cp's OEIS Frontend

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

%I A382301 #9 Mar 28 2025 22:54:32
%S A382301 1,1,2,2,3,6,8,9,14,16,25,30,41,52,69,83,105,129,164,208,263,315,388,
%T A382301 449,573,694
%N A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.
%e A382301 The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
%e A382301   {4}     {5}       {6}         {7}        {8}
%e A382301   {22}    {1}{4}    {33}        {1}{6}     {44}
%e A382301   {1}{3}  {2}{3}    {1}{5}      {2}{5}     {1}{7}
%e A382301           {11}{3}   {2}{4}      {3}{4}     {2}{6}
%e A382301           {1}{22}   {11}{4}     {11}{5}    {3}{5}
%e A382301           {2}{111}  {11}{22}    {1}{33}    {11}{6}
%e A382301                     {1}{2}{3}   {3}{22}    {2}{33}
%e A382301                     {1}{11}{3}  {1}{2}{4}  {11}{33}
%e A382301                                 {3}{1111}  {11}{222}
%e A382301                                            {1}{2}{5}
%e A382301                                            {1}{3}{4}
%e A382301                                            {1}{3}{22}
%e A382301                                            {1}{4}{111}
%e A382301                                            {1}{111}{22}
%t A382301 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A382301 pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
%t A382301 Table[Length[Select[IntegerPartitions[n],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]
%Y A382301 For distinct blocks instead of block-sums we have A000726, ranks A004709.
%Y A382301 Twice-partitions of this type (constant with distinct) are counted by A279786.
%Y A382301 MM-numbers of these multiset partitions are A326535 /\ A355743.
%Y A382301 For no choices we have A381717, ranks A381636, zeros of A381635.
%Y A382301 The Heinz numbers of these partitions are A381991, positions of 1 in A381635.
%Y A382301 Normal multiset partitions of this type are counted by A382203.
%Y A382301 For at least one choice we have A382427.
%Y A382301 For strict instead of constant blocks we have A382460, ranks A381870.
%Y A382301 A000041 counts integer partitions, strict A000009.
%Y A382301 A000688 counts factorizations into prime powers, see A381455, A381453.
%Y A382301 A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
%Y A382301 A050361 counts factorizations into distinct prime powers.
%Y A382301 Cf. A006171, A047966, A279784, A293511, A295935, A353864, A381633, A381716, A381990, A381992, A381993, A382079.
%K A382301 nonn,more
%O A382301 0,3
%A A382301 _Gus Wiseman_, Mar 26 2025