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A324145
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Minimal length of a string over the alphabet A = {1,2,...,n} that contains every derangement of A as a substring exactly once.
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1
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OFFSET
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1,2
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COMMENTS
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Such strings could be called superderangements (compare A180632).
I used the TSP (Traveling Salesman) solver in SAS, which discovered the values reported for n = 4 through 7 and proved that they are optimal.
For n = 2 and 3, the optimal solution is unique.
For n = 4, there are exactly four optimal solutions:
4321431241314234123421
4312413142341234214321
4312341231424134214321
4321431234123142413421
(End)
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LINKS
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EXAMPLE
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Examples of minimal superderangements for 2,3,4 symbols:
For n = 2: 21, length 2.
For n = 3: 2312, length = 4 (For n=3 there are just two derangements, 231 and 312, so 2312 is clearly optimal.)
For n = 4: 4312413142341234214321, length = 22 (optimality established by Rob Pratt, Feb 21 2019).
For examples for n = 5, 6, and 7 that were discovered and proved optimal by Rob Pratt using SAS, see the link.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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a(4) confirmed and a(5)-a(7) found by Rob Pratt, Feb 21 2019
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STATUS
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approved
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