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A330461
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Array read by antidiagonals where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.
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5
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 4, 7, 7, 5, 1, 1, 1, 1, 5, 12, 14, 11, 6, 1, 1, 1, 1, 6, 19, 29, 25, 16, 7, 1, 1, 1, 1, 8, 30, 57, 60, 41, 22, 8, 1, 1, 1, 1, 10, 49, 110, 141, 111, 63, 29, 9, 1, 1, 1
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OFFSET
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0,12
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LINKS
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FORMULA
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Column k is the k-th weigh transform of the all-ones sequence. The weigh transform of a sequence b has generating function Product_{i > 0} (1 + x^i)^b(i).
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EXAMPLE
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Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6
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n=0: 1 1 1 1 1 1 1
n=1: 1 1 1 1 1 1 1
n=2: 1 1 1 1 1 1 1
n=3: 1 2 3 4 5 6 7
n=4: 1 2 4 7 11 16 22
n=5: 1 3 7 14 25 41 63
n=6: 1 4 12 29 60 111 189
For example, the A(5,3) = 14 partitions are:
{{5}} {{1}}{{4}}
{{14}} {{2}}{{3}}
{{23}} {{1}}{{13}}
{{1}{4}} {{2}}{{12}}
{{2}{3}} {{1}}{{1}{3}}
{{1}{13}} {{2}}{{1}{2}}
{{2}{12}} {{1}}{{1}{12}}
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MATHEMATICA
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spl[n_, 0]:={n};
spl[n_, k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#, k-1]&/@ptn]], {ptn, IntegerPartitions[n]}], UnsameQ@@#&];
Table[Length[spl[n-k, k]], {n, 0, 10}, {k, 0, n}]
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PROG
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(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
M(n, k=n)={my(L=List(), v=vector(n, i, 1)); listput(L, concat([1], v)); for(j=1, k, v=WeighT(v); listput(L, concat([1], v))); Mat(Col(L))~}
{ my(A=M(7)); for(i=1, #A, print(A[i, ])) } \\ Andrew Howroyd, Dec 31 2019
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CROSSREFS
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Cf. A001970, A004111, A007713, A060016, A273873, A279375, A279785, A294617, A306186, A323718, A323790, A330462.
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KEYWORD
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AUTHOR
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STATUS
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approved
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