cp's OEIS Frontend

A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

%I A318562 #5 Aug 29 2018 16:50:22
%S A318562 1,4,10,32,80,239,605,1670,4251
%N A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.
%C A318562 A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.
%C A318562 A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.
%e A318562 The a(3) = 10 combinatory separations:
%e A318562   111<={111}
%e A318562   111<={1,11}
%e A318562   111<={1,1,1}
%e A318562   112<={112}
%e A318562   112<={1,11}
%e A318562   112<={1,12}
%e A318562   112<={1,1,1}
%e A318562   123<={123}
%e A318562   123<={1,12}
%e A318562   123<={1,1,1}
%t A318562 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318562 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A318562 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A318562 normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t A318562 strnormQ[m_]:=OrderedQ[Length/@Split[m],GreaterEqual];
%t A318562 Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@strnormQ/@#[[2]]&]],{n,6}]
%Y A318562 Cf. A000041, A007716, A255906, A265947, A269134.
%Y A318562 Cf. A317533, A317791, A318396, A318559, A318563, A318564, A318565, A318566.
%K A318562 nonn,more
%O A318562 1,2
%A A318562 _Gus Wiseman_, Aug 29 2018