A366127 - OEIS

%I #6 Oct 05 2023 21:30:50

%S 1,2,4,6,11,15,25,35,53,72,108

%N Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

%C A non-subset-sum of a multiset of positive integers summing to n is an element of {1..n} that is not the sum of any submultiset. A multiset is incomplete if it has at least one non-subset-sum.

%e The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).

%e The a(1) = 1 through a(6) = 15 multisets:

%e   {2}  {3}    {4}      {5}        {6}          {7}

%e        {1,3}  {1,4}    {1,5}      {1,6}        {1,7}

%e               {2,2}    {2,3}      {2,4}        {2,5}

%e               {1,1,4}  {1,1,5}    {3,3}        {3,4}

%e                        {1,2,5}    {1,1,6}      {1,1,7}

%e                        {1,1,1,5}  {1,2,6}      {1,2,7}

%e                                   {1,3,3}      {1,3,4}

%e                                   {2,2,2}      {2,2,3}

%e                                   {1,1,1,6}    {1,1,1,7}

%e                                   {1,1,2,6}    {1,1,2,7}

%e                                   {1,1,1,1,6}  {1,1,3,7}

%e                                                {1,2,2,7}

%e                                                {1,1,1,1,7}

%e                                                {1,1,1,2,7}

%e                                                {1,1,1,1,1,7}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];

%t Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n],Max@@nmz[#]==n&]],{n,5}]

%Y For least instead of greatest we have A126796, ranks A325781, strict A188431.

%Y These multisets have ranks A365830.

%Y Counts appearances of n in the rank statistic A365920.

%Y Column sums of A365921.

%Y These multisets counted by sum are A365924, strict A365831.

%Y The strict case is A366129.

%Y A000041 counts integer partitions, strict A000009.

%Y A046663 counts partitions without a submultiset summing k, strict A365663.

%Y A325799 counts non-subset-sums of prime indices.

%Y A364350 counts combination-free strict partitions, complement A364839.

%Y A365543 counts partitions with a submultiset summing to k.

%Y A365661 counts strict partitions w/ a subset summing to k.

%Y A365918 counts non-subset-sums of partitions.

%Y A365923 counts partitions by non-subset sums, strict A365545.

%Y Cf. A006827, A276024, A284640, A304792, A365658, A365919, A365925.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Sep 30 2023