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A365918 Number of distinct non-subset-sums of integer partitions of n.

%I A365918 #17 Sep 26 2023 01:58:56
%S A365918 0,1,2,6,8,19,24,46,60,101,124,206,250,378,462,684,812,1165,1380,1927,
%T A365918 2268,3108,3606,4862,5648,7474,8576,11307,12886,16652,19050,24420,
%U A365918 27584,35225,39604,49920,56370,70540,78608,98419,109666,135212,151176,185875,205308
%N A365918 Number of distinct non-subset-sums of integer partitions of n.
%C A365918 For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.
%F A365918 a(n) = (n+1)*A000041(n) - A304792(n).
%e A365918 The a(6) = 19 ways, showing each partition and its non-subset-sums:
%e A365918        (6): 1,2,3,4,5
%e A365918       (51): 2,3,4
%e A365918       (42): 1,3,5
%e A365918      (411): 3
%e A365918       (33): 1,2,4,5
%e A365918      (321):
%e A365918     (3111):
%e A365918      (222): 1,3,5
%e A365918     (2211):
%e A365918    (21111):
%e A365918   (111111):
%t A365918 Table[Total[Length[Complement[Range[n],Total/@Subsets[#]]]&/@IntegerPartitions[n]],{n,10}]
%o A365918 (Python)
%o A365918 # uses A304792_T
%o A365918 from sympy import npartitions
%o A365918 def A365918(n): return (n+1)*npartitions(n)-A304792_T(n,n,(0,),1) # _Chai Wah Wu_, Sep 25 2023
%Y A365918 Row sums of A046663, strict A365663.
%Y A365918 The zero-full complement (subset-sums) is A304792.
%Y A365918 The strict case is A365922.
%Y A365918 Weighted row-sums of A365923, rank statistic A325799, complement A365658.
%Y A365918 A000041 counts integer partitions, strict A000009.
%Y A365918 A126796 counts complete partitions, ranks A325781, strict A188431.
%Y A365918 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365918 A365924 counts incomplete partitions, ranks A365830, strict A365831.
%Y A365918 Cf. A006827, A122768, A364272, A364350, A364839, A365919, A365921.
%K A365918 nonn
%O A365918 1,3
%A A365918 _Gus Wiseman_, Sep 23 2023
%E A365918 a(21)-a(45) from _Chai Wah Wu_, Sep 25 2023