cp's OEIS Frontend

A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

%I A367411 #5 Nov 18 2023 18:07:58
%S A367411 0,0,0,0,0,0,0,1,2,2,4,5,8,10,14,16,23,27,35,42,52,61,75,89,106,126,
%T A367411 149,173,204,237,274,319,369,424,490,560,642,734,838,952,1085,1231,
%U A367411 1394,1579,1784,2011,2269,2554,2872,3225,3619,4054,4540,5077,5671,6332
%N A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.
%C A367411 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 A367411 The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
%e A367411 The a(7) = 1 through a(13) = 10 partitions:
%e A367411   (4,2,1)  (4,3,1)  (5,3,1)  (5,3,2)  (5,4,2)  (6,4,2)    (6,4,3)
%e A367411            (5,2,1)  (6,2,1)  (5,4,1)  (6,3,2)  (6,5,1)    (6,5,2)
%e A367411                              (6,3,1)  (6,4,1)  (7,3,2)    (7,4,2)
%e A367411                              (7,2,1)  (7,3,1)  (7,4,1)    (7,5,1)
%e A367411                                       (8,2,1)  (8,3,1)    (8,3,2)
%e A367411                                                (9,2,1)    (8,4,1)
%e A367411                                                (5,4,2,1)  (9,3,1)
%e A367411                                                (6,3,2,1)  (10,2,1)
%e A367411                                                           (6,4,2,1)
%e A367411                                                           (7,3,2,1)
%t A367411 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}];If[d=={},{}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,30}]
%Y A367411 For parts instead of sums we have A238007:
%Y A367411 - complement A001227
%Y A367411 - non-strict complement A034296, ranks A073491
%Y A367411 - non-strict  A239955, ranks A073492
%Y A367411 The non-strict version is A367403.
%Y A367411 The non-strict complement is A367402.
%Y A367411 The complement is counted by A367410.
%Y A367411 The non-binary version is A365831:
%Y A367411 - non-strict complement A126796, ranks A325781
%Y A367411 - complement A188431
%Y A367411 - non-strict A365924, ranks A365830
%Y A367411 A000009 counts partitions covering an initial interval, ranks A055932.
%Y A367411 A046663 counts partitions w/o submultiset summing to k, strict A365663.
%Y A367411 A365543 counts partitions w/ submultiset summing to k, strict A365661.
%Y A367411 Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.
%K A367411 nonn
%O A367411 0,9
%A A367411 _Gus Wiseman_, Nov 17 2023