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

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, 36, 21, 36, 36, 52, 11, 19, 29, 38, 31, 52, 52, 74, 29, 52, 66, 92, 52, 92, 92, 135, 19, 38, 52, 74, 52, 92, 92, 135, 38, 74, 92, 135, 74, 135, 135, 203, 15, 30, 47

Offset: 0

Author

Gus Wiseman, Apr 16 2020

Keywords

• nonn

Comments

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.

Examples

			The a(1) = 1 through a(11) = 11 multiset partitions:
  {1}  {11}    {12}    {111}      {112}      {122}      {123}
       {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{22}    {1}{23}
                       {1}{1}{1}  {2}{11}    {2}{12}    {2}{13}
                                  {1}{1}{2}  {1}{2}{2}  {3}{12}
                                                        {1}{2}{3}
  {1111}        {1112}        {1122}        {1123}
  {1}{111}      {1}{112}      {1}{122}      {1}{123}
  {11}{11}      {11}{12}      {11}{22}      {11}{23}
  {1}{1}{11}    {2}{111}      {12}{12}      {12}{13}
  {1}{1}{1}{1}  {1}{1}{12}    {2}{112}      {2}{113}
                {1}{2}{11}    {1}{1}{22}    {3}{112}
                {1}{1}{1}{2}  {1}{2}{12}    {1}{1}{23}
                              {2}{2}{11}    {1}{2}{13}
                              {1}{1}{2}{2}  {1}{3}{12}
                                            {2}{3}{11}
                                            {1}{1}{2}{3}
		

Crossrefs

The described multiset has A000120 distinct parts.

The sum of the described multiset is A029931.

Multisets of compositions are A034691.

The described multiset is a row of A095684.

Combinatory separations of normal multisets are A269134.

The product of the described multiset is A284001.

The version for prime indices is A318284.

The version counting combinatory separations is A334030.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Sum is A070939.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Length of Lyndon factorization is A329312.

- Dealings are counted by A333939.

- Distinct parts are counted by A334028.

- Length of co-Lyndon factorization is A334029.

Cf. A057335, A065609, A275692, A292884, A318560, A318563, A326774, A333764, A333765, A333940.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[Times@@Prime/@ptnToNorm[stc[n]]]],{n,0,30}]

Formula

a(n) = A001055(A057335(n)).