A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

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

Offset: 1

Author

Gus Wiseman, Sep 30 2023

Keywords

• nonn
• more

Comments

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.

Examples

			The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).
The a(1) = 1 through a(6) = 15 multisets:
  {2}  {3}    {4}      {5}        {6}          {7}
       {1,3}  {1,4}    {1,5}      {1,6}        {1,7}
              {2,2}    {2,3}      {2,4}        {2,5}
              {1,1,4}  {1,1,5}    {3,3}        {3,4}
                       {1,2,5}    {1,1,6}      {1,1,7}
                       {1,1,1,5}  {1,2,6}      {1,2,7}
                                  {1,3,3}      {1,3,4}
                                  {2,2,2}      {2,2,3}
                                  {1,1,1,6}    {1,1,1,7}
                                  {1,1,2,6}    {1,1,2,7}
                                  {1,1,1,1,6}  {1,1,3,7}
                                               {1,2,2,7}
                                               {1,1,1,1,7}
                                               {1,1,1,2,7}
                                               {1,1,1,1,1,7}
		

Crossrefs

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

These multisets have ranks A365830.

Counts appearances of n in the rank statistic A365920.

Column sums of A365921.

These multisets counted by sum are A365924, strict A365831.

The strict case is A366129.

A000041 counts integer partitions, strict A000009.

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

A325799 counts non-subset-sums of prime indices.

A364350 counts combination-free strict partitions, complement A364839.

A365543 counts partitions with a submultiset summing to k.

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

A365918 counts non-subset-sums of partitions.

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

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n],Max@@nmz[#]==n&]],{n,5}]