%I #7 Jul 06 2020 19:56:22

%S 1,1,0,2,3,5,11,23,40,80,180,344,661,1321,2657,5268,10481,20903,41572,

%T 82734,164998,328304,654510,1305421,2598811,5182174,10332978,20594318,

%U 41066611,81897091,163309679,325707492,649648912,1295827380,2584941276,5156774487

%N Number of complete compositions of n whose run-lengths 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(0) = 1 through a(6) = 11 compositions (empty column not shown):

%e () (1) (1,2) (1,1,2) (1,2,2) (1,2,3)

%e (2,1) (1,2,1) (2,1,2) (1,3,2)

%e (2,1,1) (2,2,1) (2,1,3)

%e (1,1,2,1) (2,3,1)

%e (1,2,1,1) (3,1,2)

%e (3,2,1)

%e (1,2,1,2)

%e (1,2,2,1)

%e (2,1,1,2)

%e (2,1,2,1)

%e (1,1,2,1,1)

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[#]]&]],{n,0,10}]

%Y Looking at multiplicities instead of run-lengths gives A329748.

%Y The non-complete version is A329766.

%Y Complete compositions are A107429.

%Y Cf. A000740, A008965, A098504, A244164, A274174, A329738, A329739, A329740, A329741, A329744.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 21 2019

%E a(21)-a(35) from _Alois P. Heinz_, Jul 06 2020