cp's OEIS Frontend

A367410 Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.

%I A367410 #9 Nov 18 2023 18:08:03
%S A367410 1,1,1,2,2,3,4,4,4,6,6,7,7,8,8,11,9,11,11,12,12,15,14,15,16,16,16,19,
%T A367410 18,19,22,21,21,24,22,25,26,26,26,30,28,29,32,31,32,37,35,36,38,39,39,
%U A367410 43,42,43,47,46,49,51,52,51,58
%N A367410 Number of strict integer partitions of n whose semi-sums cover an interval of positive integers.
%C A367410 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 A367410 The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is not counted under a(7).
%e A367410 The a(1) = 1 through a(9) = 6 partitions:
%e A367410   (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
%e A367410             (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
%e A367410                           (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
%e A367410                                  (3,2,1)  (6,1)  (7,1)  (7,2)
%e A367410                                                         (8,1)
%e A367410                                                         (4,3,2)
%t A367410 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#,{2}]; If[d=={},{}, Range[Min@@d, Max@@d]]==Union[d])&]], {n,0,30}]
%Y A367410 For parts instead of sums we have A001227:
%Y A367410 - non-strict A034296, ranks A073491
%Y A367410 - complement A238007
%Y A367410 - non-strict complement A239955, ranks A073492
%Y A367410 The non-binary version is A188431:
%Y A367410 - non-strict A126796, ranks A325781
%Y A367410 - complement A365831
%Y A367410 - non-strict complement A365924, ranks A365830
%Y A367410 The non-strict version is A367402.
%Y A367410 The non-strict complement is A367403.
%Y A367410 The complement is counted by A367411.
%Y A367410 A000009 counts partitions covering an initial interval, ranks A055932.
%Y A367410 A046663 counts partitions w/o submultiset summing to k, strict A365663.
%Y A367410 A365543 counts partitions w/ submultiset summing to k, strict A365661.
%Y A367410 Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.
%K A367410 nonn
%O A367410 0,4
%A A367410 _Gus Wiseman_, Nov 18 2023