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A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

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