cp's OEIS Frontend

A367398 Number of integer partitions of n whose length is not a semi-sum of the parts.

%I A367398 #13 Mar 17 2024 14:03:49
%S A367398 1,1,1,3,4,6,8,12,16,23,28,41,52,71,89,122,151,200,246,321,398,510,
%T A367398 620,794,968,1212,1474,1837,2219,2748,3302,4055,4882,5942,7094,8623,
%U A367398 10275,12376,14721,17661,20920,25011,29516,35120,41419,49053,57609,68092,79780
%N A367398 Number of integer partitions of n whose length is not a semi-sum of the parts.
%C A367398 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 A367398 For the partition y = (4,3,1) we have semi-sums {4,5,7}, which do not include 3 (the length of y), so y is counted under a(8).
%e A367398 The a(1) = 1 through a(8) = 16 partitions:
%e A367398   (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
%e A367398             (21)   (22)    (32)     (33)      (43)       (44)
%e A367398             (111)  (31)    (41)     (42)      (52)       (53)
%e A367398                    (1111)  (311)    (51)      (61)       (62)
%e A367398                            (2111)   (222)     (322)      (71)
%e A367398                            (11111)  (411)     (331)      (332)
%e A367398                                     (21111)   (511)      (422)
%e A367398                                     (111111)  (4111)     (431)
%e A367398                                               (22111)    (611)
%e A367398                                               (31111)    (4211)
%e A367398                                               (211111)   (5111)
%e A367398                                               (1111111)  (22211)
%e A367398                                                          (221111)
%e A367398                                                          (311111)
%e A367398                                                          (2111111)
%e A367398                                                          (11111111)
%t A367398 Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#,{2}],Length[#]]&]],{n,0,10}]
%Y A367398 The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
%Y A367398              sum-full  sum-free  comb-full comb-free semi-full semi-free
%Y A367398             -----------------------------------------------------------
%Y A367398 partitions:  A367212   A367213   A367218   A367219   A367394   A367398*
%Y A367398 strict:      A367214   A367215   A367220   A367221   A367395   A367399
%Y A367398 subsets:     A367216   A367217   A367222   A367223   A367396   A367400
%Y A367398 ranks:       A367224   A367225   A367226   A367227   A367397   A367401
%Y A367398 A000041 counts partitions, strict A000009.
%Y A367398 A002865 counts partitions whose length is a part, complement A229816.
%Y A367398 A236912 counts partitions containing no semi-sum, ranks A364461.
%Y A367398 A237113 counts partitions containing a semi-sum, ranks A364462.
%Y A367398 A237667 counts sum-free partitions, sum-full A237668.
%Y A367398 A366738 counts semi-sums of partitions, strict A366741.
%Y A367398 A367402 counts partitions with covering semi-sums, complement A367403.
%Y A367398 Triangles:
%Y A367398 A008284 counts partitions by length, strict A008289.
%Y A367398 A365541 counts subsets with a semi-sum k.
%Y A367398 A367404 counts partitions with a semi-sum k, strict A367405.
%Y A367398 Cf. A000700, A126796, A304792, A363225, A364272, A365543.
%K A367398 nonn
%O A367398 0,4
%A A367398 _Gus Wiseman_, Nov 19 2023