cp's OEIS Frontend

A306186 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.

%I A306186 #11 May 11 2021 17:22:31
%S A306186 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 A306186 12,1,22,910,2335,1430,455,78,14,1,30,3017,11992,10179,3505,775,105,
%U A306186 16,1,42,9945,66810,74299,31881,7297,1218,136,18,1,56
%N A306186 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 A306186 Array begins:
%e A306186       k=1:  k=2:  k=3:  k=4:  k=5:  k=6:
%e A306186   n=1:  1     1     1     1     1     1
%e A306186   n=2:  2     4     6     8    10    12
%e A306186   n=3:  3    10    21    36    55    78
%e A306186   n=4:  5    33   104   238   455   775
%e A306186   n=5:  7    91   452  1430  3505  7297
%e A306186   n=6: 11   298  2335 10179 31881 80897
%e A306186 Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:
%e A306186   {{111}}          {{112}}          {{123}}
%e A306186   {{1}{11}}        {{1}{12}}        {{1}{23}}
%e A306186   {{1}}{{11}}      {{2}{11}}        {{1}}{{23}}
%e A306186   {{1}{1}{1}}      {{1}}{{12}}      {{1}{2}{3}}
%e A306186   {{1}}{{1}{1}}    {{1}{1}{2}}      {{1}}{{2}{3}}
%e A306186   {{1}}{{1}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}
%e A306186                    {{1}}{{1}{2}}
%e A306186                    {{2}}{{1}{1}}
%e A306186                    {{1}}{{1}}{{2}}
%t A306186 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A306186 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A306186 undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
%t A306186 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 A306186 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 A306186 strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];
%t A306186 kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];
%t A306186 Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]
%Y A306186 Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).
%Y A306186 Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).
%Y A306186 Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.
%K A306186 nonn,tabl,more
%O A306186 1,2
%A A306186 _Gus Wiseman_, Jan 27 2019
%E A306186 a(46)-a(56) from _Robert Price_, May 11 2021