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A108713
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Number of possible canonical minimal transition-complete sequences over n objects.
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1
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1, 1, 3, 128, 162000, 10319560704, 50185433088000000, 26294650153960734720000000, 1991323677312505284928104038400000000, 28163375844474584946472694002483200000000000000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Definition of a canonical minimal transition-complete sequence, by example: If n=3, then 2123132312132312 is a transition-complete sequence because each element (1,2, or 3) is followed by each other element at least once.
3132123 is a minimal transition complete sequence, as each element is followed by each other element EXACTLY once.
1231321 is a canonical minimal transition-complete sequence because 1 appears before the first appearance of 2 and 2 appears before the first appearance of 3.
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FORMULA
| For n>1, a(n) = n^(n-2) * (n-2)!^(n-1). This is the number of Eulerian circuits in the complete directed graph on the vertices 1,2,...,n (starting with an arc (1,2)) divided by (n-2)! (the number of relabellings of the vertices 3,4,...,n). [From Max Alekseyev (maxale(AT)gmail.com), Feb 06 2010]
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EXAMPLE
| With n=1, there is only the possibility "1". With n=2, there is only the possibility "121". With n=3, there are the following 3 possibilities: "1213231", "1231321" and "1232131". Here is one of the 128 possibilities with n=4: "1231342143241" With n=5, I think there are over 120000 possibilities and at n=6 there may be a large number.
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CROSSREFS
| Sequence in context: A041867 A134711 A163850 * A123047 A097420 A037119
Adjacent sequences: A108710 A108711 A108712 * A108714 A108715 A108716
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KEYWORD
| nonn,nice
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AUTHOR
| Philipp G. Blume (pgblu(AT)hotmail.com), Jun 20 2005
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EXTENSIONS
| Terms a(5) onwards from Max Alekseyev (maxale(AT)gmail.com), Feb 06 2010
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