%I #14 Oct 24 2023 10:46:11
%S 0,0,0,1,1,1,3,2,4,5,7,7,12,12,17,20,26,29,39,43,54,62,77,88,107,122,
%T 148,168,200,229,267,308,360,407,476,536,623,710,812,917,1050,1190,
%U 1349,1530,1733,1944,2206,2483,2794,3138,3524
%N Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.
%e For y = (4,3,2) we can write 4 = 0*3 + 2*2, so y is counted under a(9).
%e For y = (11,5,3) we can write 11 = 1*5 + 2*3, so y is counted under a(19).
%e For y = (17,5,4,3) we can write 17 = 1*3 + 1*4 + 2*5, so y is counted under a(29).
%e The a(1) = 0 through a(12) = 12 strict partitions (A = 10, B = 11):
%e . . (21) (31) (41) (42) (61) (62) (63) (82) (A1) (84)
%e (51) (421) (71) (81) (91) (542) (93)
%e (321) (431) (432) (532) (632) (A2)
%e (521) (531) (541) (641) (B1)
%e (621) (631) (731) (642)
%e (721) (821) (651)
%e (4321) (5321) (732)
%e (741)
%e (831)
%e (921)
%e (5421)
%e (6321)
%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[combs[#[[k]], Delete[#,k]]!={}, {k,Length[#]}]&]],{n,0,15}]
%o (Python)
%o from sympy.utilities.iterables import partitions
%o def A364839(n):
%o if n <= 1: return 0
%o alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
%o for p in partitions(n,k=n-1):
%o if max(p.values(),default=0)==1:
%o s = set(p)
%o if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
%o c += 1
%o return c # _Chai Wah Wu_, Sep 23 2023
%Y For sums instead of combinations we have A364272, binary A364670.
%Y The complement in strict partitions is A364350.
%Y Non-strict versions are A364913 and the complement of A364915.
%Y For subsets instead of partitions we have A364914, complement A326083.
%Y The case of no all positive coefficients is A365006.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A116861 and A364916 count linear combinations of strict partitions.
%Y Cf. A085489, A151897, A236912, A237113, A237667, A275972, A363226, A365002.
%K nonn
%O 0,7
%A _Gus Wiseman_, Aug 19 2023