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A365378 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.

%I A365378 #15 Sep 13 2023 14:16:22
%S A365378 0,0,0,1,1,4,2,9,5,13,10,28,7,45,25,51,32,101,31,148,50,166,106,291,
%T A365378 47,374,176,450,179,721,121,963,285,1080,474,1534,200,2140,712,2407,
%U A365378 599,3539,481,4546,1014,4885
%N A365378 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.
%e A365378 The partition (5,2,2) has distinct parts {2,5} and has 11 = 3*2 + 1*5, so is not counted under a(11).
%e A365378 The partition (4,2,2) cannot be linearly combined to obtain 9, so is counted under a(9).
%e A365378 The partition (4,2,2) has distinct parts {2,4} and has 10 = 5*2 + 0*4, so is not counted under a(10).
%e A365378 The a(3) = 1 through a(10) = 10 partitions:
%e A365378   (2)  (3)  (2)   (4)  (2)    (3)   (2)     (3)
%e A365378             (3)   (5)  (3)    (5)   (4)     (4)
%e A365378             (4)        (4)    (6)   (5)     (6)
%e A365378             (22)       (5)    (7)   (6)     (7)
%e A365378                        (6)    (33)  (7)     (8)
%e A365378                        (22)         (8)     (9)
%e A365378                        (33)         (22)    (33)
%e A365378                        (42)         (42)    (44)
%e A365378                        (222)        (44)    (63)
%e A365378                                     (62)    (333)
%e A365378                                     (222)
%e A365378                                     (422)
%e A365378                                     (2222)
%t A365378 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365378 Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combs[n,Union[#]]=={}&]],{n,0,10}]
%o A365378 (Python)
%o A365378 from sympy.utilities.iterables import partitions
%o A365378 def A365378(n):
%o A365378     a = {tuple(sorted(set(p))) for p in partitions(n)}
%o A365378     return sum(1 for m in range(1,n) for b in partitions(m) if not any(set(d).issubset(set(b)) for d in a)) # _Chai Wah Wu_, Sep 13 2023
%Y A365378 The complement for subsets is A365073, positive coefficients A088314.
%Y A365378 For strict partitions we have A365312, positive coefficients A088528.
%Y A365378 For positive coefficients we have A365323.
%Y A365378 The complement is counted by A365379.
%Y A365378 The version for subsets is A365380, positive coefficients A365322.
%Y A365378 The relatively prime case is A365382.
%Y A365378 A000041 counts integer partitions, strict A000009.
%Y A365378 A008284 counts partitions by length, strict A008289.
%Y A365378 A116861 and A364916 count linear combinations of strict partitions.
%Y A365378 A364350 counts combination-free strict partitions, non-strict A364915.
%Y A365378 A364839 counts combination-full strict partitions, non-strict A364913.
%Y A365378 Cf. A237668, A363225, A364272, A364345, A364914, A365320.
%K A365378 nonn,more
%O A365378 0,6
%A A365378 _Gus Wiseman_, Sep 04 2023
%E A365378 a(21)-a(45) from _Chai Wah Wu_, Sep 13 2023