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 A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n. 10
 1, 1, 2, 6, 43, 440, 7158, 151414 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. LINKS Table of n, a(n) for n=0..7. EXAMPLE The a(2) = 2 and a(3) = 6 multisystems: {1,1} {{1},{1,1}} {1,2} {{1},{1,2}} {{1},{2,3}} {{2},{1,1}} {{2},{1,3}} {{3},{1,2}} The a(4) = 43 multisystems (commas and outer brackets elided): {{1}}{{1}{11}} {{1}}{{1}{12}} {{1}}{{1}{22}} {{1}}{{1}{23}} {{1}}{{2}{34}} {{11}}{{1}{1}} {{11}}{{1}{2}} {{11}}{{2}{2}} {{11}}{{2}{3}} {{12}}{{3}{4}} {{1}}{{2}{11}} {{1}}{{2}{12}} {{1}}{{2}{13}} {{1}}{{3}{24}} {{12}}{{1}{1}} {{12}}{{1}{2}} {{12}}{{1}{3}} {{13}}{{2}{4}} {{2}}{{1}{11}} {{2}}{{1}{12}} {{1}}{{3}{12}} {{1}}{{4}{23}} {{2}}{{2}{11}} {{13}}{{1}{2}} {{14}}{{2}{3}} {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{1}{34}} {{2}}{{3}{11}} {{2}}{{3}{14}} {{23}}{{1}{1}} {{23}}{{1}{4}} {{3}}{{1}{12}} {{2}}{{4}{13}} {{3}}{{2}{11}} {{24}}{{1}{3}} {{3}}{{1}{24}} {{3}}{{2}{14}} {{3}}{{4}{12}} {{34}}{{1}{2}} {{4}}{{1}{23}} {{4}}{{2}{13}} {{4}}{{3}{12}} MATHEMATICA strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n]; sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; totm[m_]:=Prepend[Join@@Table[totm[p], {p, Select[mps[m], 1

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Last modified November 29 13:49 EST 2023. Contains 367445 sequences. (Running on oeis4.)