{"id":3883,"date":"2018-03-20T06:00:01","date_gmt":"2018-03-20T04:00:01","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3883"},"modified":"2018-03-28T09:08:51","modified_gmt":"2018-03-28T07:08:51","slug":"particiones-primas","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/particiones-primas\/","title":{"rendered":"Particiones primas"},"content":{"rendered":"<p>Una <a href=\"http:\/\/bit.ly\/2tZYRAX\">partici\u00f3n prima<\/a> de un n\u00famero natural n es un conjunto de primos cuya suma es n. Por ejemplo, el n\u00famero 7 tiene 7 particiones primas ya que<\/p>\n<pre lang=\"text\">\n   7 = 7 = 5 + 2 = 3 + 2 + 2\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   particiones :: Int -> [[Int]]\n<\/pre>\n<p>tal que (particiones n) es el comjunto de las particiones primas de n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   particiones 7             ==  [[7],[5,2],[3,2,2]]\n   particiones 8             ==  [[5,3],[3,3,2],[2,2,2,2]]\n   particiones 9             ==  [[7,2],[5,2,2],[3,3,3],[3,2,2,2]]\n   length (particiones 100)  ==  40899\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes (primes)\nimport Data.Array          (Array, (!), array)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nparticiones1 :: Int -> [[Int]]\nparticiones1 0 = [[]]\nparticiones1 n = [x:y | x <- xs, \n                        y <- particiones1 (n-x), \n                        [x] >= take 1 y]\n  where xs = reverse (takeWhile (<= n) primes)\n\n-- 2\u00aa soluci\u00f3n (con programaci\u00f3n din\u00e1mica)\n-- =======================================\n\nparticiones2 :: Int -> [[Int]]\nparticiones2 n = (vectorParticiones n) ! n\n\n-- (vectorParticiones n) es el vector con \u00edndices de 0 a n tal que el\n-- valor del \u00edndice k es la lista de las particiones primas de k. Por\n-- ejemplo, \n--    \u03bb> mapM_ print (elems (vectorParticiones 9))\n--    [[]]\n--    []\n--    [[2]]\n--    [[3]]\n--    [[2,2]]\n--    [[5],[3,2]]\n--    [[3,3],[2,2,2]]\n--    [[7],[5,2],[3,2,2]]\n--    [[5,3],[3,3,2],[2,2,2,2]]\n--    [[7,2],[5,2,2],[3,3,3],[3,2,2,2]]\n--    \u03bb> elems (vectorParticiones 9) == map particiones1 [0..9]\n--    True\nvectorParticiones :: Int -> Array Int [[Int]]\nvectorParticiones n = v where\n  v = array (0,n) [(i,f i) | i <- [0..n]]\n    where f 0 = [[]]\n          f m = [x:y | x <- xs, \n                       y <- v ! (m-x), \n                       [x] >= take 1 y]\n            where xs = reverse (takeWhile (<= m) primes)\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> length (particiones1 35)\n--    175\n--    (5.88 secs, 2,264,266,040 bytes)\n--    \u03bb> length (particiones2 35)\n--    175\n--    (0.02 secs, 1,521,560 bytes)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Una partici\u00f3n prima de un n\u00famero natural n es un conjunto de primos cuya suma es n. Por ejemplo, el n\u00famero 7 tiene 7 particiones primas ya que 7 = 7 = 5 + 2 = 3 + 2 + 2 Definir la funci\u00f3n particiones :: Int -> [[Int]] tal que (particiones n) es el&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[4],"tags":[250,8,286,42,11,173,6,32,47,34],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3883"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=3883"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3883\/revisions"}],"predecessor-version":[{"id":3914,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3883\/revisions\/3914"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3883"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3883"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3883"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}