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A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

%I A365006 #19 Sep 20 2023 18:16:31
%S A365006 1,1,1,1,1,2,1,3,2,4,4,8,4,11,9,16,14,25,20,37,31,49,47,73,64,101,96,
%T A365006 135,133,190,181,256,253,336,342,453,452,596,609,771,803,1014,1041,
%U A365006 1309,1362,1674,1760,2151,2249,2736,2884,3449,3661,4366,4615,5486,5825
%N A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.
%C A365006 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.
%H A365006 Chai Wah Wu, <a href="/A365006/b365006.txt">Table of n, a(n) for n = 0..109</a>
%e A365006 The a(8) = 2 through a(13) = 11 partitions:
%e A365006   (8)    (9)      (10)       (11)       (12)       (13)
%e A365006   (5,3)  (5,4)    (6,4)      (6,5)      (7,5)      (7,6)
%e A365006          (7,2)    (7,3)      (7,4)      (5,4,3)    (8,5)
%e A365006          (4,3,2)  (4,3,2,1)  (8,3)      (5,4,2,1)  (9,4)
%e A365006                              (9,2)                 (10,3)
%e A365006                              (5,4,2)               (11,2)
%e A365006                              (6,3,2)               (6,4,3)
%e A365006                              (5,3,2,1)             (6,5,2)
%e A365006                                                    (7,4,2)
%e A365006                                                    (5,4,3,1)
%e A365006                                                    (6,4,2,1)
%t A365006 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365006 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combp[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,30}]
%o A365006 (Python)
%o A365006 from sympy.utilities.iterables import partitions
%o A365006 def A365006(n):
%o A365006     if n <= 1: return 1
%o A365006     alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
%o A365006     c = 1
%o A365006     for p in partitions(n,k=n-1):
%o A365006         if max(p.values()) == 1:
%o A365006             s = set(p)
%o A365006             for q in s:
%o A365006                 if tuple(sorted(s-{q})) in alist[q]:
%o A365006                     break
%o A365006             else:
%o A365006                 c += 1
%o A365006     return c # _Chai Wah Wu_, Sep 20 2023
%Y A365006 The nonnegative version for subsets appears to be A124506.
%Y A365006 For sums instead of combinations we have A364349, binary A364533.
%Y A365006 The nonnegative version is A364350, complement A364839.
%Y A365006 For subsets instead of partitions we have A365044, complement A365043.
%Y A365006 The non-strict version is A365072, nonnegative A364915.
%Y A365006 A000041 counts integer partitions, strict A000009.
%Y A365006 A008284 counts partitions by length, strict A008289.
%Y A365006 A116861 and A364916 count linear combinations of strict partitions.
%Y A365006 A364912 counts linear combinations of partitions of k.
%Y A365006 Cf. A151897, A237113, A236912, A237667, A275972, A363226, A364272, A364913, A365004, A365068.
%K A365006 nonn
%O A365006 0,6
%A A365006 _Gus Wiseman_, Aug 31 2023
%E A365006 a(31)-a(56) from _Chai Wah Wu_, Sep 20 2023