cp's OEIS Frontend

A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

%I A365831 #6 Sep 30 2023 09:21:44
%S A365831 0,0,1,1,2,3,3,4,6,8,9,11,13,16,21,25,31,36,43,50,59,69,82,96,113,131,
%T A365831 155,179,208,239,276,315,362,414,472,539,614,698,795,902,1023,1158,
%U A365831 1311,1479,1672,1881,2118,2377,2671,2991,3354,3748,4194,4679,5223,5815
%N A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.
%e A365831 The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).
%e A365831 The a(2) = 1 through a(10) = 9 strict partitions:
%e A365831   (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (10)
%e A365831             (3,1)  (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)
%e A365831                    (4,1)  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)
%e A365831                                  (6,1)  (7,1)    (7,2)    (8,2)
%e A365831                                         (4,3,1)  (8,1)    (9,1)
%e A365831                                         (5,2,1)  (4,3,2)  (5,3,2)
%e A365831                                                  (5,3,1)  (5,4,1)
%e A365831                                                  (6,2,1)  (6,3,1)
%e A365831                                                           (7,2,1)
%t A365831 nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
%t A365831 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[nmz[#]]>0&]],{n,0,15}]
%Y A365831 For parts instead of sums we have ranks A080259, A055932.
%Y A365831 The strict complement is A188431, non-strict A126796 (ranks A325781).
%Y A365831 Row sums of A365545 without the first column, non-strict A365923.
%Y A365831 The non-strict version is A365924, ranks A365830.
%Y A365831 A000041 counts integer partitions, strict A000009.
%Y A365831 A046663 counts partitions w/o a submultiset summing to k, strict A365663.
%Y A365831 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A365831 A325799 counts non-subset-sums of prime indices.
%Y A365831 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365831 Cf. A006827, A047967, A299701, A304792, A364350, A365658, A365918, A365921.
%K A365831 nonn
%O A365831 0,5
%A A365831 _Gus Wiseman_, Sep 28 2023