Wedgie/Archived version: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 03:40:22 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 03:44:07 UTC</tt>.<br>

: The original revision id was <tt>~~141665907~~</tt>.<br>

: The original revision id was <tt>141666273</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

Line 16:

f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))

where the sum is taken over S(n,m), the set of all [[http://en.wikipedia.org/wiki/Permutation|permutations]] of the first n+m integers which are an [[http://en.wikipedia.org/wiki/%28p,q%29_shuffle|(n,m) shuffles]], and sgn(t) is the [[http://en.wikipedia.org/wiki/Parity_of_a_permutation|parity of the permutation]] t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.

where the sum is taken over S(n,m), the set of all [[http://en.wikipedia.org/wiki/Permutation|permutations]] of the first n+m integers which are an [[http://en.wikipedia.org/wiki/%28p,q%29_shuffle|(n,m) shuffles]], and sgn(t) is the [[http://en.wikipedia.org/wiki/Parity_of_a_permutation|parity of the permutation]] t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.

If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).

Line 32:

f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))<br />

<br />

where the sum is taken over S(n,m), the set of all <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation" rel="nofollow">permutations</a> of the first n+m integers which are an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/%28p,q%29_shuffle" rel="nofollow">(n,m) shuffles</a>, and sgn(t) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Parity_of_a_permutation" rel="nofollow">parity of the permutation</a> t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.<br />

where the sum is taken over S(n,m), the set of all <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation" rel="nofollow">permutations</a> of the first n+m integers which are an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/%28p,q%29_shuffle" rel="nofollow">(n,m) shuffles</a>, and sgn(t) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Parity_of_a_permutation" rel="nofollow">parity of the permutation</a> t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.<br />

<br />

If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 03:40:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 03:44:07 UTC</tt>.<br>
-: The original revision id was <tt>141665907</tt>.<br>
+: The original revision id was <tt>141666273</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 16: / Line 16: @@
 f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))
-where the sum is taken over S(n,m), the set of all [[http://en.wikipedia.org/wiki/Permutation|permutations]] of the first n+m integers which are an [[http://en.wikipedia.org/wiki/%28p,q%29_shuffle|(n,m) shuffles]], and sgn(t) is the [[http://en.wikipedia.org/wiki/Parity_of_a_permutation|parity of the permutation]] t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.
+where the sum is taken over S(n,m), the set of all [[http://en.wikipedia.org/wiki/Permutation|permutations]] of the first n+m integers which are an [[http://en.wikipedia.org/wiki/%28p,q%29_shuffle|(n,m) shuffles]], and sgn(t) is the [[http://en.wikipedia.org/wiki/Parity_of_a_permutation|parity of the permutation]] t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.
 If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).
@@ Line 32: / Line 32: @@
 f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))&lt;br /&gt;
 &lt;br /&gt;
-where the sum is taken over S(n,m), the set of all &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation" rel="nofollow"&gt;permutations&lt;/a&gt; of the first n+m integers which are an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/%28p,q%29_shuffle" rel="nofollow"&gt;(n,m) shuffles&lt;/a&gt;, and sgn(t) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Parity_of_a_permutation" rel="nofollow"&gt;parity of the permutation&lt;/a&gt; t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.&lt;br /&gt;
+where the sum is taken over S(n,m), the set of all &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation" rel="nofollow"&gt;permutations&lt;/a&gt; of the first n+m integers which are an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/%28p,q%29_shuffle" rel="nofollow"&gt;(n,m) shuffles&lt;/a&gt;, and sgn(t) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Parity_of_a_permutation" rel="nofollow"&gt;parity of the permutation&lt;/a&gt; t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.&lt;br /&gt;
 &lt;br /&gt;
 If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).&lt;/body&gt;&lt;/html&gt;</pre></div>