|
%I #8 Jun 21 2019 22:44:20
%S 0,0,0,1,17,170,1455,11678,92871,752473
%N Number of unsortable normal multiset partitions of weight n.
%C A multiset partition is normal if it covers an initial interval of positive integers. It is unsortable if no permutation has an ordered concatenation, or equivalently if the concatenation of its lexicographically-ordered parts is not weakly increasing. For example, the multiset partition {{1,2},{1,1,1},{2,2,2}} is sortable because the permutation ((1,1,1),(1,2),(2,2,2)) has concatenation (1,1,1,1,2,2,2,2), which is weakly increasing.
%F A255906(n) = a(n) + A326212(n).
%e The a(3) = 1 and a(4) = 17 multiset partitions:
%e {{1,3},{2}} {{1,1,3},{2}}
%e {{1,2},{1,2}}
%e {{1,2},{1,3}}
%e {{1,2,3},{2}}
%e {{1,2,4},{3}}
%e {{1,3},{2,2}}
%e {{1,3},{2,3}}
%e {{1,3},{2,4}}
%e {{1,3,3},{2}}
%e {{1,3,4},{2}}
%e {{1,4},{2,3}}
%e {{1},{1,3},{2}}
%e {{1},{2,4},{3}}
%e {{1,3},{2},{2}}
%e {{1,3},{2},{3}}
%e {{1,3},{2},{4}}
%e {{1,4},{2},{3}}
%t lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%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 Table[Length[Select[Sort[#,lexsort]&/@Join@@mps/@allnorm[n],!OrderedQ[Join@@#]&]],{n,0,5}]
%Y Unsortable set partitions are A058681.
%Y Sortable normal multiset partitions are A326212.
%Y Non-crossing normal multiset partitions are A324171.
%Y MM-numbers of unsortable multiset partitions are A326258.
%Y Cf. A000108, A016098, A255906, A324170.
%Y Cf. A326209, A326210, A326243, A326250, A326255, A326256.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Jun 19 2019
|
