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A330461 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.

%I A330461 #9 Jan 07 2020 13:03:56
%S A330461 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,
%T A330461 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,
%U A330461 8,1,1,1,1,10,49,110,141,111,63,29,9,1,1,1
%N A330461 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.
%H A330461 Andrew Howroyd, <a href="/A330461/b330461.txt">Table of n, a(n) for n = 0..1325</a>
%F A330461 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).
%e A330461 Array begins:
%e A330461        k=0 k=1 k=2 k=3 k=4 k=5 k=6
%e A330461       -----------------------------
%e A330461   n=0:  1   1   1   1   1   1   1
%e A330461   n=1:  1   1   1   1   1   1   1
%e A330461   n=2:  1   1   1   1   1   1   1
%e A330461   n=3:  1   2   3   4   5   6   7
%e A330461   n=4:  1   2   4   7  11  16  22
%e A330461   n=5:  1   3   7  14  25  41  63
%e A330461   n=6:  1   4  12  29  60 111 189
%e A330461 For example, the A(5,3) = 14 partitions are:
%e A330461   {{5}}      {{1}}{{4}}
%e A330461   {{14}}     {{2}}{{3}}
%e A330461   {{23}}     {{1}}{{13}}
%e A330461   {{1}{4}}   {{2}}{{12}}
%e A330461   {{2}{3}}   {{1}}{{1}{3}}
%e A330461   {{1}{13}}  {{2}}{{1}{2}}
%e A330461   {{2}{12}}  {{1}}{{1}{12}}
%t A330461 spl[n_,0]:={n};
%t A330461 spl[n_,k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}],UnsameQ@@#&];
%t A330461 Table[Length[spl[n-k,k]],{n,0,10},{k,0,n}]
%o A330461 (PARI)
%o A330461 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A330461 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))~}
%o A330461 { my(A=M(7)); for(i=1, #A, print(A[i,])) } \\ _Andrew Howroyd_, Dec 31 2019
%Y A330461 Columns are A000012 (k = 0), A000009 (k = 1), A050342 (k = 2), A050343 (k = 3), A050344 (k = 4).
%Y A330461 The non-strict version is A290353.
%Y A330461 Cf. A001970, A004111, A007713, A060016, A273873, A279375, A279785, A294617, A306186, A323718, A323790, A330462.
%K A330461 nonn,tabl
%O A330461 0,12
%A A330461 _Gus Wiseman_, Dec 18 2019