|
%I #7 Jul 06 2020 20:07:25
%S 1,1,0,2,3,3,6,12,12,42,114,210,60,360,720,1320,1590,3690,6450,16110,
%T 33120,59940,61320,112980,171780,387240,803880,769440,1773240,2823240,
%U 5790960,9916200,19502280,28244160,56881440,130548600,279578880,320554080,541323720
%N Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.
%C A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.
%e The a(1) = 1 through a(8) = 12 compositions (empty column not shown):
%e (1) (12) (112) (122) (123) (1123) (1223)
%e (21) (121) (212) (132) (1132) (1232)
%e (211) (221) (213) (1213) (1322)
%e (231) (1231) (2123)
%e (312) (1312) (2132)
%e (321) (1321) (2213)
%e (2113) (2231)
%e (2131) (2312)
%e (2311) (2321)
%e (3112) (3122)
%e (3121) (3212)
%e (3211) (3221)
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[Sort[#]]]&]],{n,0,10}]
%Y Looking at run-lengths instead of multiplicities gives A329749.
%Y The non-complete version is A329741.
%Y Complete compositions are A107429.
%Y Cf. A008965, A098504, A244164, A274174, A329740, A329744, A329766.
%K nonn
%O 0,4
%A _Gus Wiseman_, Nov 21 2019
%E a(21)-a(38) from _Alois P. Heinz_, Jul 06 2020
|
