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Number of minimal complete rulers of length n.
11

%I #11 Feb 27 2022 05:46:02

%S 1,1,1,2,3,4,5,12,12,24,40,46,92,133,192,308,546,710,1108,1754,2726,

%T 3878,5928,9260,14238,20502,30812,48378,72232,105744,160308,241592,

%U 362348,540362,797750,1183984,1786714

%N Number of minimal complete rulers of length n.

%C 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.

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

%e The a(1) = 1 through a(7) = 12 rulers:

%e {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}

%e {0,2,3} {0,1,3,4} {0,1,3,5} {0,2,5,6} {0,1,2,4,7}

%e {0,2,3,4} {0,2,4,5} {0,1,2,3,6} {0,1,2,5,7}

%e {0,3,4,5} {0,1,3,5,6} {0,1,3,5,7}

%e {0,3,4,5,6} {0,1,3,6,7}

%e {0,1,4,5,7}

%e {0,1,4,6,7}

%e {0,2,3,6,7}

%e {0,2,4,6,7}

%e {0,2,5,6,7}

%e {0,3,5,6,7}

%e {0,4,5,6,7}

%e The a(1) = 1 through a(9) = 24 compositions:

%e (1) (11) (12) (112) (113) (132) (1114) (1133) (1143)

%e (21) (121) (122) (231) (1123) (1241) (1332)

%e (211) (221) (1113) (1132) (1322) (2331)

%e (311) (1221) (1222) (1412) (3411)

%e (3111) (1231) (1421) (11115)

%e (1312) (2141) (11124)

%e (1321) (2231) (11142)

%e (2131) (3311) (11241)

%e (2221) (11114) (11322)

%e (2311) (11132) (12141)

%e (3211) (23111) (12222)

%e (4111) (41111) (12231)

%e (12312)

%e (13221)

%e (14112)

%e (14121)

%e (14211)

%e (21141)

%e (21321)

%e (22221)

%e (22311)

%e (24111)

%e (42111)

%e (51111)

%t fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];

%t Table[Length[fasmin[Accumulate/@Select[Join@@Permutations/@IntegerPartitions[n],SubsetQ[ReplaceList[#,{___,s__,___}:>Plus[s]],Range[n]]&]]],{n,0,15}]

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

%K nonn,more

%O 0,4

%A _Gus Wiseman_, May 13 2019

%E a(16)-a(36) from _Fausto A. C. Cariboni_, Feb 27 2022