{"id":1227,"date":"2026-05-13T11:15:35","date_gmt":"2026-05-13T09:15:35","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=1227"},"modified":"2026-05-09T19:23:26","modified_gmt":"2026-05-09T17:23:26","slug":"mad-particiones-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=1227","title":{"rendered":"MAD: Particiones con maxima"},"content":{"rendered":"<p>Recordemos que los n\u00fameros de Bell cumplen una relaci\u00f3n de recurrencia muy interesante:\\[B_n=\\sum_{k=0}^{n-1}B_k\\binom{n-1}{k}\\]<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Utilizando la f\u00f3rmula anterior, construir un algoritmo en maxima que nos calcule cualquier n\u00famero de Bell.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv124e() {\n  var htmlShow124e = document.getElementById(\"html-show124e\");\n  if (htmlShow124e.style.display === \"none\") {\n    htmlShow124e.style.display = \"block\";\n  } else {\n    htmlShow124e.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv124ex()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show124e\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Primero defiamos la combinaciones:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">comb<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">if<\/span><span class=\"code_operator\"> (<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> or <\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> then<\/span><span class=\"code_number\"> 1 <\/span><span class=\"code_function\">else<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0 <span class=\"code_function\">comb<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\"> 1 <\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_function\">comb<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ya estamos en condiciones de definir los n\u00fameros de Bell:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">Bell<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">B<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">B<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if <\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">&lt;<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> then<\/span><span class=\"code_number\"> 1 <\/span><span class=\"code_function\">else<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if<\/span><span class=\"code_variable\"> n<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">2<\/span><span class=\"code_function\"> then <\/span><span class=\"code_number\">2<\/span><span class=\"code_function\"> else <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">for <\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">3 <\/span><span class=\"code_function\">thru<\/span><span class=\"code_variable\"> n <\/span><span class=\"code_function\">do<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">B<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">comb<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">B<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">B<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">B<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">B<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">B<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"> <span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">Bell<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ 1\\operatorname{,}1\\operatorname{,}2\\operatorname{,}5\\operatorname{,}15\\operatorname{,}52\\operatorname{,}203\\operatorname{,}877\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<p>Sin embargo, resulta m\u00e1s sencillo calcularlos con el llamado tri\u00e1ngulo de Bell.<\/p>\n<ol>\n<li>Comience con el n\u00famero uno. Pon esto en una fila por s\u00ed mismo. \\({\\displaystyle x_{0,1}=1}\\) <\/li>\n<li>Comience una nueva fila con el elemento m\u00e1s a la derecha de la fila anterior como el n\u00famero m\u00e1s a la izquierda \\(({\\displaystyle x_{i,1}\\leftarrow x_{i-1,r}}\\) donde \\(r\\) es el \u00faltimo elemento de (i-1)-fila).<\/li>\n<li>Determine los n\u00fameros que no est\u00e1n en la columna izquierda tomando la suma del n\u00famero a la izquierda y el n\u00famero sobre el n\u00famero a la izquierda, es decir, el n\u00famero diagonalmente arriba y a la izquierda del n\u00famero que estamos calculando(\\({\\displaystyle (x_{i,j}\\leftarrow x_{i,j-1}+x_{i-1,j-1}}\\))<\/li>\n<li>Repita el paso tres hasta que haya una nueva fila con un n\u00famero m\u00e1s que la fila anterior (haga el paso 3 hasta \\({\\displaystyle j=r+1}\\))<\/li>\n<li>El n\u00famero en el lado izquierdo de una fila dada es el n\u00famero de Bell para esa fila. \\({\\displaystyle B_{i}\\leftarrow x_{i,1}}\\)<\/li>\n<\/ol>\n<p>Estas son las primeras cinco filas del tri\u00e1ngulo construido por estas reglas:<\/p>\n<p>\\[\\begin{array}{rrrrr} 1&#038;&#038;&#038;&#038; \\\\  1&#038;2&#038;&#038;&#038; \\\\  2&#038;3&#038;5&#038;&#038; \\\\  5&#038;7&#038;10&#038;15&#038; \\\\  15&#038;20&#038;27&#038;37&#038;52 \\end{array}\\]<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Utilizando el tri\u00e1ngulo de Bell, construir un algoritmo con m\u00e1xima que nos devuelva \\(B_n\\).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv124e8() {\n  var htmlShow124e8 = document.getElementById(\"html-show124e8\");\n  if (htmlShow124e8.style.display === \"none\") {\n    htmlShow124e8.style.display = \"block\";\n  } else {\n    htmlShow124e8.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv124e8()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show124e8x\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">numero_bell<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_variable\">T<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Caso especial para n=0 *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if <\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> then <\/span><span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Creamos una lista de listas pre-rellenada con ceros *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* T[fila][columna] *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">T<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Paso 1: Primer elemento *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Bucle para construir las filas *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">for <\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_function\"> thru< \/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_comment\">\/* Paso 2: El primero de la fila es el \u00faltimo de la anterior *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_comment\">\/* En la fila (i-1), el \u00faltimo elemento est\u00e1 en la posici\u00f3n (i-1) *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_comment\">\/* Pasos 3 y 4: Rellenar el resto de la fila *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">for <\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_function\"> thru <\/span><span class=\"code_variable\">i<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Paso 5: El n\u00famero de Bell B_n es el primer elemento de la fila n+1 *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr \/>\n<p>Sabemos que los n\u00fameros de Stirling de segunda especie siguen la relaci\u00f3n de recurrencia<\/p>\n<p>\\[ \\left\\{{\\begin{matrix}n\\\\k\\end{matrix}}\\right\\}=\\left\\{{\\begin{matrix}n-1\\\\k-1\\end{matrix}}\\right\\}+k\\left\\{{\\begin{matrix}n-1\\\\k\\end{matrix}}\\right\\}\\]<\/p>\n<p>con \\[\\left\\{{\\begin{matrix}0\\\\0\\end{matrix}}\\right\\}=1,\\quad  \\left\\{{\\begin{matrix}n\\\\0\\end{matrix}}\\right\\}=0,\\quad  \\left\\{{\\begin{matrix}n\\\\1\\end{matrix}}\\right\\}=\\left\\{{\\begin{matrix}n\\\\n\\end{matrix}}\\right\\}=1,\\quad {\\mbox{ y }}\\quad \\left\\{{\\begin{matrix}n\\\\k\\end{matrix}}\\right\\}=0,\\ \\forall k&gt;n.\\]<\/p>\n<table style=\"border-collapse: collapse; margin: 20px auto; font-family: Arial, sans-serif;\">\n<thead>\n<tr style=\"background-color: #f0f0f0; font-weight: bold;\">\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">n \\ k<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">0<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">2<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">3<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">4<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">5<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">0<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">0<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">2<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">0<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">3<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">0<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">3<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">4<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">0<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">7<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">6<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">5<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">0<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">15<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">25<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">10<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<blockquote><p><strong>Ejercicio:<\/strong> Utilizando la recurrencia anterior, construir un algoritmo con m\u00e1xima que nos devuelva \\[ \\left\\{{\\begin{matrix}n\\\\k\\end{matrix}}\\right\\}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4r() {\n  var htmlShow4r = document.getElementById(\"html-show4r\");\n  if (htmlShow4r.style.display === \"none\") {\n    htmlShow4r.style.display = \"block\";\n  } else {\n    htmlShow4r.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4r()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4rx\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">stirling_2<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Condici\u00f3n: si k &gt; n, el resultado es 0 *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if <\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">&gt;<\/span><span class=\"code_variable\">n<\/span><span class=\"code_function\"> then <\/span><span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Condici\u00f3n: si n = 0 y k = 0, el resultado es 1 *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if <\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> and <\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> then <\/span><span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Condici\u00f3n: si n &gt; 0 y k = 0, el resultado es 0 *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if <\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">&gt;<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> and <\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> then <\/span><span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Casos base adicionales: n=k o k=1 *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">if <\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\"> or <\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">n<\/span><span class=\"code_function\"> then <\/span><span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Relaci\u00f3n de recurrencia *\/<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">stirling2<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">stirling2<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr \/>\n<p>El pasado d\u00eda tratamos los \u00e1rboles, entre ellos los \u00e1rboles enraizados. Por ejemplo, en la imagen vemos todos los posibles \u00e1rboles enraizados ordenados que tienen cuatro aristas y dos hojas.<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Unlabeled_ordered_rooted_trees_of_4_edges_and_2_leaves.svg#\/media\/File:Unlabeled_ordered_rooted_trees_of_4_edges_and_2_leaves.svg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/3\/33\/Unlabeled_ordered_rooted_trees_of_4_edges_and_2_leaves.svg\" alt=\"Unlabeled ordered rooted trees of 4 edges and 2 leaves.svg\" height=\"372\" width=\"277\"><\/a><br \/>By <a href=\"\/\/commons.wikimedia.org\/w\/index.php?title=User:%C5%98i%C5%A5opi%C4%8D&amp;action=edit&amp;redlink=1\" class=\"new\" title=\"User:\u0158i\u0165opi\u010d (page does not exist)\">\u0158i\u0165opi\u010d<\/a> &#8211; <span class=\"int-own-work\" lang=\"en\">Own work<\/span>, <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\" title=\"Creative Commons Attribution-Share Alike 4.0\">CC BY-SA 4.0<\/a>, <a href=\"https:\/\/commons.wikimedia.org\/w\/index.php?curid=92284453\">Link<\/a><\/p>\n<p>Los \u00e1rboles enraizados ordenados imponen un orden lineal (izquierda a derecha) en los hijos de cada nodo, haciendo que las permutaciones de hermanos generen \u00e1rboles distintos. Un \u00e1rbol enraizado ordenado parte de un \u00e1rbol enraizado est\u00e1ndar, pero los sub\u00e1rboles de cada nodo se ordenan secuencialmente, como \u00abprimer hijo\u00bb, \u00absegundo hijo\u00bb, etc. Esto permite referir posiciones espec\u00edficas, similar a un \u00e1rbol binario donde izquierda < ra\u00edz < derecha.\n\n\n\n<blockquote><strong>Proposici\u00f3n:<\/strong> Sea \\({\\displaystyle \\mathbf{N} (n,k),n\\in \\mathbb {N} ^{+},1\\leq k\\leq n}\\) el n\u00famero de \u00e1rboles enraizados ordenados con \\(n\\) lados y \\(k\\) hojas, entonces<br \/>\n\\[{\\displaystyle \\mathbf{N} (n,k)={\\frac {1}{n}}{n \\choose k}{n \\choose k-1}}\\]\n<\/p><\/blockquote>\n<p>La siguiente tabla est\u00e1 construida con la f\u00f3rmula anterior.<\/p>\n<table style=\"border-collapse: collapse; margin: 20px auto; font-family: Arial, sans-serif;\">\n<thead>\n<tr style=\"background-color: #f0f0f0; font-weight: bold;\">\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">n \\ k<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">2<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">3<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">4<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">5<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">6<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">7<\/th>\n<th style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">8<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">2<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">3<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">3<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">4<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">6<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">6<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">5<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">10<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">20<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">10<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">6<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">15<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">50<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">50<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">15<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">7<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">21<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">105<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">175<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">105<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">21<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center; font-weight: bold; background-color: #f9f9f9;\">8<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">28<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">196<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">490<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">490<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">196<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">28<\/td>\n<td style=\"border: 1px solid #ddd; padding: 10px; text-align: center;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<blockquote><p><strong>Proposici\u00f3n:<\/strong> Si consideramos $\\mathbf{N}(n, 1) = 1$ y $\\mathbf{N}(n, n) = 1$, entonces<br \/>\n $$ \\mathbf{N}(n, k) = \\mathbf{N}(n, k-1) \\cdot \\frac{(n-k+1)(n-k+2)}{k(k-1)} $$\n<\/p><\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Utilizando el resultado anterior, construir un algoritmo con m\u00e1xima que nos devuelva \\(\\mathbf{N}(n,k)\\).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv34() {\n  var htmlShow34 = document.getElementById(\"html-show34\");\n  if (htmlShow34.style.display === \"none\") {\n    htmlShow34.style.display = \"block\";\n  } else {\n    htmlShow34.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv34()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show34x\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">N<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">if <\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\"> or <\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">n<\/span><span class=\"code_function\"> then <\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_number\">1<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\"> else <\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">N<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr \/>\n","protected":false},"excerpt":{"rendered":"<p>Recordemos que los n\u00fameros de Bell cumplen una relaci\u00f3n de recurrencia muy interesante:\\[B_n=\\sum_{k=0}^{n-1}B_k\\binom{n-1}{k}\\] Ejercicio: Utilizando la f\u00f3rmula anterior, construir un algoritmo en maxima que nos calcule cualquier n\u00famero de Bell. Soluci\u00f3n: Primero&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[9],"class_list":["post-1227","post","type-post","status-publish","format-standard","hentry","category-matematica-discreta","tag-practicas-mad"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/1227","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1227"}],"version-history":[{"count":25,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/1227\/revisions"}],"predecessor-version":[{"id":1229,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/1227\/revisions\/1229"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}