cp's OEIS Frontend

A334030 Number of combinatory separations of a multiset whose multiplicities are the parts of the n-th composition in standard order.

%I A334030 #5 Apr 16 2020 18:48:56
%S A334030 1,1,2,2,3,4,4,3,5,7,8,8,7,9,8,5,7,12,15,14,15,17,18,13,12,17,17,16,
%T A334030 14,16,13,7,11,19,27,26,27,37,37,25,27,37,33,34,37,40,36,22,19,32,37,
%U A334030 33,37,38,40,28,26,33,34,30,25,28,22,11,15,30,44,42,51,68
%N A334030 Number of combinatory separations of a multiset whose multiplicities are the parts of the n-th composition in standard order.
%C A334030 A multiset is normal if it covers an initial interval of positive integers. 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 (3,3,5,5,5,6) is (1,1,2,2,2,3).
%C A334030 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 has two equal elements.
%e A334030 The combinatory separations for n = 1, 3, 5, 9, 10, 13 (heads not shown):
%e A334030   (1)  (12)    (112)      (1112)        (1122)        (1223)
%e A334030        (1)(1)  (1)(11)    (1)(111)      (11)(11)      (1)(112)
%e A334030                (1)(12)    (1)(112)      (1)(112)      (11)(12)
%e A334030                (1)(1)(1)  (11)(12)      (1)(122)      (1)(122)
%e A334030                           (1)(1)(11)    (12)(12)      (1)(123)
%e A334030                           (1)(1)(12)    (1)(1)(11)    (12)(12)
%e A334030                           (1)(1)(1)(1)  (1)(1)(12)    (1)(1)(11)
%e A334030                                         (1)(1)(1)(1)  (1)(1)(12)
%e A334030                                                       (1)(1)(1)(1)
%t A334030 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334030 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A334030 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A334030 normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t A334030 ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
%t A334030 Table[Length[Union[Table[Sort[normize/@m],{m,mps[ptnToNorm[stc[n]]]}]]],{n,0,100}]
%Y A334030 Multisets of compositions are A034691.
%Y A334030 The described multiset is a row of A095684.
%Y A334030 Combinatory separations of normal multisets are A269134.
%Y A334030 Shuffles of compositions are counted by A292884.
%Y A334030 Combinatory separations of prime indices are A318559.
%Y A334030 The version for prime indices is A318560.
%Y A334030 Combinatory separations of strongly normal multisets are A318563.
%Y A334030 Multiset partitions of the described multiset are A333942.
%Y A334030 All of the following pertain to compositions in standard order (A066099):
%Y A334030 - Length is A000120.
%Y A334030 - Sum is A070939.
%Y A334030 - Strict compositions are A233564.
%Y A334030 - Constant compositions are A272919.
%Y A334030 - Length of Lyndon factorization is A329312.
%Y A334030 - Dealings are counted by A333939.
%Y A334030 Cf. A065609, A275692, A318284, A326774, A333764, A333765, A333940.
%K A334030 nonn
%O A334030 0,3
%A A334030 _Gus Wiseman_, Apr 16 2020