|
%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
|
