cp's OEIS Frontend

A325788 Number of complete strict necklace compositions of n.

%I A325788 #12 Aug 24 2020 23:06:30
%S A325788 1,0,1,0,0,2,2,0,0,4,4,4,4,0,20,6,16,12,10,0,84,40,74,42,66,38,22,254,
%T A325788 238,188,356,242,272,150,148,1140,1058,1208,1546,1288
%N A325788 Number of complete strict necklace compositions of n.
%C A325788 A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part (counted by A032153). A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.
%e A325788 The a(1) = 1 through a(16) = 6 complete strict necklace compositions (empty columns not shown):
%e A325788   (1)  (12)  (123)  (124)  (1234)  (1253)  (1245)  (1264)  (12345)  (12634)
%e A325788              (132)  (142)  (1324)  (1325)  (1326)  (1327)  (12354)  (13624)
%e A325788                            (1423)  (1352)  (1542)  (1462)  (12435)  (14263)
%e A325788                            (1432)  (1523)  (1623)  (1723)  (12453)  (14326)
%e A325788                                                            (12543)  (14362)
%e A325788                                                            (13254)  (16234)
%e A325788                                                            (13425)
%e A325788                                                            (13452)
%e A325788                                                            (13524)
%e A325788                                                            (13542)
%e A325788                                                            (14235)
%e A325788                                                            (14253)
%e A325788                                                            (14325)
%e A325788                                                            (14523)
%e A325788                                                            (14532)
%e A325788                                                            (15234)
%e A325788                                                            (15243)
%e A325788                                                            (15324)
%e A325788                                                            (15342)
%e A325788                                                            (15432)
%t A325788 neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A325788 subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];
%t A325788 Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,30}]
%Y A325788 Cf. A000740, A002033, A008965, A032153, A103295, A126796, A188431, A325684, A325785, A325786, A325787, A325790, A325791.
%K A325788 nonn,more
%O A325788 1,6
%A A325788 _Gus Wiseman_, May 22 2019