{"id":2890,"date":"2023-05-27T06:04:09","date_gmt":"2023-05-26T22:04:09","guid":{"rendered":"https:\/\/www.shuizilong.com\/house\/?p=2890"},"modified":"2023-05-27T23:21:54","modified_gmt":"2023-05-27T15:21:54","slug":"luogu-p5827-%e7%82%b9%e5%8f%8c%e8%bf%9e%e9%80%9a%e5%9b%be%e8%ae%a1%e6%95%b0","status":"publish","type":"post","link":"https:\/\/www.shuizilong.com\/house\/archives\/luogu-p5827-%e7%82%b9%e5%8f%8c%e8%bf%9e%e9%80%9a%e5%9b%be%e8%ae%a1%e6%95%b0\/","title":{"rendered":"Luogu P5827. \u70b9\u53cc\u8fde\u901a\u56fe\u8ba1\u6570"},"content":{"rendered":"<ul>\n<li><a href=\"https:\/\/www.luogu.com.cn\/problem\/P5827\">https:\/\/www.luogu.com.cn\/problem\/P5827<\/a><\/li>\n<\/ul>\n<p>\u6709\u4e00\u4e2a\u4e1c\u897f\u53eb\u505a labeled counting lemma\u3002\u5176\u5b9e\u5c31\u662f\u4ea4\u4ee3\u4e86 EGF \u5377\u79ef\u7684\u7ec4\u5408\u610f\u4e49\u3002<\/p>\n<p>\u8bf4\u7684\u662f\u7ed9\u4e24\u4e2a EGF\uff0cF(x), G(x)\uff0c\u90a3\u4e48 F(x)*G(x) \u5c31\u662f\u8fd9\u4e24\u4e2a\u5bf9\u8c61\u91cd\u65b0\u6807\u53f7\u4e4b\u540e\u6240\u751f\u6210\u7684\u5bf9\u8c61\u7684\u6709\u5e8f\u5bf9\uff0c\u6bd4\u5982\u5982\u679c C(x) \u662f\u8fde\u901a\u56fe\u3002\u90a3\u4e48 C^k(x)\/k! \u5c31\u662f\u6709 k \u4e2a\u8fde\u901a\u5206\u91cf\u7684\u56fe\uff0c\u8fdb\u4e00\u6b65\u5c31\u80fd\u63a8\u51fa EGF \u91cc exp \u7684\u7ec4\u5408\u610f\u4e49\u3002<\/p>\n<p>\u8fd9\u4e2a\u9898\u591a\u9879\u5f0f\u5f88\u5bb9\u6613\u6784\u5efa\uff0c\u56e0\u4e3a\u4efb\u610f\u56fe\u90fd\u53ef\u4ee5\u6784\u5efa\u51fa block-cut tree~<br \/>\n\u65b9\u6cd5\u662f\u6211\u4eec\u601d\u8003\u6709\u6839\u7248\u672c\u7684 C(x)\uff0c\u90a3\u4e48\u8fd9\u4e2a\u6839\u8282\u70b9\u4e00\u5b9a\u5728\u4e00\u4e2a block \u91cc\uff0c\u6211\u4eec\u679a\u4e3e\u8fd9\u4e2a block \u7684 size i\uff0c\u90a3\u4e48\u8fd9\u4e2a block \u5269\u4e0b\u7684\u90e8\u5206\u90fd\u80fd\u8fde\u4e00\u4e2a C(x)\u3002\u3002\u8fd9\u90e8\u5206\u5c31\u662f C(x)^(i-1)\uff0c\u7136\u540e\u518d\u4e58\u4ee5 b_i \u5c31\u597d\u3002\u3002\u3002\u8fd9\u4e2a\u90e8\u5206\u53ef\u4ee5\u7528 exp \u8868\u8fbe\u3002<\/p>\n<p>\u5269\u4e0b\u7684\u662f\u8fd9\u4e2a\u9898\u7684\u96be\u70b9\uff0c\u5c31\u662f\u62c9\u683c\u6717\u65e5\u5b9a\u7406\u3002\u3002\u5176\u5b9e\u6709\u6807\u53f7\u6811\u5c31\u5e94\u8be5\u8981\u5b66\u4f1a\u8fd9\u4e2a\u3002\u3002\u53ea\u662f\u90a3\u4e2a\u95ee\u9898\u91cc\u6211\u4eec\u6709\u592a\u591a\u65b9\u6cd5\u53ef\u4ee5\u9003\u8bfe\u3002\u3002<\/p>\n<p>\u5728\u6709\u6839\u6811\u95ee\u9898\u91cc\uff0c\u79fb\u9664\u4e86\u6839\u8282\u70b9\u4e4b\u540e\uff0c\u5269\u4e0b\u7684\u5c31\u662f\u6709\u6839\u6811\u672c\u8eab\u7684\u751f\u6210\u96c6\u65cf\uff0c\u6240\u4ee5\u6709\u9012\u5f52\u516c\u5f0f\uff1a<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 18px;\"><span class=\"ql-right-eqno\"> (1) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.shuizilong.com\/house\/wp-content\/ql-cache\/quicklatex.com-402237d7ca6368d1e199ceb1a4c761e4_l3.png\" height=\"18\" width=\"58\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#121;&#32;&#38;&#61;&#32;&#120;&#101;&#94;&#121;&#32; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>\u5269\u4e0b\u7684\u5168\u90fd\u662f\u62c9\u683c\u6717\u65e5\u53cd\u6f14\u7684\u4ee3\u6570\u6280\u5de7\uff0c\u6211\u4eec\u7c7b\u6bd4\u8fd9\u79cd\u505a\u6cd5\u6765\u89e3\u51b3\u70b9\u53cc\u548c\u8fb9\u53cc\u7684\u8ba1\u6570\u95ee\u9898\uff0c\u6838\u5fc3\u90fd\u662f\u5efa\u7acb\u5404\u81ea\u7684 EGF \u4e0e \u6709\u6839\u8fde\u901a\u56fe EGF \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u6211\u4eec\u4e0d\u59a8\u4e5f\u8bbe\u5b83\u4eec\u5206\u522b\u4e3a b = B(x)\u3001y = Y(x)\u3002<\/p>\n<p>\u70b9\u53cc\u548c\u8fb9\u53cc\u6700\u5927\u4e0d\u540c\u4e4b\u5904\u5728\u4e8e\uff0c\u4e00\u4e2a\u5272\u70b9\u53ef\u80fd\u540c\u65f6\u5b58\u5728\u4e8e\u591a\u4e2a\u70b9\u53cc\u8fde\u901a\u5206\u91cf\u91cc\u3002<\/p>\n<p>\u8003\u5bdf\u70b9\u53cc\uff0c\u6211\u4eec\u5220\u9664\u6839\u8282\u70b9 x \u540e\uff0c\u6240\u5269\u4e0b\u7684\u90e8\u5206\u662f\u4e00\u4e9b\u5c11\u4e86\u6839\u8282\u70b9\u7684\u8fde\u901a\u5206\u91cf t\uff0c\u90a3\u4e48 y \u5c31\u662f e^t\uff0c\u6240\u4ee5\u6211\u4eec\u53ea\u9700\u8003\u5bdf t \u7684 EGF\u3002<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u679a\u4e3e t \u4e2d\u88ab\u5220\u9664\u7684\u6839\u8282\u70b9 x \u540e\u6240\u5728\u7684\u53cc\u8fde\u901a\u5206\u91cf\u7684\u5927\u5c0f i\uff0c\u90a3\u4e48\u8fd9\u4e2a\u8fde\u901a\u5206\u91cf\u4e2d\uff0c\u9664\u4e86 x \u4e4b\u5916\u7684\u5176\u5b83\u8282\u70b9\u90fd\u5bf9\u5e94\u7740\u4e00\u4e2a\u72ec\u7acb\u7684 y\uff0c\u4e8e\u662f\u6709\uff1a<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 138px;\"><span class=\"ql-right-eqno\"> (2) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.shuizilong.com\/house\/wp-content\/ql-cache\/quicklatex.com-bd1ae124c9c9eb9bb288bc294ccf2120_l3.png\" height=\"138\" width=\"110\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#121;&#32;&#38;&#61;&#32;&#120;&#101;&#94;&#116;&#32;&#92;&#92; &#116;&#32;&#38;&#61;&#32;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#32;&#98;&#95;&#123;&#105;&#43;&#49;&#125;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#94;&#105;&#125;&#123;&#105;&#33;&#125;&#32;&#92;&#92; &#32;&#32;&#38;&#61;&#32;&#66;&#39;&#40;&#121;&#41;&#32;&#92;&#92; &#121;&#32;&#38;&#61;&#32;&#120;&#101;&#94;&#123;&#66;&#39;&#40;&#121;&#41;&#125;&#32;&#92;&#92; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>\u518d\u8003\u5bdf\u8fb9\u53cc\uff0c\u6211\u4eec\u540c\u6837\u679a\u4e3e\u6839\u8282\u70b9\u6240\u5728\u53cc\u8fde\u901a\u5206\u91cf\u7684\u5927\u5c0f\uff0c\u90a3\u4e48\u5220\u9664\u8fd9\u4e2a\u53cc\u8fde\u901a\u5206\u91cf\u540e\uff0c\u5269\u4e0b\u7684\u8fde\u901a\u5206\u91cf\u7684\u6bcf\u4e00\u4e2a\u90fd\u4f1a\u901a\u8fc7\u6865\u6302\u8f7d\u5230\u8fd9\u4e2a\u53cc\u8fde\u901a\u5206\u91cf\u4e2d\u7684\u4efb\u610f\u8282\u70b9\uff0c\u8bbe\u8fd9\u4e2a\u90e8\u5206\u662f t\u3002<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 75px;\"><span class=\"ql-right-eqno\"> (3) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.shuizilong.com\/house\/wp-content\/ql-cache\/quicklatex.com-1d013b71faa9718f736d47bb642bb688_l3.png\" height=\"75\" width=\"87\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#121;&#32;&#38;&#61;&#32;&#98;&#32;&#101;&#94;&#116;&#32;&#92;&#92; &#116;&#32;&#38;&#61;&#32;&#105;&#121;&#32;&#92;&#92; &#121;&#32;&#38;&#61;&#32;&#66;&#40;&#120;&#101;&#94;&#121;&#41; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<pre class=\"brush: cpp; light: false; title: ; toolbar: true; notranslate\" title=\"\">\r\n#include &lt;lastweapon\/poly&gt;\r\nusing namespace lastweapon;\r\n\r\nPoly H, HH;\r\nint n;\r\n\r\nLL C2(LL n) {\r\n    return n*(n-1)\/2;\r\n}\r\n\r\nint b(int n) {\r\n    --n; if (!n) return 1;\r\n    Poly A(n); REP(i, n) A&#x5B;i] = H&#x5B;i] * -n;\r\n    Poly B = HH.mod(n) * A.exp();\r\n    return (B&#x5B;n-1] * fac&#x5B;n-1]).x;\r\n}\r\n\r\nint main(){\r\n\r\n#ifndef ONLINE_JUDGE\r\n    freopen(&quot;in.txt&quot;, &quot;r&quot;, stdin);\r\n#endif\r\n\r\n    vector&lt;int&gt; q; DO(5) q.PB(RD()); n = *max_element(ALL(q)) + 1;\r\n    Poly C(n), G(n); REP(i, n) G&#x5B;i] = Mint(2).pow(C2(i)), G&#x5B;i] *= invFac&#x5B;i]; C = G.log();\r\n    H = C.D().log(); HH = H.D();\r\n\r\n    for (auto i: q) printf(&quot;%d\\n&quot;, b(i));\r\n}\r\n\r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>https:\/\/www.luogu.com.cn\/problem\/P5827 \u6709\u4e00\u4e2a\u4e1c\u897f\u53eb\u505a labeled counting lemma\u3002\u5176\u5b9e\u5c31\u662f\u4ea4\u4ee3\u4e86 EGF \u5377\u79ef\u7684\u7ec4\u5408\u610f\u4e49\u3002 \u8bf4\u7684\u662f\u7ed9\u4e24\u4e2a EGF\uff0cF(x), G(x)\uff0c\u90a3\u4e48 F(x)*G(x) \u5c31\u662f\u8fd9\u4e24\u4e2a\u5bf9\u8c61\u91cd\u65b0\u6807\u53f7\u4e4b\u540e\u6240\u751f\u6210\u7684\u5bf9\u8c61\u7684\u6709\u5e8f\u5bf9\uff0c\u6bd4\u5982\u5982\u679c C(x) \u662f\u8fde\u901a\u56fe\u3002\u90a3\u4e48 C^k(x)\/k! \u5c31\u662f\u6709 k \u4e2a\u8fde\u901a\u5206\u91cf\u7684\u56fe\uff0c\u8fdb\u4e00\u6b65\u5c31\u80fd\u63a8\u51fa EGF \u91cc exp \u7684\u7ec4\u5408\u610f\u4e49\u3002 \u8fd9\u4e2a\u9898\u591a\u9879\u5f0f\u5f88\u5bb9\u6613\u6784\u5efa\uff0c\u56e0\u4e3a\u4efb\u610f\u56fe\u90fd\u53ef\u4ee5\u6784\u5efa\u51fa block-cut tree~ \u65b9\u6cd5\u662f\u6211\u4eec\u601d\u8003\u6709\u6839\u7248\u672c\u7684 C(x)\uff0c\u90a3\u4e48\u8fd9\u4e2a\u6839\u8282\u70b9\u4e00\u5b9a\u5728\u4e00\u4e2a block \u91cc\uff0c\u6211\u4eec\u679a\u4e3e\u8fd9\u4e2a block \u7684 size i\uff0c\u90a3\u4e48\u8fd9\u4e2a block \u5269\u4e0b\u7684\u90e8\u5206\u90fd\u80fd\u8fde\u4e00\u4e2a C(x)\u3002\u3002\u8fd9\u90e8\u5206\u5c31\u662f C(x)^(i-1)\uff0c\u7136\u540e\u518d\u4e58\u4ee5 b_i \u5c31\u597d\u3002\u3002\u3002\u8fd9\u4e2a\u90e8\u5206\u53ef\u4ee5\u7528 exp \u8868\u8fbe\u3002 \u5269\u4e0b\u7684\u662f\u8fd9\u4e2a\u9898\u7684\u96be\u70b9\uff0c\u5c31\u662f\u62c9\u683c\u6717\u65e5\u5b9a\u7406\u3002\u3002\u5176\u5b9e\u6709\u6807\u53f7\u6811\u5c31\u5e94\u8be5\u8981\u5b66\u4f1a\u8fd9\u4e2a\u3002\u3002\u53ea\u662f\u90a3\u4e2a\u95ee\u9898\u91cc\u6211\u4eec\u6709\u592a\u591a\u65b9\u6cd5\u53ef\u4ee5\u9003\u8bfe\u3002\u3002 \u5728\u6709\u6839\u6811\u95ee\u9898\u91cc\uff0c\u79fb\u9664\u4e86\u6839\u8282\u70b9\u4e4b\u540e\uff0c\u5269\u4e0b\u7684\u5c31\u662f\u6709\u6839\u6811\u672c\u8eab\u7684\u751f\u6210\u96c6\u65cf\uff0c\u6240\u4ee5\u6709\u9012\u5f52\u516c\u5f0f\uff1a (1) &nbsp; \u5269\u4e0b\u7684\u5168\u90fd\u662f\u62c9\u683c\u6717\u65e5\u53cd\u6f14\u7684\u4ee3\u6570\u6280\u5de7\uff0c\u6211\u4eec\u7c7b\u6bd4\u8fd9\u79cd\u505a\u6cd5\u6765\u89e3\u51b3\u70b9\u53cc\u548c\u8fb9\u53cc\u7684\u8ba1\u6570\u95ee\u9898\uff0c\u6838\u5fc3\u90fd\u662f\u5efa\u7acb\u5404\u81ea\u7684 EGF \u4e0e \u6709\u6839\u8fde\u901a\u56fe EGF \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u6211\u4eec\u4e0d\u59a8\u4e5f\u8bbe\u5b83\u4eec\u5206\u522b\u4e3a b = B(x)\u3001y = [&hellip;]<\/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,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[1],"tags":[],"class_list":["post-2890","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2tdP7-KC","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/2890","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/comments?post=2890"}],"version-history":[{"count":13,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/2890\/revisions"}],"predecessor-version":[{"id":2909,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/2890\/revisions\/2909"}],"wp:attachment":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/media?parent=2890"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/categories?post=2890"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/tags?post=2890"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}