under the action of the product of the
symmetric groups of the sets of variables represented in these <var>p</var>
lists.
</p>
<pre class="example">(%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]);
(%o1)                [b y + a x, a y + b x]
(%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]);
(%o2)        [y + x + 2 c, y + x + 2 b, y + x + 2 a]
</pre>
<p>Also see: <code>orbit</code> for the action of a single symmetric group.
</p>
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<p><a name="Item_003a-multsym"></a>
</p><dl>
<dt><u>Function:</u> <b>multsym</b><i> (<var>ppart_1</var>, <var>ppart_2</var>, <var>n</var>)</i>
<a name="IDX1204"></a>
</dt>
<dd><p>returns the product of the two symmetric polynomials in <var>n</var>
variables by working only modulo the action of the symmetric group of
order <var>n</var>. The polynomials are in their partitioned form.
</p>
<p>Given the 2 symmetric polynomials in <var>x</var>, <var>y</var>:  <code>3*(x + y)
+ 2*x*y</code> and <code>5*(x^2 + y^2)</code> whose partitioned forms are <code>[[3,
1], [2, 1, 1]]</code> and <code>[[5, 2]]</code>, their product will be
</p>
<pre class="example">(%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2);
(%o1)         [[10, 3, 1], [15, 3, 0], [15, 2, 1]]
</pre>
<p>that is <code>10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3)</code>.
</p>
<p>Functions for changing the representations of a symmetric polynomial:
</p>
<p><code>contract</code>, <code>cont2part</code>, <code>explose</code>, <code>part2cont</code>,
<code>partpol</code>, <code>tcontract</code>, <code>tpartpol</code>.
</p>
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<p><a name="Item_003a-orbit"></a>
</p><dl>
<dt><u>Function:</u> <b>orbit</b><i> (<var>P</var>, <var>lvar</var>)</i>
<a name="IDX1205"></a>
</dt>
<dd><p>computes the orbit of the polynomial <var>P</var> in the variables in the list
<var>lvar</var> under the action of the symmetric group of the set of
variables in the list <var>lvar</var>.
</p>
<pre class="example">(%i1) orbit (a*x + b*y, [x, y]);
(%o1)                [a y + b x, b y + a x]
(%i2) orbit (2*x + x^2, [x, y]);
                        2         2
(%o2)                 [y  + 2 y, x  + 2 x]
</pre>
<p>See also <code>multi_orbit</code> for the action of a product of symmetric
groups on a polynomial.
</p>
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<p><a name="Item_003a-pui_005fdirect"></a>
</p><dl>
<dt><u>Function:</u> <b>pui_direct</b><i> (<var>orbite</var>, [<var>lvar_1</var>, ..., <var>lvar_n</var>], [<var>d_1</var>, <var>d_2</var>, ..., <var>d_n</var>])</i>
<a name="IDX1206"></a>
</dt>
<dd><p>Let <var>f</var> be a polynomial in <var>n</var> blocks of variables <var>lvar_1</var>,
..., <var>lvar_n</var>.  Let <var>c_i</var> be the number of variables in
<var>lvar_i</var>, and <var>SC</var> be the product of <var>n</var> symmetric groups of
degree <var>c_1</var>, ..., <var>c_n</var>. This group acts naturally on <var>f</var>.
The list <var>orbite</var> is the orbit, denoted <code><var>SC</var>(<var>f</var>)</code>, of
the function <var>f</var> under the action of <var>SC</var>. (This list may be
obtained by the function <code>multi_orbit</code>.)  The <var>di</var> are integers
s.t.
<em>c_1 &lt;= d_1, c_2 &lt;= d_2, ..., c_n &lt;= d_n</em>.
</p>
<p>Let <var>SD</var> be the product of the symmetric groups <em>S_[d_1] x
S_[d_2] x ... x S_[d_n]</em>.
The function <code>pui_direct</code> returns
the first <var>n</var> power functions of <code><var>SD</var>(<var>f</var>)</code> deduced
from the power functions of <code><var>SC</var>(<var>f</var>)</code>, where <var>n</var> is
the size of <code><var>SD</var>(<var>f</var>)</code>.
</p>
<p>The result is in multi-contracted form w.r.t. <var>SD</var>, i.e. only one
element is kept per orbit, under the action of <var>SD</var>.
</p>
<pre class="example">(%i1) l: [[x, y], [a, b]];
(%o1)                   [[x, y], [a, b]]
(%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
                                       2  2
(%o2)               [a x, 4 a b x y + a  x ]
(%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
                             2  2     2    2        3  3
(%o3) [2 a x, 4 a b x y + 2 a  x , 3 a  b x  y + 2 a  x , 

    2  2  2  2      3    3        4  4
12 a  b  x  y  + 4 a  b x  y + 2 a  x , 

    3  2  3  2      4    4        5  5
10 a  b  x  y  + 5 a  b x  y + 2 a  x , 

    3  3  3  3       4  2  4  2      5    5        6  6
40 a  b  x  y  + 15 a  b  x  y  + 6 a  b x  y + 2 a  x ]
(%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],
      [[x, y], [a, b, c]], [2, 3]);
                             2              2
(%o4) [3 x + 2 a, 6 x y + 3 x  + 4 a x + 4 a , 

                 2                   3        2       2        3
              9 x  y + 12 a x y + 3 x  + 6 a x  + 12 a  x + 8 a ]
</pre>
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<h3 class="subsection"> 32.2.4 Partitions </h3>

<p><a name="Item_003a-kostka"></a>
</p><dl>
<dt><u>Function:</u> <b>kostka</b><i> (<var>part_1</var>, <var>part_2</var>)</i>
<a name="IDX1207"></a>
</dt>
<dd><p>written by P. Esperet, calculates the Kostka number of the partition
<var>part_1</var> and <var>part_2</var>.
</p>
<pre class="example">(%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]);
(%o1)                           6
</pre>
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<p><a name="Item_003a-lgtreillis"></a>
</p><dl>
<dt><u>Function:</u> <b>lgtreillis</b><i> (<var>n</var>, <var>m</var>)</i>
<a name="IDX1208"></a>
</dt>
<dd><p>returns the list of partitions of weight <var>n</var> and length <var>m</var>.
</p>
<pre class="example">(%i1) lgtreillis (4, 2);
(%o1)                   [[3, 1], [2, 2]]
</pre>
<p>Also see: <code>ltreillis</code>, <code>treillis</code> and <code>treinat</code>.
</p>
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<p><a name="Item_003a-ltreillis"></a>
</p><dl>
<dt><u>Function:</u> <b>ltreillis</b><i> (<var>n</var>, <var>m</var>)</i>
<a name="IDX1209"></a>
</dt>
<dd><p>returns the list of partitions of weight <var>n</var> and length less than or
equal to <var>m</var>.
</p>
<pre class="example">(%i1) ltreillis (4, 2);
(%o1)               [[4, 0], [3, 1], [2, 2]]
</pre>
<p>Also see: <code>lgtreillis</code>, <code>treillis</code> and <code>treinat</code>.
</p>
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<p><a name="Item_003a-treillis"></a>
</p><dl>
<dt><u>Function:</u> <b>treillis</b><i> (<var>n</var>)</i>
<a name="IDX1210"></a>
</dt>
<dd><p>returns all partitions of weight <var>n</var>.
</p>
<pre class="example">(%i1) treillis (4);
(%o1)    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
</pre>
<p>See also: <code>lgtreillis</code>, <code>ltreillis</code> and <code>treinat</code>.
</p>
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<p><a name="Item_003a-treinat"></a>
</p><dl>
<dt><u>Function:</u> <b>treinat</b><i> (<var>part</var>)</i>
<a name="IDX1211"></a>
</dt>
<dd><p>retruns the list of partitions inferior to the partition <var>part</var> w.r.t.
the natural order.
</p>
<pre class="example">(%i1) treinat ([5]);
(%o1)                         [[5]]
(%i2) treinat ([1, 1, 1, 1, 1]);
(%o2) [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], 

                                                 [1, 1, 1, 1, 1]]
(%i3) treinat ([3, 2]);
(%o3)                 [[5], [4, 1], [3, 2]]
</pre>
<p>See also: <code>lgtreillis</code>, <code>ltreillis</code> and <code>treillis</code>.
</p>
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<h3 class="subsection"> 32.2.5 Polynomials and their roots </h3>

<p><a name="Item_003a-ele2polynome"></a>
</p><dl>
<dt><u>Function:</u> <b>ele2polynome</b><i> (<var>L</var>, <var>z</var>)</i>
<a name="IDX1212"></a>
</dt>
<dd><p>returns the polynomial in <var>z</var> s.t. the elementary symmetric
functions of its roots are in the list <code><var>L</var> = [<var>n</var>,
<var>e_1</var>, ..., <var>e_n</var>]</code>, where <var>n</var> is the degree of the
polynomial and <var>e_i</var> the <var>i</var>-th elementary symmetric function.
</p>
<pre class="example">(%i1) ele2polynome ([2, e1, e2], z);
                          2
(%o1)                    z  - e1 z + e2
(%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3  - 56*x + 22, x);
(%o2)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
(%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                  7       5       3
(%o3)            x  - 14 x  + 56 x  - 56 x + 22
</pre>
<p>The inverse: <code>polynome2ele (<var>P</var>, <var>z</var>)</code>.
</p>
<p>Also see:
<code>polynome2ele</code>, <code>pui2polynome</code>.
</p>
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<p><a name="Item_003a-polynome2ele"></a>
</p><dl>
<dt><u>Function:</u> <b>polynome2ele</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1213"></a>
</dt>
<dd><p>gives the list <code><var>l</var> = [<var>n</var>, <var>e_1</var>, ..., <var>e_n</var>]</code>
where <var>n</var> is the degree of the polynomial <var>P</var> in the variable
<var>x</var> and <var>e_i</var> is the <var>i</var>-the elementary symmetric function
of the roots of <var>P</var>.
</p>
<pre class="example">(%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
(%o1)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
(%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                  7       5       3
(%o2)            x  - 14 x  + 56 x  - 56 x + 22
</pre>
<p>The inverse: <code>ele2polynome (<var>l</var>, <var>x</var>)</code>
</p>
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<p><a name="Item_003a-prodrac"></a>
</p><dl>
<dt><u>Function:</u> <b>prodrac</b><i> (<var>L</var>, <var>k</var>)</i>
<a name="IDX1214"></a>
</dt>
<dd><p><var>L</var> is a list containing the elementary symmetric functions 
on a set <var>A</var>. <code>prodrac</code> returns the polynomial whose roots
are the <var>k</var> by <var>k</var> products of the elements of <var>A</var>.
</p>
<p>Also see <code>somrac</code>.
</p>
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<p><a name="Item_003a-pui2polynome"></a>
</p><dl>
<dt><u>Function:</u> <b>pui2polynome</b><i> (<var>x</var>, <var>lpui</var>)</i>
<a name="IDX1215"></a>
</dt>
<dd><p>calculates the polynomial in <var>x</var> whose power functions of the roots
are given in the list <var>lpui</var>.
</p>
<pre class="example">(%i1) pui;
(%o1)                           1
(%i2) kill(labels);
(%o0)                         done
(%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x);
(%o1)                     [3, 4, 5, 1]
(%i2) ele2pui (3, %);
(%o2)                     [3, 4, 6, 7]
(%i3) pui2polynome (x, %);
                        3      2
(%o3)                  x  - 4 x  + 5 x - 1
</pre>
<p>See also:
<code>polynome2ele</code>, <code>ele2polynome</code>.
</p>
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<p><a name="Item_003a-somrac"></a>
</p><dl>
<dt><u>Function:</u> <b>somrac</b><i> (<var>L</var>, <var>k</var>)</i>
<a name="IDX1216"></a>
</dt>
<dd><p>The list <var>L</var> contains elementary symmetric functions of a polynomial
<var>P</var> . The function computes the polynomial whose roots are the 
<var>k</var> by <var>k</var> distinct sums of the roots of <var>P</var>. 
</p>
<p>Also see <code>prodrac</code>.
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<h3 class="subsection"> 32.2.6 Resolvents </h3>

<p><a name="Item_003a-resolvante"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante</b><i> (<var>P</var>, <var>x</var>, <var>f</var>, [<var>x_1</var>,..., <var>x_d</var>]) </i>
<a name="IDX1217"></a>
</dt>
<dd><p>calculates the resolvent of the polynomial <var>P</var> in <var>x</var> of degree
<var>n</var> &gt;= <var>d</var> by the function <var>f</var> expressed in the variables
<var>x_1</var>, ..., <var>x_d</var>.  For efficiency of computation it is
important to not include in the list <code>[<var>x_1</var>, ..., <var>x_d</var>]</code>
variables which do not appear in the transformation function <var>f</var>.
</p>
<p>To increase the efficiency of the computation one may set flags in
<code>resolvante</code> so as to use appropriate algorithms:
</p>
<p>If the function <var>f</var> is unitary:
</p><ul>
<li>
A polynomial in a single variable,
</li><li>
  linear,
</li><li>
  alternating,
</li><li>
  a sum,
</li><li>
  symmetric,
</li><li>
  a product,
</li><li>
the function of the Cayley resolvent (usable up to degree 5)

<pre class="example">(x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 -
     (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2
</pre>
<p>general,
</p></li></ul>
<p>the flag of <code>resolvante</code> may be, respectively:
</p><ul>
<li>
  unitaire,
</li><li>
  lineaire,
</li><li>
  alternee,
</li><li>
  somme,
</li><li>
  produit,
</li><li>
  cayley,
</li><li>
  generale.
</li></ul>

<pre class="example">(%i1) resolvante: unitaire$
(%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
      [x]);

&quot; resolvante unitaire &quot; [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
                         - 2772, 56448, - 33880, 

413952, - 352352, 3076668, - 3363360, 23114112, - 30494464, 

175230832, - 267412992, 1338886528, - 2292126760] 
  3       6      3       9      6      3
[x  - 1, x  - 2 x  + 1, x  - 3 x  + 3 x  - 1, 

 12      9      6      3       15      12       9       6      3
x   - 4 x  + 6 x  - 4 x  + 1, x   - 5 x   + 10 x  - 10 x  + 5 x

       18      15       12       9       6      3
 - 1, x   - 6 x   + 15 x   - 20 x  + 15 x  - 6 x  + 1, 

 21      18       15       12       9       6      3
x   - 7 x   + 21 x   - 35 x   + 35 x  - 21 x  + 7 x  - 1] 
[- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011] 
       7      6        5         4          3           2
(%o2) y  + 7 y  - 539 y  - 1841 y  + 51443 y  + 315133 y

                                              + 376999 y + 125253
(%i3) resolvante: lineaire$
(%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

&quot; resolvante lineaire &quot; 
       24       20         16            12             8
(%o4) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i5) resolvante: general$
(%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

&quot; resolvante generale &quot; 
       24       20         16            12             8
(%o6) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);

&quot; resolvante generale &quot; 
       24       20         16            12             8
(%o7) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
       24       20         16            12             8
(%o8) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i9) resolvante :lineaire$
(%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

&quot; resolvante lineaire &quot; 
                              4
(%o10)                       y  - 1
(%i11) resolvante: symetrique$
(%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

&quot; resolvante symetrique &quot; 
                              4
(%o12)                       y  - 1
(%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

&quot; resolvante symetrique &quot; 
                           6      2
(%o13)                    y  - 4 y  - 1
(%i14) resolvante: alternee$
(%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

&quot; resolvante alternee &quot; 
            12      8       6        4        2
(%o15)     y   + 8 y  + 26 y  - 112 y  + 216 y  + 229
(%i16) resolvante: produit$
(%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

&quot; resolvante produit &quot;
        35      33         29        28         27        26
(%o17) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

         24           23          22            21           20
 + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

          19           18             17              15
 - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

             14              12             11            10
 - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

              9            8            7             6
 - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

            5             3
 - 3720087 y  + 26040609 y  + 14348907
(%i18) resolvante: symetrique$
(%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

&quot; resolvante symetrique &quot; 
        35      33         29        28         27        26
(%o19) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

         24           23          22            21           20
 + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

          19           18             17              15
 - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

             14              12             11            10
 - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

              9            8            7             6
 - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

            5             3
 - 3720087 y  + 26040609 y  + 14348907
(%i20) resolvante: cayley$
(%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);

&quot; resolvante de Cayley &quot;
        6       5         4          3            2
(%o21) x  - 40 x  + 4080 x  - 92928 x  + 3772160 x  + 37880832 x

                                                       + 93392896
</pre>
<p>For the Cayley resolvent, the 2 last arguments are neutral and the input
polynomial must necessarily be of degree 5.
</p>
<p>See also:
</p>
<p><code>resolvante_bipartite</code>, <code>resolvante_produit_sym</code>,
<code>resolvante_unitaire</code>, <code>resolvante_alternee1</code>, <code>resolvante_klein</code>, 
<code>resolvante_klein3</code>, <code>resolvante_vierer</code>, <code>resolvante_diedrale</code>. 
</p>
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<p><a name="Item_003a-resolvante_005falternee1"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_alternee1</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1218"></a>
</dt>
<dd><p>calculates the transformation
<code><var>P</var>(<var>x</var>)</code> of degree <var>n</var> by the function
<em>product(x_i - x_j, 1 &lt;= i &lt; j &lt;= n - 1)</em>.
</p>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante</code> , <code>resolvante_klein</code>, <code>resolvante_klein3</code>,
<code>resolvante_vierer</code>, <code>resolvante_diedrale</code>, <code>resolvante_bipartite</code>.
</p>
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<p><a name="Item_003a-resolvante_005fbipartite"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_bipartite</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1219"></a>
</dt>
<dd><p>calculates the transformation of
<code><var>P</var>(<var>x</var>)</code> of even degree <var>n</var> by the function 
<em>x_1 x_2 ... x_[n/2] + x_[n/2 + 1] ... x_n</em>.
</p>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante</code> , <code>resolvante_klein</code>, <code>resolvante_klein3</code>,
<code>resolvante_vierer</code>, <code>resolvante_diedrale</code>, <code>resolvante_alternee1</code>.
</p>
<pre class="example">(%i1) resolvante_bipartite (x^6 + 108, x);
              10        8           6             4
(%o1)        y   - 972 y  + 314928 y  - 34012224 y
</pre>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante</code>, <code>resolvante_klein</code>, <code>resolvante_klein3</code>,
<code>resolvante_vierer</code>, <code>resolvante_diedrale</code>,
<code>resolvante_alternee1</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>



<p><a name="Item_003a-resolvante_005fdiedrale"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_diedrale</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1220"></a>
</dt>
<dd><p>calculates the transformation of <code><var>P</var>(<var>x</var>)</code> by the function
<code><var>x_1</var> <var>x_2</var> + <var>x_3</var> <var>x_4</var></code>.
</p>
<pre class="example">(%i1) resolvante_diedrale (x^5 - 3*x^4 + 1, x);
       15       12       11       10        9         8         7
(%o1) x   - 21 x   - 81 x   - 21 x   + 207 x  + 1134 x  + 2331 x

        6         5          4          3          2
 - 945 x  - 4970 x  - 18333 x  - 29079 x  - 20745 x  - 25326 x

 - 697
</pre>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante_alternee1</code>, <code>resolvante_klein</code>, <code>resolvante_klein3</code>,
<code>resolvante_vierer</code>, <code>resolvante</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>



<p><a name="Item_003a-resolvante_005fklein"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_klein</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1221"></a>
</dt>
<dd><p>calculates the transformation of <code><var>P</var>(<var>x</var>)</code> by the function
<code><var>x_1</var> <var>x_2</var> <var>x_4</var> + <var>x_4</var></code>.
</p>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante_alternee1</code>, <code>resolvante</code>, <code>resolvante_klein3</code>,
<code>resolvante_vierer</code>, <code>resolvante_diedrale</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-resolvante_005fklein3"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_klein3</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1222"></a>
</dt>
<dd><p>calculates the transformation of <code><var>P</var>(<var>x</var>)</code> by the function
<code><var>x_1</var> <var>x_2</var> <var>x_4</var> + <var>x_4</var></code>.
</p>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante_alternee1</code>, <code>resolvante_klein</code>, <code>resolvante</code>,
<code>resolvante_vierer</code>, <code>resolvante_diedrale</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>



<p><a name="Item_003a-resolvante_005fproduit_005fsym"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_produit_sym</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1223"></a>
</dt>
<dd><p>calculates the list of all product resolvents of the polynomial
<code><var>P</var>(<var>x</var>)</code>.
</p>
<pre class="example">(%i1) resolvante_produit_sym (x^5 + 3*x^4 + 2*x - 1, x);
        5      4             10      8       7       6       5
(%o1) [y  + 3 y  + 2 y - 1, y   - 2 y  - 21 y  - 31 y  - 14 y

    4       3      2       10      8       7    6       5       4
 - y  + 14 y  + 3 y  + 1, y   + 3 y  + 14 y  - y  - 14 y  - 31 y

       3      2       5      4
 - 21 y  - 2 y  + 1, y  - 2 y  - 3 y - 1, y - 1]
(%i2) resolvante: produit$
(%i3) resolvante (x^5 + 3*x^4 + 2*x - 1, x, a*b*c, [a, b, c]);

&quot; resolvante produit &quot;
       10      8       7    6        5       4       3     2
(%o3) y   + 3 y  + 14 y  - y  - 14 y  - 31 y  - 21 y  - 2 y  + 1
</pre>
<p>See also:
</p>
<p><code>resolvante</code>, <code>resolvante_unitaire</code>,
<code>resolvante_alternee1</code>, <code>resolvante_klein</code>,
<code>resolvante_klein3</code>, <code>resolvante_vierer</code>,
<code>resolvante_diedrale</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>



<p><a name="Item_003a-resolvante_005funitaire"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_unitaire</b><i> (<var>P</var>, <var>Q</var>, <var>x</var>)</i>
<a name="IDX1224"></a>
</dt>
<dd><p>computes the resolvent of the polynomial <code><var>P</var>(<var>x</var>)</code> by the
polynomial <code><var>Q</var>(<var>x</var>)</code>. 
</p>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante</code>,
<code>resolvante_alternee1</code>, <code>resolvante_klein</code>, <code>resolvante_klein3</code>,
<code>resolvante_vierer</code>, <code>resolvante_diedrale</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-resolvante_005fvierer"></a>
</p><dl>
<dt><u>Function:</u> <b>resolvante_vierer</b><i> (<var>P</var>, <var>x</var>)</i>
<a name="IDX1225"></a>
</dt>
<dd><p>computes the transformation of
<code><var>P</var>(<var>x</var>)</code> by the function <code><var>x_1</var> <var>x_2</var> -
<var>x_3</var> <var>x_4</var></code>.
</p>
<p>See also:
</p>
<p><code>resolvante_produit_sym</code>, <code>resolvante_unitaire</code>,
<code>resolvante_alternee1</code>, <code>resolvante_klein</code>, <code>resolvante_klein3</code>,
<code>resolvante</code>, <code>resolvante_diedrale</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>




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<h3 class="subsection"> 32.2.7 Miscellaneous </h3>


<p><a name="Item_003a-multinomial"></a>
</p><dl>
<dt><u>Function:</u> <b>multinomial</b><i> (<var>r</var>, <var>part</var>)</i>
<a name="IDX1226"></a>
</dt>
<dd><p>where <var>r</var> is the weight of the partition <var>part</var>.  This function
returns the associate multinomial coefficient: if the parts of
<var>part</var> are <var>i_1</var>, <var>i_2</var>, ..., <var>i_k</var>, the result is
<code><var>r</var>!/(<var>i_1</var>! <var>i_2</var>! ... <var>i_k</var>!)</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-permut"></a>
</p><dl>
<dt><u>Function:</u> <b>permut</b><i> (<var>L</var>)</i>
<a name="IDX1227"></a>
</dt>
<dd><p>returns the list of permutations of the list <var>L</var>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-sym">Package sym</a>
 &middot;
<a href="maxima_95.html#Category_003a-Lists">Lists</a>
</p>
</div>

</dd></dl>

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