{"id":204,"date":"2025-02-12T12:51:31","date_gmt":"2025-02-12T04:51:31","guid":{"rendered":"https:\/\/blog.seekrealthing.net\/blog\/?p=204"},"modified":"2025-02-12T14:24:43","modified_gmt":"2025-02-12T06:24:43","slug":"b-kyoya-and-permutation-%e8%a7%a3%e6%9e%90%e6%80%9d%e7%b6%ad%e3%80%81%e6%95%b8%e8%ab%96%e3%80%81dp","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=204","title":{"rendered":"B. Kyoya and Permutation \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001DP)"},"content":{"rendered":"\n<div class=\"wp-block-jetpack-markdown\"><p>\u4eca\u5929\u6211\u5011\u4f86\u770b\u770bCF553B\n<a href=\"https:\/\/codeforces.com\/contest\/553\/problem\/B\">\u984c\u76ee\u9023\u7d50<\/a><\/p>\n<h3>\u984c\u76ee\u63cf\u8ff0<\/h3>\n<p>\u7d66\u5b9a\u4e00\u500b\u9577\u5ea6\u70ba $n$ \u7684\u6392\u5217 $p = [p_1, p_2, \\dots, p_n]$\uff08\u7531 $1$ \u5230 $n$ \u7684\u4e92\u4e0d\u76f8\u540c\u7684\u6574\u6578\u7d44\u6210\uff09\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5176\u5206\u89e3\u6210\u82e5\u5e72\u500b\u5faa\u74b0\uff0c\u6bcf\u500b\u5faa\u74b0\u8868\u793a\u6578\u5b57\u4e4b\u9593\u7684\u6620\u5c04\u95dc\u4fc2\u3002\u5177\u9ad4\u5730\uff0c\u5c0d\u65bc\u4e00\u500b\u5faa\u74b0\uff1a<\/p>\n<ul>\n<li>\u5c07\u5faa\u74b0\u5167\u7684\u5143\u7d20\u91cd\u65b0\u6392\u5217\uff0c\u4f7f\u5f97\u6700\u5927\u7684\u5143\u7d20\u4f4d\u65bc\u7b2c\u4e00\u4f4d\uff1b<\/li>\n<li>\u5c07\u6240\u6709\u5faa\u74b0\u6309\u7167\u5176\u7b2c\u4e00\u500b\u5143\u7d20\uff08\u4e5f\u5c31\u662f\u6bcf\u500b\u5faa\u74b0\u4e2d\u7684\u6700\u5927\u5143\u7d20\uff09\u5f9e\u5c0f\u5230\u5927\u6392\u5e8f\uff1b<\/li>\n<li>\u5c07\u6a19\u6e96\u5faa\u74b0\u8868\u793a\u6cd5\u4e2d\u7684\u62ec\u865f\u53bb\u9664\u5f8c\uff0c\u5f97\u5230\u4e00\u500b\u65b0\u7684\u6392\u5217\u3002<\/li>\n<\/ul>\n<p>\u5982\u679c\u4e00\u500b\u6392\u5217\u5728\u7d93\u904e\u4e0a\u8ff0\u64cd\u4f5c\u5f8c\u4fdd\u6301\u4e0d\u8b8a\uff0c\u5247\u7a31\u9019\u500b\u6392\u5217\u70ba\u300c\u7a69\u5b9a\u6392\u5217\u300d\u3002Kyoya \u5c07\u6240\u6709\u9577\u5ea6\u70ba $n$ \u7684\u7a69\u5b9a\u6392\u5217\u6309\u7167\u5b57\u5178\u5e8f\u6392\u5e8f\uff0c\u4e26\u5e0c\u671b\u91cd\u73fe\u9019\u500b\u5e8f\u5217\u3002\u7d66\u5b9a $n$ \u548c $k$\uff0c\u8acb\u6c42\u51fa\u5b57\u5178\u5e8f\u4e2d\u7b2c $k$ \u500b\u7a69\u5b9a\u6392\u5217\u3002<\/p>\n<hr>\n<h3>\u8f38\u5165\u683c\u5f0f<\/h3>\n<p>\u7b2c\u4e00\u884c\u5305\u542b\u5169\u500b\u6574\u6578 $n$ \u8207 $k$ ($1 \\leq n \\leq 50,\\ 1 \\leq k \\leq \\min{10^{18}, l}$)\uff0c\u5176\u4e2d $l$ \u8868\u793a\u6240\u6709\u7a69\u5b9a\u6392\u5217\u7684\u7e3d\u6578\u3002<\/p>\n<hr>\n<h3>\u8f38\u51fa\u683c\u5f0f<\/h3>\n<p>\u8f38\u51fa\u4e00\u500b\u5305\u542b $n$ \u500b\u6574\u6578\u7684\u6392\u5217\uff0c\u5404\u6578\u5b57\u4e4b\u9593\u4ee5\u7a7a\u683c\u5206\u9694\uff0c\u5373\u5b57\u5178\u5e8f\u4e2d\u7b2c $k$ \u500b\u7a69\u5b9a\u6392\u5217\u3002<\/p>\n<hr>\n<h2>\u524d\u8a00<\/h2>\n<p>\u6c92\u60f3\u5230\u5c45\u7136\u7531\u6211\u81ea\u5df1\u60f3\u51fa\u4f86\u4e86\uff0c\u6211\u9084\u7279\u5730\u56de\u53bb\u770b\u6211\u4ee5\u524d\u4ee3\u6578\u7b46\u8a18\u8907\u7fd2permutation\u6709\u4ec0\u9ebc\u6027\u8cea\uff0c\u7d50\u679c\u5176\u5be6\u6c92\u4ec0\u9ebc\u7528\ud83e\udd23\ud83e\udd23<br>\n\u9019\u984c\u6211\u7279\u5730\u7528\u6b63\u5e38\u4e00\u9ede\u7684\u65b9\u5f0f\u5bebcode\uff0c\u6c92\u6709\u7528\u6211\u7684macro\u90a3\u4e9b\u4e82\u4e03\u516b\u7cdf\u7684\u4ec0\u9ebcrep,per,pb,ll,SZ(),VI<br>\n\u5176\u5be6\u4e5f\u9084\u597d\uff0c\u53cd\u6b63\u5bebLeetCode\u4e5f\u90fd\u9019\u6a23\u5beb<\/p>\n<h3>\u60f3\u6cd5<\/h3>\n<p>\u5beb\u9019\u984c\u8b93\u6211\u5c0dpermutation\u7684\u8a8d\u8b58\u66f4\u6e05\u6670\u4e86\uff0c\u6211\u539f\u672c\u9023\u5230\u5e95\u8981\u600e\u9ebc\u628a\u4e00\u500bpermutation\u8b8a\u6210cyclic presentation\u90fd\u4e0d\u77e5\u9053\u3002<br>\n\u5efa\u8b70\u53ef\u4ee5\u5b78\u984c\u76ee\u6558\u8ff0\u9019\u6a23\u770b\u4e00\u500bpermutation\uff1a<br>\n\u5c0d\u65bc$\\sigma=(4,1,6,2,5,3)$\uff0c\u6211\u5011\u770b\u6210\u4e00\u500bGraph\uff1a<br>\n$$\n\\begin{aligned}\n1\\rightarrow 4\\\n2\\rightarrow 1\\\n3\\rightarrow 6\\\n4\\rightarrow 2\\\n5\\rightarrow 5\\\n6\\rightarrow 3\n\\end{aligned}\n$$\n\u53ef\u4ee5\u756b\u6210\u9019\u6a23\uff1a<br>\n<img decoding=\"async\" src=\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/upload_ecee56408634d207b5951b38ee59d1bd.png\" alt=\"image\"><br>\n\u4e5f\u5c31\u662f$p=(4,1,6,2,5,3)=(4,2,1)(5)(6,3)$<br>\n\u7136\u5f8c\u6211\u5011\u4f86\u770b\u770b\u5982\u679c\u984c\u76ee\u8981\u6c42\u7684\u689d\u4ef6\u6210\u7acb\uff0c\u6703\u600e\u9ebc\u6a23\uff1a<br>\n\u5047\u8a2d$p=(p(1),p(2),p(3),p(4))$\uff0c\u5148\u4f86\u770b\u770b\u5982\u679c$p=(p(1),p(2))(p(3),p(4))$<br>\n\u4ee3\u8868$p(3)&gt;p(4)&gt;p(1)&gt;p(2)$\n\u5716\u6703\u9577\u9019\u6a23\uff1a<br>\n<img decoding=\"async\" src=\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/upload_2e5f410a45bbf9fc91db816aec17837e.png\" alt=\"image\"><br>\n\u540c\u6642\u5716\u4e5f\u6703\u9577\u9019\u6a23\uff1a<br>\n<img decoding=\"async\" src=\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/upload_ba049546d979998c5c55bbbe4fd06063.png\" alt=\"image\"><br>\n\u6240\u4ee5\u4ee3\u8868$p(2)=1,p(1)=2,p(3)=4,p(4)=3$<br>\n\u6211\u5011\u518d\u4f86\u770b$n=5$\u7684\u60c5\u6cc1\uff1a<br>\n\u5047\u8a2d$p=(p(1),p(2),p(3),p(4),p(5))=(p(1),p(2),p(3))(p(4),p(5))$<br>\n\u4ee5\u4e0b\u5169\u5f35\u5716\u540c\u6642\u6210\u7acb\uff1a<br>\n<img decoding=\"async\" src=\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/upload_207bf7587631c98b46f8498cea1bb2ac.png\" alt=\"image\"><br>\n<img decoding=\"async\" src=\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/upload_8abed6e97e94118e463624b791699ff1.png\" alt=\"image\"><br>\n\u56e0\u6b64\u6703\u767c\u73fe\uff0c$p(3)=1,p(2)=3,p(1)=2$<br>\n\u7136\u800c\u6839\u64da\u984c\u76ee\u689d\u4ef6\uff0c\u6211\u5011\u61c9\u8a72\u8981\u6709$p(4)&gt;p(5)&gt;p(1)&gt;p(2)&gt;p(3)$\uff0c\u6211\u5011\u73fe\u5728\u9055\u53cd\u4e86\uff01<br>\n\u7a0d\u5fae\u89c0\u5bdf\uff0c\u6703\u767c\u73fe\u540c\u500bcycle\u88e1\u7684$(p(i),p(i+1),p(i+2))$\uff0c\u5176\u5be6\u5c31\u6703\u662f$(i+1,i+2,i)$\uff0c\u4f46\u662f$i+1&lt;i+2$\uff0c\u9055\u53cd\u984c\u76ee\u7684\u689d\u4ef6\u3002<br>\n\u56e0\u6b64\uff0c\u6211\u5011\u767c\u73fe\u53ea\u6709$2-$cycle\u6216\u8005$1-$cycle(\u50cf\u662f(5)\u5c31\u662f$1-$cycle)\u662f\u53ef\u80fd\u6eff\u8db3\u984c\u76ee\u689d\u4ef6\u7684\u3002<br>\n\u56e0\u6b64\u6211\u5011\u73fe\u5728\u984c\u76ee\u7684\u89e3\u6cd5\u5c31\u662f\uff0c\u5c0d\u65bc$p=(1,2,3,\u2026)$\uff0c\u6211\u5011\u8981\u627e\u5230\u7b2c$k$\u500b\u5b57\u5178\u5e8f\u7684$1-,2-$cycle\u6392\u5217\u9806\u5e8f\u3002<br>\n\u5c0d\u65bc\u9577\u5ea6$n$\u7684\u5e8f\u5217\uff0c\u8981\u600e\u9ebc\u6c42\u6709\u5e7e\u7a2e\u5206\u914d$1-,2-$cycle\u7684\u65b9\u6cd5\u5462\uff1f\u6211\u5011\u53ef\u4ee5\u7528dp\uff0c\u8f49\u79fb\u5f0f\u662f\uff1a\n$$\n\\text{dp}[i] = \\text{dp}[i-2] + \\text{dp}[i-1]\n$$\n$\\text{dp}[i]$\u4ee3\u8868\u9577\u5ea6\u70ba$i$\u7684\u5e8f\u5217\u6709\u5e7e\u7a2e\u5206\u914d\u65b9\u5f0f\u3002(\u4e5f\u53ef\u4ee5\u767c\u73fe\u9019\u5176\u5be6\u5c31\u662f\u6590\u6ce2\u90a3\u5951\u6578\u5217)<br>\n\u77e5\u9053\u9019\u500b\u4ee5\u5f8c\uff0c\u6211\u5011\u4e5f\u77e5\u9053\uff0c\u5982\u679c\u4f60\u73fe\u5728\u53ef\u4ee5\u9078\u64c7\u7528$1-$cycle\u6216\u8005$2-$cycle\uff0c$1-$cycle\u5b57\u5178\u5e8f\u662f\u6bd4\u8f03\u5c0f\u7684\uff0c\u56e0\u70ba\u6700\u524d\u9762\u7684\u6578\u5b57\u6bd4\u8f03\u5c0f\u3002<br>\n\u6240\u4ee5\u6211\u5011\u5047\u8a2d\u5df2\u7d93\u770b\u5b8c\u524d$i$\u500b\u6578\u5b57\u4e86\uff0c\u5982\u679c\u6392$1-$cycle\u5269\u4e0b\u7684\u53ef\u80fdcycle\u5206\u914d\u65b9\u5f0f\u6703$&gt;=k$\uff0c\u90a3\u5c31\u7b54\u6848\u653e\u5165\u4e00\u500b$1-$cycle\uff1b\u53cd\u4e4b\u5982\u679c\u5269\u4e0b\u7684\u53ef\u80fdcycle\u5206\u914d\u65b9\u5f0f\u6703$&lt;k$\uff0c\u6211\u5011\u5c31\u628a$k$\u6e1b\u6389\u300c\u6392$1-$cycle\u5269\u4e0b\u7684\u53ef\u80fdcycle\u5206\u914d\u65b9\u5f0f\u300d\uff0c\u7136\u5f8c\u7b54\u6848\u653e\u5165\u4e00\u500b$2-$cycle\u3002<\/p>\n<h3>\u7a0b\u5f0f\u78bc:<\/h3>\n<pre><code class=\"language-cpp\">const int _n = 51;\nll n, k, possibility[_n];\nmain(void) {\n  ios_base::sync_with_stdio(0);\n  cin.tie(0);\n  cout.tie(0);\n  cin &gt;&gt; n &gt;&gt; k;\n  \/\/ 1-indexed\n  possibility[0] = 1;\n  possibility[1] = 1;\n  possibility[2] = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    possibility[i] = possibility[i - 1] + possibility[i - 2];\n  }\n  vector&lt;int&gt; ansseq;  \/\/ 1 for 1 cycle, 2 for 2 cycle\n  ll k2 = k;\n  int i = 0;\n  while (i &lt; n) {\n    \/\/ i = idx before making new cycle length decision\n    if (possibility[n - i - 1] &gt;= k2) {\n      ansseq.push_back(1);\n      i += 1;\n    } else {\n      k2 -= possibility[n - i - 1];\n      ansseq.push_back(2);\n      i += 2;\n    }\n  }\n  vector&lt;int&gt; ans;\n  for (int i = 1; i &lt;= n; i++) {\n    ans.push_back(i);\n  }\n  int idx = 0;\n  for (int i = 0; i &lt; (int)ansseq.size(); i++) {\n    if (ansseq[i] == 1) {\n      idx++;\n    } else {\n      swap(ans[idx], ans[idx + 1]);\n      idx += 2;\n    }\n  }\n  for (int x : ans) {\n    cout &lt;&lt; x &lt;&lt; ' ';\n  }\n  cout &lt;&lt; '\\n';\n  return 0;\n}\n<\/code><\/pre>\n<p>\u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b\n<a href=\"https:\/\/codeforces.com\/contest\/553\/submission\/305592389\">Submission<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_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},"version":2}},"categories":[3],"tags":[10,4,9,8,21],"class_list":["post-204","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-codeforce","tag-dp","tag-ojcode","tag-8","tag-21"],"jetpack_publicize_connections":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>B. Kyoya and Permutation \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001DP) - Logic Formatting<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blog.seekrealthing.net\/blog\/?p=204\" \/>\n<meta property=\"og:locale\" content=\"zh_TW\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"B. Kyoya and Permutation \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001DP) - Logic Formatting\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blog.seekrealthing.net\/blog\/?p=204\" \/>\n<meta property=\"og:site_name\" content=\"Logic Formatting\" \/>\n<meta property=\"article:published_time\" content=\"2025-02-12T04:51:31+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-02-12T06:24:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/upload_ecee56408634d207b5951b38ee59d1bd.png\" \/>\n<meta name=\"author\" content=\"jeff\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005:\" \/>\n\t<meta name=\"twitter:data1\" content=\"jeff\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9810\u4f30\u95b1\u8b80\u6642\u9593\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 \u5206\u9418\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=204#article\",\"isPartOf\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=204\"},\"author\":{\"name\":\"jeff\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/233e80a0fee3d4706f067f2e88c0b454\"},\"headline\":\"B. 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