%I #9 Jan 07 2020 13:03:56

%S 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 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 8,1,1,1,1,10,49,110,141,111,63,29,9,1,1,1

%N 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 Andrew Howroyd, <a href="/A330461/b330461.txt">Table of n, a(n) for n = 0..1325</a>

%F 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 Array begins:

%e k=0 k=1 k=2 k=3 k=4 k=5 k=6

%e -----------------------------

%e n=0: 1 1 1 1 1 1 1

%e n=1: 1 1 1 1 1 1 1

%e n=2: 1 1 1 1 1 1 1

%e n=3: 1 2 3 4 5 6 7

%e n=4: 1 2 4 7 11 16 22

%e n=5: 1 3 7 14 25 41 63

%e n=6: 1 4 12 29 60 111 189

%e For example, the A(5,3) = 14 partitions are:

%e {{5}} {{1}}{{4}}

%e {{14}} {{2}}{{3}}

%e {{23}} {{1}}{{13}}

%e {{1}{4}} {{2}}{{12}}

%e {{2}{3}} {{1}}{{1}{3}}

%e {{1}{13}} {{2}}{{1}{2}}

%e {{2}{12}} {{1}}{{1}{12}}

%t spl[n_,0]:={n};

%t spl[n_,k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}],UnsameQ@@#&];

%t Table[Length[spl[n-k,k]],{n,0,10},{k,0,n}]

%o (PARI)

%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}

%o 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 { my(A=M(7)); for(i=1, #A, print(A[i,])) } \\ _Andrew Howroyd_, Dec 31 2019

%Y Columns are A000012 (k = 0), A000009 (k = 1), A050342 (k = 2), A050343 (k = 3), A050344 (k = 4).

%Y The non-strict version is A290353.

%Y Cf. A001970, A004111, A007713, A060016, A273873, A279375, A279785, A294617, A306186, A323718, A323790, A330462.

%K nonn,tabl

%O 0,12

%A _Gus Wiseman_, Dec 18 2019