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A325684 Number of minimal complete rulers of length n. 11
1, 1, 1, 2, 3, 4, 5, 12, 12, 24, 40, 46, 92, 133, 192, 308 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

LINKS

Table of n, a(n) for n=0..15.

EXAMPLE

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)

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}]

CROSSREFS

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

Sequence in context: A193679 A066574 A240304 * A339453 A281597 A039007

Adjacent sequences:  A325681 A325682 A325683 * A325685 A325686 A325687

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 13 2019

STATUS

approved

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)