%I #6 Apr 16 2020 18:48:48
%S 1,1,2,2,3,4,4,5,5,7,9,11,7,11,11,15,7,12,16,21,16,26,26,36,12,21,26,
%T 36,21,36,36,52,11,19,29,38,31,52,52,74,29,52,66,92,52,92,92,135,19,
%U 38,52,74,52,92,92,135,38,74,92,135,74,135,135,203,15,30,47
%N Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.
%C A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%F a(n) = A001055(A057335(n)).
%e The a(1) = 1 through a(11) = 11 multiset partitions:
%e {1} {11} {12} {111} {112} {122} {123}
%e {1}{1} {1}{2} {1}{11} {1}{12} {1}{22} {1}{23}
%e {1}{1}{1} {2}{11} {2}{12} {2}{13}
%e {1}{1}{2} {1}{2}{2} {3}{12}
%e {1}{2}{3}
%e {1111} {1112} {1122} {1123}
%e {1}{111} {1}{112} {1}{122} {1}{123}
%e {11}{11} {11}{12} {11}{22} {11}{23}
%e {1}{1}{11} {2}{111} {12}{12} {12}{13}
%e {1}{1}{1}{1} {1}{1}{12} {2}{112} {2}{113}
%e {1}{2}{11} {1}{1}{22} {3}{112}
%e {1}{1}{1}{2} {1}{2}{12} {1}{1}{23}
%e {2}{2}{11} {1}{2}{13}
%e {1}{1}{2}{2} {1}{3}{12}
%e {2}{3}{11}
%e {1}{1}{2}{3}
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[facs[Times@@Prime/@ptnToNorm[stc[n]]]],{n,0,30}]
%Y The described multiset has A000120 distinct parts.
%Y The sum of the described multiset is A029931.
%Y Multisets of compositions are A034691.
%Y The described multiset is a row of A095684.
%Y Combinatory separations of normal multisets are A269134.
%Y The product of the described multiset is A284001.
%Y The version for prime indices is A318284.
%Y The version counting combinatory separations is A334030.
%Y All of the following pertain to compositions in standard order (A066099):
%Y - Length is A000120.
%Y - Sum is A070939.
%Y - Strict compositions are A233564.
%Y - Constant compositions are A272919.
%Y - Length of Lyndon factorization is A329312.
%Y - Dealings are counted by A333939.
%Y - Distinct parts are counted by A334028.
%Y - Length of co-Lyndon factorization is A334029.
%Y Cf. A057335, A065609, A275692, A292884, A318560, A318563, A326774, A333764, A333765, A333940.
%K nonn
%O 0,3
%A _Gus Wiseman_, Apr 16 2020