A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

1, 1, 2, 2, 3, 6, 8, 9, 14, 16, 25, 30, 41, 52, 69, 83, 105, 129, 164, 208, 263, 315, 388, 449, 573, 694

Offset: 0

Author

Gus Wiseman, Mar 26 2025

Keywords

• nonn
• more

Examples

			The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
  {4}     {5}       {6}         {7}        {8}
  {22}    {1}{4}    {33}        {1}{6}     {44}
  {1}{3}  {2}{3}    {1}{5}      {2}{5}     {1}{7}
          {11}{3}   {2}{4}      {3}{4}     {2}{6}
          {1}{22}   {11}{4}     {11}{5}    {3}{5}
          {2}{111}  {11}{22}    {1}{33}    {11}{6}
                    {1}{2}{3}   {3}{22}    {2}{33}
                    {1}{11}{3}  {1}{2}{4}  {11}{33}
                                {3}{1111}  {11}{222}
                                           {1}{2}{5}
                                           {1}{3}{4}
                                           {1}{3}{22}
                                           {1}{4}{111}
                                           {1}{111}{22}
		

Crossrefs

For distinct blocks instead of block-sums we have A000726, ranks A004709.

Twice-partitions of this type (constant with distinct) are counted by A279786.

MM-numbers of these multiset partitions are A326535 /\ A355743.

For no choices we have A381717, ranks A381636, zeros of A381635.

The Heinz numbers of these partitions are A381991, positions of 1 in A381635.

Normal multiset partitions of this type are counted by A382203.

For at least one choice we have A382427.

For strict instead of constant blocks we have A382460, ranks A381870.

A000041 counts integer partitions, strict A000009.

A000688 counts factorizations into prime powers, see A381455, A381453.

A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.

A050361 counts factorizations into distinct prime powers.

Cf. A006171, A047966, A279784, A293511, A295935, A353864, A381633, A381716, A381990, A381992, A381993, A382079.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]