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.
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
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 13 2019
EXTENSIONS
a(16)-a(36) from Fausto A. C. Cariboni, Feb 27 2022
STATUS
approved
