cp's OEIS Frontend

A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.

%I A367402 #8 Nov 18 2023 18:18:53
%S A367402 1,1,2,3,5,6,9,10,13,17,20,26,31,38,44,58,64,81,95,116,137,166,192,
%T A367402 233,278,330,385,459,542,636,759,879,1038,1211,1418,1656,1942,2242,
%U A367402 2618,3029,3535,4060,4735,5429,6299,7231,8346,9556,11031,12593,14482,16525
%N A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.
%C A367402 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 A367402 The partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
%e A367402 The a(1) = 1 through a(8) = 13 partitions:
%e A367402   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A367402        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A367402              (111)  (31)    (41)     (42)      (52)       (53)
%e A367402                     (211)   (221)    (51)      (61)       (62)
%e A367402                     (1111)  (2111)   (222)     (322)      (71)
%e A367402                             (11111)  (321)     (2221)     (332)
%e A367402                                      (2211)    (3211)     (2222)
%e A367402                                      (21111)   (22111)    (3221)
%e A367402                                      (111111)  (211111)   (22211)
%e A367402                                                (1111111)  (32111)
%e A367402                                                           (221111)
%e A367402                                                           (2111111)
%e A367402                                                           (11111111)
%t A367402 Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]==Union[d])&]], {n,0,15}]
%Y A367402 For parts instead of sums we have A034296, ranks A073491.
%Y A367402 For all subset-sums we have A126796, ranks A325781, strict A188431.
%Y A367402 The complement for parts instead of sums is A239955, ranks A073492.
%Y A367402 The complement for all sub-sums is A365924, ranks A365830, strict A365831.
%Y A367402 The complement is counted by A367403.
%Y A367402 The strict case is A367410, complement A367411.
%Y A367402 A000009 counts partitions covering an initial interval, ranks A055932.
%Y A367402 A086971 counts semi-sums of prime indices.
%Y A367402 A261036 counts complete partitions by maximum.
%Y A367402 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A367402 Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365661, A365918, A365921.
%K A367402 nonn
%O A367402 0,3
%A A367402 _Gus Wiseman_, Nov 17 2023