cp's OEIS Frontend

A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

%I A364913 #7 Oct 24 2023 10:45:29
%S A364913 0,0,1,2,4,5,10,12,20,27,39,51,74,95,130,169,225,288,378,479,617,778,
%T A364913 990,1239,1560,1938,2419,2986,3696,4538,5575,6810,8319,10102,12274,
%U A364913 14834,17932,21587,25963,31120,37275,44513,53097,63181,75092,89030,105460,124647
%N A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.
%C A364913 Includes all non-strict partitions (A047967).
%F A364913 a(n) + A364915(n) = A000041(n).
%e A364913 The a(0) = 0 through a(7) = 12 partitions:
%e A364913   .  .  (11)  (21)   (22)    (41)     (33)      (61)
%e A364913               (111)  (31)    (221)    (42)      (322)
%e A364913                      (211)   (311)    (51)      (331)
%e A364913                      (1111)  (2111)   (222)     (421)
%e A364913                              (11111)  (321)     (511)
%e A364913                                       (411)     (2221)
%e A364913                                       (2211)    (3211)
%e A364913                                       (3111)    (4111)
%e A364913                                       (21111)   (22111)
%e A364913                                       (111111)  (31111)
%e A364913                                                 (211111)
%e A364913                                                 (1111111)
%e A364913 The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(12).
%e A364913 The partition (6,4,3,2) has 6 = 4+2, or 6 = 3+3, or 6 = 2+2+2, or 4 = 2+2, so is counted under a(15).
%t A364913 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A364913 Table[Length[Select[IntegerPartitions[n],!UnsameQ@@#||Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,15}]
%Y A364913 The strict case is A364839.
%Y A364913 For sums instead of combinations we have A364272, binary A364670.
%Y A364913 The complement in strict partitions is A364350.
%Y A364913 For subsets instead of partitions we have A364914, complement A326083.
%Y A364913 Allowing equal parts gives A365068, complement A364915.
%Y A364913 A000041 counts integer partitions, strict A000009.
%Y A364913 A008284 counts partitions by length, strict A008289.
%Y A364913 A116861 and A364916 count linear combinations of strict partitions.
%Y A364913 A365006 = no strict partitions w/ pos linear combination.
%Y A364913 Cf. A085489, A151897, A236912, A237113, A237667, A275972, A364346.
%K A364913 nonn
%O A364913 0,4
%A A364913 _Gus Wiseman_, Aug 20 2023