http://openproblemgarden.org/op/a_sequence_defined_on_multisets Comments for "Sequence defined on multisets" en http://openproblemgarden.org/op/a_sequence_defined_on_multisets#comment-6995 <p>This problem has recently been solved by the author.</p> Mon, 18 Jul 2011 15:26:40 +0200 Anonymous comment 6995 at http://openproblemgarden.org http://openproblemgarden.org/op/a_sequence_defined_on_multisets <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/erickson_martin">Erickson</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Define a <img class="teximage" src="/files/tex/16b6ab80cd3957eacdb4bc181037a6d86005b580.png" alt="$ 2 \times n $" /> array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array <img class="teximage" src="/files/tex/73f8649de444361674c157a2fe98e0c5783f1c46.png" alt="$ [1; 1] $" />, the sequence is: <img class="teximage" src="/files/tex/73f8649de444361674c157a2fe98e0c5783f1c46.png" alt="$ [1; 1] $" /> -> <img class="teximage" src="/files/tex/83c3d9d7589f716ed6b0f05d26d36dabe8ba47aa.png" alt="$ [1; 2] $" /> -> <img class="teximage" src="/files/tex/a6a696aec4e84df6bc046cf6d30a4df80e156a14.png" alt="$ [1, 2; 1, 1] $" /> -> <img class="teximage" src="/files/tex/98b2f3e4134422c8a286f0326fc2f57ca9be2ab7.png" alt="$ [1, 2; 3, 1] $" /> -> <img class="teximage" src="/files/tex/19a9d24510f4551fa45b950aed32efed0636b355.png" alt="$ [1, 2, 3; 2, 1, 1] $" /> -> <img class="teximage" src="/files/tex/b00d573110ba8c2472820409103dd4cbd7bff7cc.png" alt="$ [1, 2, 3; 3, 2, 1] $" /> -> <img class="teximage" src="/files/tex/2aae90b65ae3f116402a1c0143127f80d86acdf0.png" alt="$ [1, 2, 3; 2, 2, 2] $" /> -> <img class="teximage" src="/files/tex/d614399c4078a70fbffb24eb99816a0974423175.png" alt="$ [1, 2, 3; 1, 4, 1] $" /> -> <img class="teximage" src="/files/tex/bf4dd82eaf95bd71a88cb78d3367efa2c7e4c942.png" alt="$ [1, 2, 3, 4; 3, 1, 1, 1] $" /> -> <img class="teximage" src="/files/tex/a4ef5900a9dde9b32311afe7dee2b143f42c405f.png" alt="$ [1, 2, 3, 4; 4, 1, 2, 1] $" /> -> <img class="teximage" src="/files/tex/ff367376ed7bb74fc6207273d7157062b83a1be6.png" alt="$ [1, 2, 3, 4; 3, 2, 1, 2] $" /> -> <img class="teximage" src="/files/tex/81ba2b3608ddba2fc95d32b74b70279f3f5adc5b.png" alt="$ [1, 2, 3, 4; 2, 3, 2, 1] $" />, and we now have a fixed point (loop of one array).</p> <p>The process always results in a loop of 1, 2, or 3 arrays. </div> </tr></td> </table> </td> </tr> </table> Erickson, Martin multiset sequence Combinatorics http://openproblemgarden.org/op/a_sequence_defined_on_multisets#comment Tue, 29 Jun 2010 16:25:12 +0200 Martin Erickson 37226 at http://openproblemgarden.org