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A364350
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Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.
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81
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1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 6, 9, 7, 11, 10, 14, 12, 16, 15, 20, 17, 24, 22, 27, 29, 32, 30, 41, 36, 49, 45, 50, 52, 65, 63, 70, 77, 80, 83, 104, 98, 107, 116, 126, 134, 152, 148, 162, 180, 196, 195, 227, 227, 238, 272, 271, 293, 333, 325
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OFFSET
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0,6
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COMMENTS
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A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
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LINKS
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EXAMPLE
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The a(16) = 6 through a(22) = 12 strict partitions:
(16) (17) (18) (19) (20) (21) (22)
(9,7) (9,8) (10,8) (10,9) (11,9) (12,9) (13,9)
(10,6) (10,7) (11,7) (11,8) (12,8) (13,8) (14,8)
(11,5) (11,6) (13,5) (12,7) (13,7) (15,6) (15,7)
(13,3) (12,5) (14,4) (13,6) (14,6) (16,5) (16,6)
(7,5,4) (13,4) (7,6,5) (14,5) (17,3) (17,4) (17,5)
(14,3) (8,7,3) (15,4) (8,7,5) (19,2) (18,4)
(15,2) (16,3) (9,6,5) (11,10) (19,3)
(7,6,4) (17,2) (9,7,4) (8,7,6) (12,10)
(8,6,5) (11,5,4) (9,7,5) (9,7,6)
(9,6,4) (10,7,4) (9,8,5)
(10,8,3) (7,6,5,4)
(11,6,4)
(11,7,3)
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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[IntegerPartitions[n], UnsameQ@@#&&And@@Table[combs[#[[k]], Delete[#, k]]=={}, {k, Length[#]}]&]], {n, 0, 15}]
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PROG
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(Python)
from sympy.utilities.iterables import partitions
if n <= 1: return 1
alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
for p in partitions(n, k=n-1):
if max(p.values(), default=0)==1:
s = set(p)
if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
c += 1
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CROSSREFS
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For sums of subsets instead of combinations of partitions we have A151897.
For subsets instead of partitions we have A326083, complement A364914.
A more strict variation is A364915.
The case of all positive coefficients is A365006.
A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.
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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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