{"id":5118,"date":"2018-05-24T10:49:33","date_gmt":"2018-05-24T08:49:33","guid":{"rendered":"http:\/\/pimedios.es\/?p=5118"},"modified":"2018-05-24T10:49:33","modified_gmt":"2018-05-24T08:49:33","slug":"probleme-des-menages","status":"publish","type":"post","link":"https:\/\/pimedios.jesussoto.es\/?p=5118","title":{"rendered":"Probl\u00e8me des m\u00e9nages"},"content":{"rendered":"<p>Conservando su t\u00edtulo original propuesto por el matem\u00e1tico franc\u00e9s Fran\u00e7ois \u00c9douard Anatole Lucas (1842-1891), en <em>Th\u00e9orie des nombres<\/em>, Gauthier-Villars, Paris, 1891, el problema pretende contar el n\u00famero de formas en las que puedes sentar a una mesa redonda $n$ parejas de comensales de forma que no coincida junta una pareja.<\/p>\n<p>Por ejemplo, consideremos 3 parejas, $A_1A_2$, $B_1B_2$ y $C_1C_2$ y las sentamos en una mesa circular de esta forma:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-5119\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2018\/05\/Menageproblem.png\" alt=\"\" width=\"236\" height=\"216\" \/><\/p>\n<p>pretendiendo no sentar a nadie al lado de su pareja. La pregunta ser\u00eda: \u00bfde cu\u00e1ntas formas podemos sentarlas sin que coincidan\u00a0\u00a0$A_1A_2$ o $B_1B_2$ o $C_1C_2$ al lado?<\/p>\n<p>Un problema equivalente hab\u00eda sido formulado, independientemente, por el matem\u00e1tico escoc\u00e9s Peter Guthrie Tait, que estaba estudiando la teor\u00eda de nudos, en <em>On knots, i, ii, iii<\/em>, in Scientific Papers, pages 273-347. Cambridge Univ. Press, Cambridge, 1898.<\/p>\n<p>La primera f\u00f3rmula expl\u00edcita que daba el n\u00famero fue publicada por el franc\u00e9s Jacques Touchard en 1934,<br \/>\n$$M_{n}=2\\cdot n!\\sum _{{k=0}}^{n}(-1)^{k}{\\frac {2n}{2n-k}}{2n-k \\choose k}(n-k)!.$$<br \/>\nAunque no dio su prueba. Tendr\u00edamos que esperar a 1943 para que Kaplansky (<em>Solution of the probl\u00e8me des m\u00e9nages<\/em>. Bull. Amer. Math. Soc., 49:784-785) diese una demostraci\u00f3n.<\/p>\n<p>A partir de aqu\u00ed se han obtenido otras f\u00f3rmulas, como la de Wyman &amp; Moser de 1958(<em>On the probl\u00e8me des m\u00e9nages<\/em>, Canadian Journal of Mathematics, 10 (3): 468\u2013480, doi:10.4153\/cjm-1958-045-6, MR 0095127).<\/p>\n<p>En <a href=\"https:\/\/www.math.dartmouth.edu\/~doyle\/docs\/menage\/menage\/menage.html\">Non-sexist solution of the m\u00e9nage problem<\/a>, pod\u00e9is encontrar la de Kenneth P. Bogart and Peter G. Doyle, que lo resuelve utilizando el principio de inclusi\u00f3n-exclusi\u00f3n.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Historia del Probl\u00e8me des m\u00e9nages<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-5118","post","type-post","status-publish","format-standard","hentry","category-historia","entry"],"_links":{"self":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/5118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5118"}],"version-history":[{"count":1,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/5118\/revisions"}],"predecessor-version":[{"id":5120,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/5118\/revisions\/5120"}],"wp:attachment":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}