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.

1, 1, 1, 2, 4, 20, 82, 252, 1074, 7912, 39552, 152680, 776094, 5550310, 30026848, 108376910

Offset: 0

Author

Gus Wiseman, May 23 2019

Keywords

• nonn
• more

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).

Examples

			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)
		

Crossrefs

Cf. A002033, A008965, A032153, A103295, A126796, A304793, A325684, A325781, A325786, A325788, A325789, A325790.

Programs

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}]

Extensions

a(11)-a(15) from Bert Dobbelaere, Nov 01 2020