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A365379
Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.
10
0, 1, 3, 5, 10, 14, 27, 35, 61, 83, 128, 166, 264, 327, 482, 632, 882, 1110, 1565, 1938, 2663, 3339, 4401, 5471, 7290, 8921, 11555, 14291, 18280, 22303, 28507, 34507, 43534, 52882, 65798, 79621, 98932, 118629, 146072, 175562, 214708, 256351, 312583, 371779
OFFSET
0,3
EXAMPLE
The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).
The a(1) = 1 through a(5) = 14 partitions:
(1) (1) (1) (1) (1)
(2) (3) (2) (5)
(11) (11) (4) (11)
(21) (11) (21)
(111) (21) (31)
(22) (32)
(31) (41)
(111) (111)
(211) (211)
(1111) (221)
(311)
(1111)
(2111)
(11111)
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[Join@@Array[IntegerPartitions, n], combs[n, Union[#]]!={}&]], {n, 0, 10}]
PROG
(Python)
from sympy.utilities.iterables import partitions
def A365379(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1, n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
CROSSREFS
For subsets with positive coefficients we have A088314, complement A088528.
The case of strict partitions with positive coefficients is also A088314.
The version for subsets is A365073, complement A365380.
The case of strict partitions is A365311, complement A365312.
The complement is counted by A365378.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Sequence in context: A319066 A306319 A182722 * A089483 A323858 A190484
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2023
EXTENSIONS
a(21)-a(43) from Chai Wah Wu, Sep 13 2023
STATUS
approved