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A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence. 8
1, 1, 1, 2, 4, 20, 82, 252, 1074, 7912, 39552, 152680, 776094, 5550310, 30026848, 108376910 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A necklace permutation is a permutation that is either empty or whose first part is the minimum. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. The only circular subsequence of maximum length is the sequence itself, not any rotation of it. For example, the circular subsequences of (1,3,2) are: (), (1), (2), (3), (1,3), (2,1), (3,2), (1,3,2).
LINKS
EXAMPLE
The a(1) = 1 through a(5) = 20 permutations:
(1) (1,2) (1,2,3) (1,2,3,4) (1,2,3,4,5)
(1,3,2) (1,3,2,4) (1,2,3,5,4)
(1,4,2,3) (1,2,4,3,5)
(1,4,3,2) (1,2,4,5,3)
(1,2,5,4,3)
(1,3,2,5,4)
(1,3,4,2,5)
(1,3,4,5,2)
(1,3,5,2,4)
(1,3,5,4,2)
(1,4,2,3,5)
(1,4,2,5,3)
(1,4,3,2,5)
(1,4,5,2,3)
(1,4,5,3,2)
(1,5,2,3,4)
(1,5,2,4,3)
(1,5,3,2,4)
(1,5,3,4,2)
(1,5,4,3,2)
MATHEMATICA
subalt[q_]:=Union[ReplaceList[q, {___, s__, ___}:>{s}], DeleteCases[ReplaceList[q, {t___, __, u___}:>{u, t}], {}]];
Table[Length[Select[Permutations[Range[n]], #=={}||First[#]==1&&Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]], {n, 0, 5}]
CROSSREFS
Sequence in context: A009291 A081440 A204438 * A188326 A158094 A108879
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 23 2019
EXTENSIONS
a(11)-a(15) from Bert Dobbelaere, Nov 01 2020
STATUS
approved

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Last modified March 28 13:35 EDT 2024. Contains 371254 sequences. (Running on oeis4.)