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

0, 0, 0, 1, 2, 4, 7, 10, 16, 23, 34, 44, 67, 85, 119, 157, 210, 268, 360, 453, 592, 748, 956, 1195, 1520, 1883, 2365, 2920, 3628, 4451, 5494, 6702, 8211, 9976, 12147, 14666, 17776, 21389, 25774, 30887, 37035, 44224, 52819, 62836, 74753, 88614, 105062, 124160

Offset: 0

Author

Gus Wiseman, Aug 27 2023

Keywords

• nonn

Comments

These may be called "non-binary nonnegative combination-full" partitions.

Does not necessarily include all non-strict partitions (A047967).

Examples

			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).
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.
The a(3) = 1 through a(8) = 16 partitions:
  (21)  (31)   (41)    (42)     (61)      (62)
        (211)  (221)   (51)     (331)     (71)
               (311)   (321)    (421)     (422)
               (2111)  (411)    (511)     (431)
                       (2211)   (2221)    (521)
                       (3111)   (3211)    (611)
                       (21111)  (4111)    (3221)
                                (22111)   (3311)
                                (31111)   (4211)
                                (211111)  (5111)
                                          (22211)
                                          (32111)
                                          (41111)
                                          (221111)
                                          (311111)
                                          (2111111)
		

Crossrefs

The complement for sums instead of combinations is A237667, binary A236912.

For sums instead of combinations we have A237668, binary A237113.

The strict case is A364839, complement A364350.

Allowing equal parts in the combination gives A364913.

For subsets instead of partitions we have A364914, complement A326083.

The complement is A364915.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A116861 and A364916 count linear combinations of strict partitions.

A323092 counts double-free partitions, ranks A320340.

A364912 counts linear combinations of partitions of k.

Cf. A108917, A151897, A364272, A364910, A364911, A365006.

Programs

Mathematica

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], Function[ptn,Or@@Table[combs[ptn[[k]], DeleteCases[ptn,ptn[[k]]]]!={}, {k,Length[ptn]}]]]],{n,0,5}]

Python

from sympy.utilities.iterables import partitions
def A365068(n):
    if n <= 1: return 0
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
    for p in partitions(n,k=n-1):
        s = set(p)
        if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

Extensions

a(31)-a(47) from Chai Wah Wu, Sep 20 2023