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%I A329741 #8 Nov 21 2019 18:59:19
%S A329741 1,1,1,3,6,11,14,34,52,114,225,464,539,1183,1963,3753,6120,11207,
%T A329741 19808,38254,77194,147906,224853,374216,611081,1099933,2129347,
%U A329741 3336099,5816094,9797957,17577710,29766586,53276392,93139668,163600815,324464546,637029845,1010826499
%N A329741 Number of compositions of n whose multiplicities cover an initial interval of positive integers.
%C A329741 A composition of n is a finite sequence of positive integers with sum n.
%e A329741 The a(1) = 1 through a(6) = 14 compositions:
%e A329741   (1)  (2)  (3)    (4)      (5)      (6)
%e A329741             (1,2)  (1,3)    (1,4)    (1,5)
%e A329741             (2,1)  (3,1)    (2,3)    (2,4)
%e A329741                    (1,1,2)  (3,2)    (4,2)
%e A329741                    (1,2,1)  (4,1)    (5,1)
%e A329741                    (2,1,1)  (1,1,3)  (1,1,4)
%e A329741                             (1,2,2)  (1,2,3)
%e A329741                             (1,3,1)  (1,3,2)
%e A329741                             (2,1,2)  (1,4,1)
%e A329741                             (2,2,1)  (2,1,3)
%e A329741                             (3,1,1)  (2,3,1)
%e A329741                                      (3,1,2)
%e A329741                                      (3,2,1)
%e A329741                                      (4,1,1)
%t A329741 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A329741 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[Length/@Split[Sort[#]]]&]],{n,20}]
%Y A329741 Looking at run-lengths instead of multiplicities gives A329766.
%Y A329741 The complete case is A329748.
%Y A329741 Complete compositions are A107429.
%Y A329741 Cf. A000740, A008965, A098504, A242882, A244164, A329738, A329739, A329740.
%K A329741 nonn
%O A329741 0,4
%A A329741 _Gus Wiseman_, Nov 20 2019
%E A329741 a(0), a(21)-a(37) from _Alois P. Heinz_, Nov 21 2019

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