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.

1, 1, 2, 6, 16, 100, 492, 1764, 8592, 71208, 395520, 1679480, 9313128, 72154030, 420375872, 1625653650

Offset: 0

Author

Gus Wiseman, May 23 2019

Keywords

• nonn
• more

Comments

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).

Examples

			The a(1) = 1 through a(4) = 16 permutations:
  (1)  (1,2)  (1,2,3)  (1,2,3,4)
       (2,1)  (1,3,2)  (1,3,2,4)
              (2,1,3)  (1,4,2,3)
              (2,3,1)  (1,4,3,2)
              (3,1,2)  (2,1,4,3)
              (3,2,1)  (2,3,1,4)
                       (2,3,4,1)
                       (2,4,1,3)
                       (3,1,4,2)
                       (3,2,1,4)
                       (3,2,4,1)
                       (3,4,1,2)
                       (4,1,2,3)
                       (4,1,3,2)
                       (4,2,3,1)
                       (4,3,2,1)
		

Crossrefs

Cf. A002033, A008965, A032153, A103295, A103300, A126796, A325684, A325781, A325788, A325791.

Programs

Mathematica

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Permutations[Range[n]],Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]],{n,0,5}]

PARI

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<Andrew Howroyd, Aug 16 2019

Extensions

a(10)-a(12) from Andrew Howroyd, Aug 18 2019

a(13)-a(15) from Bert Dobbelaere, Nov 01 2020