cp's OEIS Frontend

A318563 Number of combinatory separations of strongly normal multisets of weight n.

%I A318563 #5 Aug 29 2018 16:50:32
%S A318563 1,4,10,33,85,272,730,2197,6133
%N A318563 Number of combinatory separations of strongly normal multisets of weight n.
%C A318563 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 A318563 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 A318563 The a(3) = 10 combinatory separations:
%e A318563   111<={111}
%e A318563   111<={1,11}
%e A318563   111<={1,1,1}
%e A318563   112<={112}
%e A318563   112<={1,11}
%e A318563   112<={1,12}
%e A318563   112<={1,1,1}
%e A318563   123<={123}
%e A318563   123<={1,12}
%e A318563   123<={1,1,1}
%t A318563 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318563 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A318563 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A318563 normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t A318563 Table[Length[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}]],{n,7}]
%Y A318563 Cf. A000041, A007716, A255906, A265947, A269134.
%Y A318563 Cf. A317533, A317791, A318396, A318559, A318562, A318564, A318565, A318566.
%K A318563 nonn,more
%O A318563 1,2
%A A318563 _Gus Wiseman_, Aug 29 2018