A325684 Number of minimal complete rulers of length n.

1, 1, 1, 2, 3, 4, 5, 12, 12, 24, 40, 46, 92, 133, 192, 308, 546, 710, 1108, 1754, 2726, 3878, 5928, 9260, 14238, 20502, 30812, 48378, 72232, 105744, 160308, 241592, 362348, 540362, 797750, 1183984, 1786714

Offset: 0

Author

Gus Wiseman, May 13 2019

Keywords

• nonn
• more

Comments

A complete ruler of length n is a subset of {0..n} containing 0 and n and such that the differences of distinct terms (up to sign) cover an initial interval of positive integers.

Also the number of maximal (most coarse) compositions of n whose consecutive subsequence-sums cover an initial interval of positive integers.

Examples

			The a(1) = 1 through a(7) = 12 rulers:
  {0,1}  {0,1,2}  {0,1,3}  {0,1,2,4}  {0,1,2,5}  {0,1,4,6}    {0,1,2,3,7}
                  {0,2,3}  {0,1,3,4}  {0,1,3,5}  {0,2,5,6}    {0,1,2,4,7}
                           {0,2,3,4}  {0,2,4,5}  {0,1,2,3,6}  {0,1,2,5,7}
                                      {0,3,4,5}  {0,1,3,5,6}  {0,1,3,5,7}
                                                 {0,3,4,5,6}  {0,1,3,6,7}
                                                              {0,1,4,5,7}
                                                              {0,1,4,6,7}
                                                              {0,2,3,6,7}
                                                              {0,2,4,6,7}
                                                              {0,2,5,6,7}
                                                              {0,3,5,6,7}
                                                              {0,4,5,6,7}
The a(1) = 1 through a(9) = 24 compositions:
  (1)  (11)  (12)  (112)  (113)  (132)   (1114)  (1133)   (1143)
             (21)  (121)  (122)  (231)   (1123)  (1241)   (1332)
                   (211)  (221)  (1113)  (1132)  (1322)   (2331)
                          (311)  (1221)  (1222)  (1412)   (3411)
                                 (3111)  (1231)  (1421)   (11115)
                                         (1312)  (2141)   (11124)
                                         (1321)  (2231)   (11142)
                                         (2131)  (3311)   (11241)
                                         (2221)  (11114)  (11322)
                                         (2311)  (11132)  (12141)
                                         (3211)  (23111)  (12222)
                                         (4111)  (41111)  (12231)
                                                          (12312)
                                                          (13221)
                                                          (14112)
                                                          (14121)
                                                          (14211)
                                                          (21141)
                                                          (21321)
                                                          (22221)
                                                          (22311)
                                                          (24111)
                                                          (42111)
                                                          (51111)
		

Crossrefs

Cf. A000079, A103295, A126796, A143823, A169942, A325677, A325683, A325685.

Programs

Mathematica

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Accumulate/@Select[Join@@Permutations/@IntegerPartitions[n],SubsetQ[ReplaceList[#,{_,s__,_}:>Plus[s]],Range[n]]&]]],{n,0,15}]

Extensions

a(16)-a(36) from Fausto A. C. Cariboni, Feb 27 2022