cp's OEIS Frontend

A325790 Number of permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

%I A325790 #18 Nov 01 2020 03:53:56
%S A325790 1,1,2,6,16,100,492,1764,8592,71208,395520,1679480,9313128,72154030,
%T A325790 420375872,1625653650
%N A325790 Number of permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.
%C A325790 A circular subsequence is a sequence of consecutive non-overlapping 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 (2,1,3) are: (), (1), (2), (3), (1,3), (2,1), (3,2), (2,1,3).
%e A325790 The a(1) = 1 through a(4) = 16 permutations:
%e A325790   (1)  (1,2)  (1,2,3)  (1,2,3,4)
%e A325790        (2,1)  (1,3,2)  (1,3,2,4)
%e A325790               (2,1,3)  (1,4,2,3)
%e A325790               (2,3,1)  (1,4,3,2)
%e A325790               (3,1,2)  (2,1,4,3)
%e A325790               (3,2,1)  (2,3,1,4)
%e A325790                        (2,3,4,1)
%e A325790                        (2,4,1,3)
%e A325790                        (3,1,4,2)
%e A325790                        (3,2,1,4)
%e A325790                        (3,2,4,1)
%e A325790                        (3,4,1,2)
%e A325790                        (4,1,2,3)
%e A325790                        (4,1,3,2)
%e A325790                        (4,2,3,1)
%e A325790                        (4,3,2,1)
%t A325790 subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];
%t A325790 Table[Length[Select[Permutations[Range[n]],Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]],{n,0,5}]
%o A325790 (PARI)
%o A325790 weigh(p)={my(b=0); for(i=1, #p, my(s=0,j=i); for(k=1, #p, s+=p[j]; if(!bittest(b,s), b=bitor(b,1<<s)); j=if(j==#p, 1, j+1))); hammingweight(b)}
%o A325790 a(n)={my(e=n*(n+1)/2, c=0); forperm(n, p, if(weigh(p)==e, c++)); c} \\ _Andrew Howroyd_, Aug 16 2019
%Y A325790 Cf. A002033, A008965, A032153, A103295, A103300, A126796, A325684, A325781, A325788, A325791.
%K A325790 nonn,more
%O A325790 0,3
%A A325790 _Gus Wiseman_, May 23 2019
%E A325790 a(10)-a(12) from _Andrew Howroyd_, Aug 18 2019
%E A325790 a(13)-a(15) from _Bert Dobbelaere_, Nov 01 2020