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A365323 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.

%I A365323 #16 Dec 30 2023 17:01:08
%S A365323 0,0,1,1,4,3,9,7,15,16,29,23,47,43,74,65,114,100,174,153,257,228,368,
%T A365323 312,530,454,736,645,1025,902,1402,1184,1909,1626,2618,2184,3412,2895,
%U A365323 4551,3887,5966,5055,7796,6509,10244,8462,13060,10881,16834,14021,21471
%N A365323 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.
%H A365323 Chai Wah Wu, <a href="/A365323/b365323.txt">Table of n, a(n) for n = 1..95</a>
%e A365323 The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
%e A365323 The a(3) = 1 through a(10) = 16 partitions:
%e A365323   (2)  (3)  (2)    (4)    (2)      (3)    (2)        (3)
%e A365323             (3)    (5)    (3)      (5)    (4)        (4)
%e A365323             (4)    (3,2)  (4)      (6)    (5)        (6)
%e A365323             (2,2)         (5)      (7)    (6)        (7)
%e A365323                           (6)      (3,3)  (7)        (8)
%e A365323                           (2,2)    (4,3)  (8)        (9)
%e A365323                           (3,3)    (5,2)  (2,2)      (3,3)
%e A365323                           (4,2)           (4,2)      (4,4)
%e A365323                           (2,2,2)         (4,3)      (5,2)
%e A365323                                           (4,4)      (5,3)
%e A365323                                           (5,3)      (5,4)
%e A365323                                           (6,2)      (6,3)
%e A365323                                           (2,2,2)    (7,2)
%e A365323                                           (4,2,2)    (3,3,3)
%e A365323                                           (2,2,2,2)  (4,3,2)
%e A365323                                                      (5,2,2)
%t A365323 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365323 Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]
%o A365323 (Python)
%o A365323 from sympy.utilities.iterables import partitions
%o A365323 def A365323(n):
%o A365323     a = {tuple(sorted(set(p))) for p in partitions(n)}
%o A365323     return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # _Chai Wah Wu_, Sep 12 2023
%Y A365323 Complement for subsets: A088314 or A365042, nonnegative A365073 or A365542.
%Y A365323 For strict partitions we have A088528, nonnegative coefficients A365312.
%Y A365323 For length-2 subsets we have A365321 (we use n instead of n-1).
%Y A365323 For subsets we have A365322 or A365045, nonnegative coefficients A365380.
%Y A365323 For nonnegative coefficients we have A365378, complement A365379.
%Y A365323 A000041 counts integer partitions, strict A000009.
%Y A365323 A008284 counts partitions by length, strict A008289.
%Y A365323 A116861 and A364916 count linear combinations of strict partitions.
%Y A365323 A364350 counts combination-free strict partitions, non-strict A364915.
%Y A365323 A364839 counts combination-full strict partitions, non-strict A364913.
%Y A365323 Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.
%K A365323 nonn
%O A365323 1,5
%A A365323 _Gus Wiseman_, Sep 12 2023
%E A365323 a(21)-a(51) from _Chai Wah Wu_, Sep 12 2023