{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":55280,"title":"Count estrangements","description":"Recently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a derangement, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an estrangement, and although I later learned of a more technical and mathematical description, I will keep my name. \r\nWrite a function to count estrangements—i.e., the permutations of elements in a 1x vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 219px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 109.5px; transform-origin: 407px 109.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 63px; text-align: left; transform-origin: 384px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 7.79167px; transform-origin: 384px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eRecently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2417px 7.79167px; transform-origin: 41.2417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ederangement\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.875px 7.79167px; transform-origin: 306.875px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42.7917px 7.79167px; transform-origin: 42.7917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eestrangement\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 165.708px 7.79167px; transform-origin: 165.708px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and although I later learned of a more technical and mathematical description, I will keep my name. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 256.575px 7.79167px; transform-origin: 256.575px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to count estrangements—i.e., the permutations of elements in a 1x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.175px 7.79167px; transform-origin: 122.175px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = estrangements(n)\r\n  y = nchoosek(n,n-3);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = '0';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = '0';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 4;\r\ny_correct = '6';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 8;\r\ny_correct = '8988';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 10;\r\ny_correct = '809856';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 12;\r\ny_correct = '106877320';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 15;\r\ny_correct = '291781655984';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 17;\r\ny_correct = '79364592318720';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 19;\r\ny_correct = '27142690734936864';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 22;\r\ny_correct = '250798462399300784640';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 24;\r\ny_correct = '138440751242507472273856';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 26;\r\ny_correct = '89986488307675206245836800';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 32;\r\ny_correct = '58712425785005411876628940337660160';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 23;\r\nassert(isequal(sum(factor(sum(estrangements(n)-'0'))),32))\r\n\r\n%%\r\nfiletext = fileread('estrangements.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp');\r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":46909,"edited_by":46909,"edited_at":"2023-04-24T19:49:33.000Z","deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":"2023-04-24T19:23:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-08-09T02:45:28.000Z","updated_at":"2026-02-03T16:41:09.000Z","published_at":"2022-08-09T02:45:54.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRecently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ederangement\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eestrangement\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and although I later learned of a more technical and mathematical description, I will keep my name. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to count estrangements—i.e., the permutations of elements in a 1x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":55280,"title":"Count estrangements","description":"Recently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a derangement, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an estrangement, and although I later learned of a more technical and mathematical description, I will keep my name. \r\nWrite a function to count estrangements—i.e., the permutations of elements in a 1x vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 219px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 109.5px; transform-origin: 407px 109.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 63px; text-align: left; transform-origin: 384px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 7.79167px; transform-origin: 384px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eRecently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2417px 7.79167px; transform-origin: 41.2417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ederangement\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.875px 7.79167px; transform-origin: 306.875px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42.7917px 7.79167px; transform-origin: 42.7917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eestrangement\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 165.708px 7.79167px; transform-origin: 165.708px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and although I later learned of a more technical and mathematical description, I will keep my name. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 256.575px 7.79167px; transform-origin: 256.575px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to count estrangements—i.e., the permutations of elements in a 1x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.175px 7.79167px; transform-origin: 122.175px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = estrangements(n)\r\n  y = nchoosek(n,n-3);\r\nend","test_suite":"%%\r\nn = 1;\r\ny_correct = '0';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 2;\r\ny_correct = '0';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 4;\r\ny_correct = '6';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 8;\r\ny_correct = '8988';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 10;\r\ny_correct = '809856';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 12;\r\ny_correct = '106877320';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 15;\r\ny_correct = '291781655984';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 17;\r\ny_correct = '79364592318720';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 19;\r\ny_correct = '27142690734936864';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 22;\r\ny_correct = '250798462399300784640';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 24;\r\ny_correct = '138440751242507472273856';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 26;\r\ny_correct = '89986488307675206245836800';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 32;\r\ny_correct = '58712425785005411876628940337660160';\r\nassert(isequal(estrangements(n),y_correct))\r\n\r\n%%\r\nn = 23;\r\nassert(isequal(sum(factor(sum(estrangements(n)-'0'))),32))\r\n\r\n%%\r\nfiletext = fileread('estrangements.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp');\r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":46909,"edited_by":46909,"edited_at":"2023-04-24T19:49:33.000Z","deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":"2023-04-24T19:23:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-08-09T02:45:28.000Z","updated_at":"2026-02-03T16:41:09.000Z","published_at":"2022-08-09T02:45:54.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRecently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ederangement\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eestrangement\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and although I later learned of a more technical and mathematical description, I will keep my name. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to count estrangements—i.e., the permutations of elements in a 1x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string. 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