cp's OEIS Frontend

A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

%I A333942 #6 Apr 16 2020 18:48:48
%S A333942 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 A333942 36,21,36,36,52,11,19,29,38,31,52,52,74,29,52,66,92,52,92,92,135,19,
%U A333942 38,52,74,52,92,92,135,38,74,92,135,74,135,135,203,15,30,47
%N A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.
%C A333942 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 A333942 a(n) = A001055(A057335(n)).
%e A333942 The a(1) = 1 through a(11) = 11 multiset partitions:
%e A333942   {1}  {11}    {12}    {111}      {112}      {122}      {123}
%e A333942        {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{22}    {1}{23}
%e A333942                        {1}{1}{1}  {2}{11}    {2}{12}    {2}{13}
%e A333942                                   {1}{1}{2}  {1}{2}{2}  {3}{12}
%e A333942                                                         {1}{2}{3}
%e A333942   {1111}        {1112}        {1122}        {1123}
%e A333942   {1}{111}      {1}{112}      {1}{122}      {1}{123}
%e A333942   {11}{11}      {11}{12}      {11}{22}      {11}{23}
%e A333942   {1}{1}{11}    {2}{111}      {12}{12}      {12}{13}
%e A333942   {1}{1}{1}{1}  {1}{1}{12}    {2}{112}      {2}{113}
%e A333942                 {1}{2}{11}    {1}{1}{22}    {3}{112}
%e A333942                 {1}{1}{1}{2}  {1}{2}{12}    {1}{1}{23}
%e A333942                               {2}{2}{11}    {1}{2}{13}
%e A333942                               {1}{1}{2}{2}  {1}{3}{12}
%e A333942                                             {2}{3}{11}
%e A333942                                             {1}{1}{2}{3}
%t A333942 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333942 ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
%t A333942 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A333942 Table[Length[facs[Times@@Prime/@ptnToNorm[stc[n]]]],{n,0,30}]
%Y A333942 The described multiset has A000120 distinct parts.
%Y A333942 The sum of the described multiset is A029931.
%Y A333942 Multisets of compositions are A034691.
%Y A333942 The described multiset is a row of A095684.
%Y A333942 Combinatory separations of normal multisets are A269134.
%Y A333942 The product of the described multiset is A284001.
%Y A333942 The version for prime indices is A318284.
%Y A333942 The version counting combinatory separations is A334030.
%Y A333942 All of the following pertain to compositions in standard order (A066099):
%Y A333942 - Length is A000120.
%Y A333942 - Sum is A070939.
%Y A333942 - Strict compositions are A233564.
%Y A333942 - Constant compositions are A272919.
%Y A333942 - Length of Lyndon factorization is A329312.
%Y A333942 - Dealings are counted by A333939.
%Y A333942 - Distinct parts are counted by A334028.
%Y A333942 - Length of co-Lyndon factorization is A334029.
%Y A333942 Cf. A057335, A065609, A275692, A292884, A318560, A318563, A326774, A333764, A333765, A333940.
%K A333942 nonn
%O A333942 0,3
%A A333942 _Gus Wiseman_, Apr 16 2020