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Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.
21

%I #11 May 11 2021 17:22:31

%S 1,2,1,3,4,1,5,10,6,1,7,33,21,8,1,11,91,104,36,10,1,15,298,452,238,55,

%T 12,1,22,910,2335,1430,455,78,14,1,30,3017,11992,10179,3505,775,105,

%U 16,1,42,9945,66810,74299,31881,7297,1218,136,18,1,56

%N Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

%e Array begins:

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

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

%e n=2: 2 4 6 8 10 12

%e n=3: 3 10 21 36 55 78

%e n=4: 5 33 104 238 455 775

%e n=5: 7 91 452 1430 3505 7297

%e n=6: 11 298 2335 10179 31881 80897

%e Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:

%e {{111}} {{112}} {{123}}

%e {{1}{11}} {{1}{12}} {{1}{23}}

%e {{1}}{{11}} {{2}{11}} {{1}}{{23}}

%e {{1}{1}{1}} {{1}}{{12}} {{1}{2}{3}}

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

%e {{1}}{{1}}{{1}} {{2}}{{11}} {{1}}{{2}}{{3}}

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

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

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

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];

%t expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Apply[Rule,Table[{undats[m][[i]],i},{i,Length[undats[m]]}],{1}]],First[Sort[expnorm[m,1]]]]];

%t expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#1>=aft&]}]},Union@@(expnorm[#1,aft+1]&)/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,_[___]]],{par,First/@Position[mx,Max[mx]]}]]]];

%t strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];

%t kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];

%t Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]

%Y Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).

%Y Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).

%Y Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

%K nonn,tabl,more

%O 1,2

%A _Gus Wiseman_, Jan 27 2019

%E a(46)-a(56) from _Robert Price_, May 11 2021