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A354152 a(n) is the number of permutations pi in S_n such that pi(i) - i != 1 (mod n) and pi(i) - i != -1 (mod n) for all i. 2
1, 0, 1, 1, 4, 13, 82, 579, 4740, 43387, 439794, 4890741, 59216644, 775596313, 10927434466, 164806435783, 2649391469060, 45226435601207, 817056406224418, 15574618910994665, 312400218671253764, 6577618644576902053, 145051250421230224306, 3343382818203784146955 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
For n > 1, this is the number of ways of rearranging guests sitting at a circular table such that a guest may stay in the same seat, but cannot move exactly one seat to their left or right.
The recurrence comes from Doron Zeilberger's MENAGE program, available via the arXiv reference.
LINKS
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.
FORMULA
a(n) = n*a(n-1) + 3*a(n-2) + (-2n+6)*a(n-3) - 3*a(n-4) + (n-6)*a(n-5) + a(n-6) for n > 8.
a(2k+1) = A000179(2k+1) for k > 1.
Conjecture: a(2k) = A000179(2k) + 2 for k > 1.
EXAMPLE
For n = 5, the a(5) = 13 permutations are 12345, 12543, 14325, 14523, 15342, 32145, 34125, 34512, 35142, 42315, 42513, 45123, and 45312.
The first letter is never 2 or 5, the second letter is never 1 or 3, the third letter is never 2 or 4, the fourth letter is never 3 or 5 and the fifth letter is never 1 or 4.
CROSSREFS
Sequence in context: A301396 A221512 A061143 * A354409 A123333 A197752
KEYWORD
nonn
AUTHOR
Peter Kagey, May 27 2022
STATUS
approved

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Last modified April 16 17:08 EDT 2024. Contains 371749 sequences. (Running on oeis4.)