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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
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).
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 23 2019
EXTENSIONS
a(11)-a(15) from Bert Dobbelaere, Nov 01 2020
STATUS
approved