{"id":536,"date":"2012-12-27T01:52:15","date_gmt":"2012-12-26T16:52:15","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=536"},"modified":"2021-10-31T07:03:30","modified_gmt":"2021-10-30T22:03:30","slug":"tkb201103","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkb201103\/","title":{"rendered":"\u6771\u5927\u6587\u7cfb2011\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(p , q\\) \u3092 \\(2\\) \u3064\u306e\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u6574\u6570 \\(a , b , c\\) \u3067\u6761\u4ef6\r\n\\[\r\n-q \\leqq b \\leqq 0 \\leqq a \\leqq p , \\quad b \\leqq c \\leqq a\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u3082\u306e\u3092\u8003\u3048, \u3053\u306e\u3088\u3046\u306a \\(a , b , c\\) \u3092 \\([ a , b ; c ]\\) \u306e\u5f62\u306b\u4e26\u3079\u305f\u3082\u306e\u3092 \\(( p , q )\\) \u30d1\u30bf\u30fc\u30f3\u3068\u547c\u3076. \u5404 \\(( p , q )\\) \u30d1\u30bf\u30fc\u30f3 \\([ a , b ; c ]\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nw \\left( [ a , b ; c ] \\right) = p -q -( a+b )\r\n\\]\r\n\u3068\u304a\u304f.<\/p>\r\n

    \r\n
  1. (1)<\/strong>\u3000\\(( p , q )\\) \u30d1\u30bf\u30fc\u30f3\u306e\u3046\u3061, \\(w \\left( [ a , b ; c ] \\right) = -q\\) \u3068\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u3092\u6c42\u3081\u3088. \u307e\u305f, \\(w \\left( [ a , b ; c ] \\right) = p\\) \u3068\u306a\u308b \\(( p , q )\\) \u30d1\u30bf\u30fc\u30f3\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0b \\(p=q\\) \u306e\u5834\u5408\u3092\u8003\u3048\u308b.<\/p>\r\n

      \r\n
    1. (2)<\/strong>\u3000\\(s\\) \u3092\u6574\u6570\u3068\u3059\u308b. \\(( p , p )\\) \u30d1\u30bf\u30fc\u30f3\u3067 \\(w \\left( [ a , b ; c ] \\right) = -p+s\\) \u3068\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/li>\r\n<\/ol>\r\n
      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

      \\(w \\left( [ a , b ; c ] \\right) = p -q -( a+b ) = -q\\) \u306e\u3068\u304d\r\n\\[\r\na+b = p\r\n\\]\r\n\\(b \\leqq 0 \\leqq a \\leqq p\\) \u306a\u306e\u3067\r\n\\[\r\n( a , b ) = ( p , 0 )\r\n\\]\r\n\\(b \\leqq c \\leqq a\\) \u3059\u306a\u308f\u3061 \\(0 \\leqq c \\leqq p\\) \u306a\u306e\u3067, \\(c = 0 , 1 , \\cdots p\\) .
      \r\n\u3088\u3063\u3066, \\(( p , q )\\) \u30d1\u30bf\u30fc\u30f3\u306f \\(\\underline{p+1}\\) \u500b.
      \r\n\u307e\u305f, \\(w \\left( [ a , b ; c ] \\right) = p -q -( a+b ) = p\\) \u306e\u3068\u304d\r\n\\[\r\na+b = -q\r\n\\]\r\n\\(-q \\leqq b \\leqq 0 \\leqq a\\) \u306a\u306e\u3067\r\n\\[\r\n( a , b ) = ( 0 , -q )\r\n\\]\r\n\\(b \\leqq c \\leqq a\\) \u3059\u306a\u308f\u3061 \\(-q \\leqq c \\leqq 0\\) \u306a\u306e\u3067, \\(c = -q , -q+1 , \\cdots 0\\) .
      \r\n\u3088\u3063\u3066, \\(( p , q )\\) \u30d1\u30bf\u30fc\u30f3\u306f \\(\\underline{q+1}\\) \u500b.<\/p>\r\n

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

      \\(p=q\\) \u304b\u3064 \\(w \\left( [ a , b ; c ] \\right) = -p+s\\) \u306e\u3068\u304d\r\n\\[\r\na+b = -p+s\r\n\\]\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(0 \\leqq s \\leqq p\\) \u306e\u3068\u304d
        \r\n\\(-p \\leqq b \\leqq 0 \\leqq a \\leqq p\\) \u306a\u306e\u3067\r\n\\[\r\n( a , b ) = ( k , -p+s-k ) \\quad ( k = 0 , 1 , \\cdots , s )\r\n\\]\r\n\\(b \\leqq c \\leqq a\\) \u3059\u306a\u308f\u3061 \\(-p+s-k \\leqq c \\leqq k\\) \u306a\u306e\u3067, \\(c = -p+s-k , -p+s+k-1 , \\cdots , k\\) \u3060\u304b\u3089, \\(c\\) \u306e\u5024\u306f\r\n\\[\r\nk -( -p+s-k ) +1 = 2k+p-s+1 \\ \\text{\u901a\u308a}\r\n\\]\r\n\u3086\u3048\u306b, \\(( p , p )\\) \u30d1\u30bf\u30fc\u30f3\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=0}^s ( 2k+p-s+1 ) & = s( s+1 ) +( s+1 )( p-s+1 ) \\\\\r\n& = ( p+1 )( s+1 ) \\ \\text{\u500b}\r\n\\end{align}\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(p+1 \\leqq s \\leqq 2p\\) \u306e\u3068\u304d
        \r\n\\(-p \\leqq b \\leqq 0 \\leqq a \\leqq p\\) \u306a\u306e\u3067\r\n\\[\r\n( a , b ) = ( s-p+k , -k ) \\quad ( k = 0 , 1 , \\cdots , 2p-s )\r\n\\]\r\n\\(b \\leqq c \\leqq a\\) \u3059\u306a\u308f\u3061 \\(-k \\leqq c \\leqq -p+s+k\\) \u306a\u306e\u3067 \\(c = -k , -k+1 , \\cdots , -p+s-k\\) \u3060\u304b\u3089, \\(c\\) \u306e\u5024\u306f\r\n\\[\r\n( -p+s+k ) -( -k ) +1 = 2k-p+s+1 \\ \\text{\u901a\u308a}\r\n\\]\r\n\u3086\u3048\u306b, \\(( p , p )\\) \u30d1\u30bf\u30fc\u30f3\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=0}^{2p-s} & ( 2k-p+s+1 ) \\\\\r\n& = ( 2p-s )( 2p-s+1 ) +( 2p-s+1 )( s-p+1 ) \\\\\r\n& = ( p+1 )( 2p-s+1 ) \\quad \\text{\u500b}\r\n\\end{align}\\]<\/li>\r\n

      3. 3*<\/strong>\u3000\\(s \\lt 0 , 2p \\lt s\\) \u306e\u3068\u304d
        \r\n\\(( p , p )\\) \u30d1\u30bf\u30fc\u30f3\u306f \\(0\\) \u500b.<\/p><\/li>\r\n<\/ol>\r\n

        1*<\/strong>\uff5e3*<\/strong>\u3088\u308a\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} ( p+1 )( s+1 ) & ( \\ 0 \\leqq s \\leqq p \\ \\text{\u306e\u3068\u304d} \\ ) \\\\ ( p+1 )( 2p-s+1 ) & ( \\ p+1 \\leqq s \\leqq 2p \\ \\text{\u306e\u3068\u304d} \\ ) \\\\ 0 & ( \\ s \\lt 0 , 2p \\lt s \\ \\text{\u306e\u3068\u304d} \\ ) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(p , q\\) \u3092 \\(2\\) \u3064\u306e\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u6574\u6570 \\(a , b , c\\) \u3067\u6761\u4ef6 \\[ -q \\leqq b \\leqq 0 \\leqq a \\leqq p , \\quad b \\leqq c \\leq … \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":[79],"tags":[149,13],"class_list":["post-536","post","type-post","status-publish","format-standard","hentry","category-tokyo_b_2011","tag-tokyo_b","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/536","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=536"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/536\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=536"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=536"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=536"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}