cp's OEIS Frontend

A365068 Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.

%I A365068 #16 Dec 30 2023 21:23:13
%S A365068 0,0,0,1,2,4,7,10,16,23,34,44,67,85,119,157,210,268,360,453,592,748,
%T A365068 956,1195,1520,1883,2365,2920,3628,4451,5494,6702,8211,9976,12147,
%U A365068 14666,17776,21389,25774,30887,37035,44224,52819,62836,74753,88614,105062,124160
%N A365068 Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.
%C A365068 These may be called "non-binary nonnegative combination-full" partitions.
%C A365068 Does not necessarily include all non-strict partitions (A047967).
%e A365068 The partition (5,4,3,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(15).
%e A365068 The partition (6,4,3,2) has 6 = 1*2 + 1*4, so is counted under a(15). The combinations 6 = 2*3 = 3*2 and 4 = 2*2 can also be used.
%e A365068 The a(3) = 1 through a(8) = 16 partitions:
%e A365068   (21)  (31)   (41)    (42)     (61)      (62)
%e A365068         (211)  (221)   (51)     (331)     (71)
%e A365068                (311)   (321)    (421)     (422)
%e A365068                (2111)  (411)    (511)     (431)
%e A365068                        (2211)   (2221)    (521)
%e A365068                        (3111)   (3211)    (611)
%e A365068                        (21111)  (4111)    (3221)
%e A365068                                 (22111)   (3311)
%e A365068                                 (31111)   (4211)
%e A365068                                 (211111)  (5111)
%e A365068                                           (22211)
%e A365068                                           (32111)
%e A365068                                           (41111)
%e A365068                                           (221111)
%e A365068                                           (311111)
%e A365068                                           (2111111)
%t A365068 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365068 Table[Length[Select[IntegerPartitions[n], Function[ptn,Or@@Table[combs[ptn[[k]], DeleteCases[ptn,ptn[[k]]]]!={}, {k,Length[ptn]}]]]],{n,0,5}]
%o A365068 (Python)
%o A365068 from sympy.utilities.iterables import partitions
%o A365068 def A365068(n):
%o A365068     if n <= 1: return 0
%o A365068     alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
%o A365068     for p in partitions(n,k=n-1):
%o A365068         s = set(p)
%o A365068         if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
%o A365068             c += 1
%o A365068     return c # _Chai Wah Wu_, Sep 20 2023
%Y A365068 The complement for sums instead of combinations is A237667, binary A236912.
%Y A365068 For sums instead of combinations we have A237668, binary A237113.
%Y A365068 The strict case is A364839, complement A364350.
%Y A365068 Allowing equal parts in the combination gives A364913.
%Y A365068 For subsets instead of partitions we have A364914, complement A326083.
%Y A365068 The complement is A364915.
%Y A365068 A000041 counts integer partitions, strict A000009.
%Y A365068 A008284 counts partitions by length, strict A008289.
%Y A365068 A116861 and A364916 count linear combinations of strict partitions.
%Y A365068 A323092 counts double-free partitions, ranks A320340.
%Y A365068 A364912 counts linear combinations of partitions of k.
%Y A365068 Cf. A108917, A151897, A364272, A364910, A364911, A365006.
%K A365068 nonn
%O A365068 0,5
%A A365068 _Gus Wiseman_, Aug 27 2023
%E A365068 a(31)-a(47) from _Chai Wah Wu_, Sep 20 2023