cp's OEIS Frontend

A367403 Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.

%I A367403 #6 Nov 18 2023 18:18:46
%S A367403 0,0,0,0,0,1,2,5,9,13,22,30,46,63,91,118,167,216,290,374,490,626,810,
%T A367403 1022,1297,1628,2051,2551,3176,3929,4845,5963,7311,8932,10892,13227,
%U A367403 16035,19395,23397,28156,33803,40523,48439,57832,68876,81903,97212,115198
%N A367403 Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.
%C A367403 We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
%e A367403 The a(0) = 0 through a(9) = 13 partitions:
%e A367403   .  .  .  .  .  (311)  (411)   (331)    (422)     (441)
%e A367403                         (3111)  (421)    (431)     (522)
%e A367403                                 (511)    (521)     (531)
%e A367403                                 (4111)   (611)     (621)
%e A367403                                 (31111)  (3311)    (711)
%e A367403                                          (4211)    (4311)
%e A367403                                          (5111)    (5211)
%e A367403                                          (41111)   (6111)
%e A367403                                          (311111)  (33111)
%e A367403                                                    (42111)
%e A367403                                                    (51111)
%e A367403                                                    (411111)
%e A367403                                                    (3111111)
%t A367403 Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,15}]
%Y A367403 The complement for parts instead of sums is A034296, ranks A073491.
%Y A367403 The complement for all sub-sums is A126796, ranks A325781, strict A188431.
%Y A367403 For parts instead of sums we have A239955, ranks A073492.
%Y A367403 For all subset-sums we have A365924, ranks A365830, strict A365831.
%Y A367403 The complement is counted by A367402.
%Y A367403 The strict case is A367411, complement A367410.
%Y A367403 A000009 counts partitions covering an initial interval, ranks A055932.
%Y A367403 A086971 counts semi-sums of prime indices.
%Y A367403 A261036 counts complete partitions by maximum.
%Y A367403 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A367403 Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365658.
%K A367403 nonn
%O A367403 0,7
%A A367403 _Gus Wiseman_, Nov 17 2023