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Number of compositions of n whose multiplicities cover an initial interval of positive integers.
7

%I #8 Nov 21 2019 18:59:19

%S 1,1,1,3,6,11,14,34,52,114,225,464,539,1183,1963,3753,6120,11207,

%T 19808,38254,77194,147906,224853,374216,611081,1099933,2129347,

%U 3336099,5816094,9797957,17577710,29766586,53276392,93139668,163600815,324464546,637029845,1010826499

%N Number of 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.

%e The a(1) = 1 through a(6) = 14 compositions:

%e (1) (2) (3) (4) (5) (6)

%e (1,2) (1,3) (1,4) (1,5)

%e (2,1) (3,1) (2,3) (2,4)

%e (1,1,2) (3,2) (4,2)

%e (1,2,1) (4,1) (5,1)

%e (2,1,1) (1,1,3) (1,1,4)

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

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

%e (2,1,2) (1,4,1)

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

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

%e (3,1,2)

%e (3,2,1)

%e (4,1,1)

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

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[Length/@Split[Sort[#]]]&]],{n,20}]

%Y Looking at run-lengths instead of multiplicities gives A329766.

%Y The complete case is A329748.

%Y Complete compositions are A107429.

%Y Cf. A000740, A008965, A098504, A242882, A244164, A329738, A329739, A329740.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 20 2019

%E a(0), a(21)-a(37) from _Alois P. Heinz_, Nov 21 2019