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Number of compositions of n whose multiplicities are distinct and cover an initial interval of positive integers.
12

%I #8 Nov 21 2019 22:14:53

%S 1,1,1,1,4,7,4,10,10,10,73,196,133,379,319,379,502,805,562,1108,13648,

%T 51448,51691,115174,140011,178597,203617,329737,292300,456703,456160,

%U 608386,633466,898186,823009,39014392,190352269,266293795,493345615,834326995,947714938

%N Number of compositions of n whose multiplicities are distinct and 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(9) = 10 compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9)

%e (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6) (1,1,7)

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

%e (2,1,1) (1,3,1) (4,1,1) (1,5,1) (2,2,4) (1,7,1)

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

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

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

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

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

%e (5,1,1) (6,1,1) (7,1,1)

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

%Y The version allowing repeated multiplicities is A329741.

%Y Complete compositions are A107429.

%Y Compositions whose multiplicities are distinct are A242882.

%Y Cf. A059966, A098504, A244164, A274174, A329739, A329766, A329748.

%K nonn

%O 0,5

%A _Gus Wiseman_, Nov 21 2019

%E a(21)-a(40) from _Alois P. Heinz_, Nov 21 2019