{"id":163,"date":"2011-12-02T22:07:08","date_gmt":"2011-12-02T13:07:08","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=163"},"modified":"2021-03-16T16:12:54","modified_gmt":"2021-03-16T07:12:54","slug":"tkr200901","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200901\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2009\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u81ea\u7136\u6570 \\(m \\geqq 2\\) \u306b\u5bfe\u3057, \\(m-1\\) \u500b\u306e\u4e8c\u9805\u4fc2\u6570\r\n\\[\r\n{} _ {m} \\text{C} {} _ {1} , {} _ {m} \\text{C} {} _ {2} , \\cdots , {} _ {m} \\text{C} {} _ {m-1}\r\n\\]\r\n\u3092\u8003\u3048, \u3053\u308c\u3089\u306e\u3059\u3079\u3066\u306e\u6700\u5927\u516c\u7d04\u6570\u3092 \\(d _ m\\) \u3068\u3059\u308b. \u3059\u306a\u308f\u3061 \\(d _ m\\) \u306f\u3053\u308c\u3089\u3059\u3079\u3066\u3092\u5272\u308a\u5207\u308b\u6700\u5927\u306e\u81ea\u7136\u6570\u3067\u3042\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(m\\) \u304c\u7d20\u6570\u306a\u3089\u3070, \\(d _ m=m\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(k\\) \u306b\u5bfe\u3057, \\(k^m-k\\) \u304c \\(d _ m\\) \u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092, \\(k\\) \u306b\u95a2\u3059\u308b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u3063\u3066\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(m\\) \u304c\u5076\u6570\u306e\u3068\u304d \\(d _ m\\) \u306f \\(1\\) \u307e\u305f\u306f \\(2\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\({} _ m \\text{C} {} _ 1 =m\\) \u306a\u306e\u3067, \\(d _ m =1 , m\\) \u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b.\r\n\\[\r\n{} _ m \\text{C} {} _ k = \\dfrac{m!}{(m-k)! k!}\r\n\\]\r\n\u306f\u81ea\u7136\u6570\u3067\u3042\u308a, \\(1 \\leqq k \\leqq m-1\\) \u3060\u304b\u3089, \u53f3\u8fba\u306e\u5206\u6bcd\u306f \\(m\\) \u306e\u500d\u6570\u3067\u306f\u306a\u3044\u306e\u3067, \\({} _ m \\text{C} {} _ k\\) \u306f \\(m\\) \u306e\u500d\u6570\u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066\r\n\\[\r\nd _ m = m\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \u300c \\(k^m-k\\) \u304c \\(d _ m\\) \u3067\u5272\u308a\u5207\u308c\u308b\u300d... [\uff0a]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(k=1\\) \u306e\u3068\u304d
      \r\n\\(1^m-1 =0\\) \u306a\u306e\u3067, [\uff0a] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(k= \\ell\\) \u306e\u3068\u304d, [\uff0a] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n(\\ell +1)^m -(\\ell +1) & = {\\ell}^m +1 +\\textstyle\\sum\\limits _ {i=1}^{m-1} {} _ m \\text{C} {} _ i {\\ell}^i -(\\ell +1) \\\\\r\n& = {\\ell}^m-\\ell +\\textstyle\\sum\\limits _ {i=1}^{m-1} {} _ m \\text{C} {} _ i {\\ell}^i\r\n\\end{align}\\]\r\n\u4eee\u5b9a\u3088\u308a, \u3053\u308c\u3082 \\(d _ m\\) \u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067, \\(k=\\ell +1\\) \u306e\u3068\u304d\u3082[\uff0a]\u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong> \u3088\u308a, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3088\u308a\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(k\\) \u306b\u3064\u3044\u3066, [\uff0a]\u304c\u6210\u7acb\u3059\u308b.<\/p>\r\n

      (3)<\/strong><\/p>\r\n

      \\(k =d _ m-1\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nk^m-k & = (d _ m-1)^m -(d _ m-1) \\\\\r\n& ={d _ m}^m +1 +\\textstyle\\sum\\limits _ {i=1}^{m-1} {} _ m \\text{C} {} _ i (-d _ m)^i -d _ m+1 \\\\\r\n& = \\left\\{ {d _ m}^m -d _ m +\\textstyle\\sum\\limits _ {i=1}^{m-1} {} _ m \\text{C} {} _ i (-d _ m)^i \\right\\} +2\r\n\\end{align}\\]\r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u3053\u308c\u306f \\(d _ m\\) \u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3060\u304b\u3089, \\(2\\) \u306f \\(d _ m\\) \u3067\u5272\u308a\u5207\u308c\u308b.
      \r\n\u3088\u3063\u3066\r\n\\[\r\nd _ m= 1, 2\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u81ea\u7136\u6570 \\(m \\geqq 2\\) \u306b\u5bfe\u3057, \\(m-1\\) \u500b\u306e\u4e8c\u9805\u4fc2\u6570 \\[ {} _ {m} \\text{C} {} _ {1} , {} _ {m} \\text{C} {} _ {2} , \\cdots , {} … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[18],"tags":[139,15],"class_list":["post-163","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2009","tag-tokyo_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/163","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=163"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/163\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}