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A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

%I A365072 #14 Sep 20 2023 18:12:33
%S A365072 1,1,2,2,3,3,4,5,6,8,9,17,15,31,34,53,65,109,117,196,224,328,405,586,
%T A365072 673,968,1163,1555,1889,2531,2986,3969,4744,6073,7333,9317,11053,
%U A365072 14011,16710,20702,24714,30549,36127,44413,52561,63786,75583,91377,107436,129463
%N A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.
%C A365072 We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.
%e A365072 The a(1) = 1 through a(8) = 6 partitions:
%e A365072   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A365072        (11)  (111)  (22)    (32)     (33)      (43)       (44)
%e A365072                     (1111)  (11111)  (222)     (52)       (53)
%e A365072                                      (111111)  (322)      (332)
%e A365072                                                (1111111)  (2222)
%e A365072                                                           (11111111)
%e A365072 The a(11) = 17 partitions:
%e A365072   (11)  (9,2)  (7,2,2)  (5,3,2,1)  (4,3,2,1,1)  (1,1,1,1,1,1,1,1,1,1,1)
%e A365072         (8,3)  (6,3,2)  (5,2,2,2)  (3,2,2,2,2)
%e A365072         (7,4)  (5,4,2)  (4,3,2,2)
%e A365072         (6,5)  (5,3,3)  (3,3,3,2)
%e A365072                (4,4,3)
%t A365072 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365072 Table[Length[Select[Union/@IntegerPartitions[n], Function[ptn,!Or@@Table[combp[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]],{n,0,15}]
%o A365072 (Python)
%o A365072 from sympy.utilities.iterables import partitions
%o A365072 def A365072(n):
%o A365072     if n <= 1: return 1
%o A365072     alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
%o A365072     c = 1
%o A365072     for p in partitions(n,k=n-1):
%o A365072         s = set(p)
%o A365072         for q in s:
%o A365072             if tuple(sorted(s-{q})) in alist[q]:
%o A365072                 break
%o A365072         else:
%o A365072             c += 1
%o A365072     return c # _Chai Wah Wu_, Sep 20 2023
%Y A365072 The nonnegative version is A364915, strict A364350.
%Y A365072 The strict case is A365006.
%Y A365072 For subsets instead of partitions we have A365044, complement A365043.
%Y A365072 A000041 counts integer partitions, strict A000009.
%Y A365072 A008284 counts partitions by length, strict A008289.
%Y A365072 A116861 and A364916 count linear combinations of strict partitions.
%Y A365072 A237667 counts sum-free partitions, binary A236912.
%Y A365072 A364912 counts positive linear combinations of partitions.
%Y A365072 A365068 counts combination-full partitions, strict A364839.
%Y A365072 Cf. A085489, A108917, A151897, A325862, A364272, A364910, A364911, A364913.
%K A365072 nonn
%O A365072 0,3
%A A365072 _Gus Wiseman_, Aug 31 2023
%E A365072 a(31)-a(49) from _Chai Wah Wu_, Sep 20 2023