

A108713


Number of possible canonical minimal transitioncomplete sequences over n objects.


1



1, 1, 3, 128, 162000, 10319560704, 50185433088000000, 26294650153960734720000000, 1991323677312505284928104038400000000, 28163375844474584946472694002483200000000000000000
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OFFSET

1,3


COMMENTS

Definition of a canonical minimal transitioncomplete sequence, by example: If n=3, then 2123132312132312 is a transitioncomplete 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 transitioncomplete sequence because 1 appears before the first appearance of 2 and 2 appears before the first appearance of 3.


LINKS



FORMULA

For n > 1, a(n) = n^(n2) * (n2)!^(n1). 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 (n2)! (the number of relabelings of the vertices 3,4,...,n).  Max Alekseyev, Feb 06 2010


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.


MATHEMATICA

Join[{1}, Table[n^(n2) ((n2)!)^(n1), {n, 2, 10}]] (* Harvey P. Dale, Dec 29 2013 *)


CROSSREFS



KEYWORD

nonn,nice


AUTHOR

Philipp G. Blume (pgblu(AT)hotmail.com), Jun 20 2005


EXTENSIONS



STATUS

approved



