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A365312
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Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.
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18
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0, 0, 0, 1, 1, 3, 2, 6, 4, 8, 7, 16, 6, 24, 17, 24, 20, 46, 22, 62, 31, 63, 57, 106, 35, 122, 90, 137, 88, 212, 74, 262, 134, 267, 206, 345, 121, 476, 294, 484, 232, 698, 242, 837, 389, 763, 571, 1185, 318, 1327, 634, 1392, 727, 1927, 640, 2056, 827, 2233, 1328
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OFFSET
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0,6
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LINKS
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EXAMPLE
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The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
The a(0) = 0 through a(11) = 16 strict partitions:
. . . (2) (3) (2) (4) (2) (3) (2) (3) (2)
(3) (5) (3) (5) (4) (4) (3)
(4) (4) (6) (5) (6) (4)
(5) (7) (6) (7) (5)
(6) (7) (8) (6)
(4,2) (8) (9) (7)
(4,2) (6,3) (8)
(6,2) (9)
(10)
(4,2)
(5,4)
(6,2)
(6,3)
(6,4)
(7,3)
(8,2)
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MATHEMATICA
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combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions, n], UnsameQ@@#&], combs[n, #]=={}&]], {n, 0, 10}]
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PROG
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(Python)
from math import isqrt
from sympy.utilities.iterables import partitions
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, m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
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CROSSREFS
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The complement for positive coefficients is counted by A088314.
For positive coefficients we have A088528.
The complement is counted by A365311.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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