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Number of finite sets of integer partitions with all equal sums and total sum n.
1

%I #10 Dec 14 2022 10:56:13

%S 1,1,2,3,6,7,14,15,32,31,63,56,142,101,240,211,467,297,985,490,1524,

%T 1247,2542,1255,6371,1979,7486,7070,14128,4565,32953,6842,42229,37863,

%U 56266,17887,192914,21637,145820,197835,371853,44583,772740,63261,943966,1124840

%N Number of finite sets of integer partitions with all equal sums and total sum n.

%F a(n) = Sum_{d|n} binomial(A000041(d),n/d).

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

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

%e {(11)} {(21)} {(22)} {(32)} {(33)}

%e {(111)} {(31)} {(41)} {(42)}

%e {(211)} {(221)} {(51)}

%e {(1111)} {(311)} {(222)}

%e {(2),(11)} {(2111)} {(321)}

%e {(11111)} {(411)}

%e {(2211)}

%e {(3111)}

%e {(21111)}

%e {(111111)}

%e {(3),(21)}

%e {(3),(111)}

%e {(21),(111)}

%t Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]

%o (PARI) a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ _Michel Marcus_, Dec 14 2022

%Y This is the constant-sum case of A261049, ordered A358906.

%Y The version for all different sums is A271619, ordered A336342.

%Y Allowing repetition gives A305551, ordered A279787.

%Y The version for compositions instead of partitions is A358904.

%Y A001970 counts multisets of partitions.

%Y A034691 counts multisets of compositions, ordered A133494.

%Y A098407 counts sets of compositions, ordered A358907.

%Y Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 14 2022