A364913
Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.
Original entry on oeis.org
0, 0, 1, 2, 4, 5, 10, 12, 20, 27, 39, 51, 74, 95, 130, 169, 225, 288, 378, 479, 617, 778, 990, 1239, 1560, 1938, 2419, 2986, 3696, 4538, 5575, 6810, 8319, 10102, 12274, 14834, 17932, 21587, 25963, 31120, 37275, 44513, 53097, 63181, 75092, 89030, 105460, 124647
Offset: 0
The a(0) = 0 through a(7) = 12 partitions:
. . (11) (21) (22) (41) (33) (61)
(111) (31) (221) (42) (322)
(211) (311) (51) (331)
(1111) (2111) (222) (421)
(11111) (321) (511)
(411) (2221)
(2211) (3211)
(3111) (4111)
(21111) (22111)
(111111) (31111)
(211111)
(1111111)
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(12).
The partition (6,4,3,2) has 6 = 4+2, or 6 = 3+3, or 6 = 2+2+2, or 4 = 2+2, so is counted under a(15).
The complement in strict partitions is
A364350.
For subsets instead of partitions we have
A364914, complement
A326083.
A365006 = no strict partitions w/ pos linear combination.
-
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@@#||Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,15}]
A364915
Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.
Original entry on oeis.org
1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 12, 10, 16, 16, 19, 21, 29, 25, 37, 35, 44, 46, 60, 55, 75, 71, 90, 90, 114, 110, 140, 138, 167, 163, 217, 201, 248, 241, 298, 303, 359, 355, 425, 422, 520, 496, 594, 603, 715, 706, 834, 826, 968, 972, 1153, 1147, 1334, 1315, 1530
Offset: 0
The a(1) = 1 through a(10) = 8 partitions (A=10):
1 2 3 4 5 6 7 8 9 A
11 111 22 32 33 43 44 54 55
1111 11111 222 52 53 72 64
111111 322 332 333 73
1111111 2222 522 433
11111111 3222 3322
111111111 22222
1111111111
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is counted under a(12).
The partition (6,4,3,2) has 6=4+2, or 6=3+3, or 6=2+2+2, or 4=2+2, so is not counted under a(15).
For subsets instead of partitions we have
A326083, complement
A364914.
A007865 counts binary sum-free sets w/ re-usable parts, complement
A093971.
A364912 counts linear combinations of partitions of k.
-
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]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]], {n,0,15}]
-
from sympy.utilities.iterables import partitions
def A364915(n):
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):
s = set(p)
if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
c += 1
return c # Chai Wah Wu, Sep 23 2023
A365006
Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 8, 4, 11, 9, 16, 14, 25, 20, 37, 31, 49, 47, 73, 64, 101, 96, 135, 133, 190, 181, 256, 253, 336, 342, 453, 452, 596, 609, 771, 803, 1014, 1041, 1309, 1362, 1674, 1760, 2151, 2249, 2736, 2884, 3449, 3661, 4366, 4615, 5486, 5825
Offset: 0
The a(8) = 2 through a(13) = 11 partitions:
(8) (9) (10) (11) (12) (13)
(5,3) (5,4) (6,4) (6,5) (7,5) (7,6)
(7,2) (7,3) (7,4) (5,4,3) (8,5)
(4,3,2) (4,3,2,1) (8,3) (5,4,2,1) (9,4)
(9,2) (10,3)
(5,4,2) (11,2)
(6,3,2) (6,4,3)
(5,3,2,1) (6,5,2)
(7,4,2)
(5,4,3,1)
(6,4,2,1)
The nonnegative version for subsets appears to be
A124506.
For subsets instead of partitions we have
A365044, complement
A365043.
A364912 counts linear combinations of partitions of k.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combp[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,30}]
-
from sympy.utilities.iterables import partitions
def A365006(n):
if n <= 1: return 1
alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
c = 1
for p in partitions(n,k=n-1):
if max(p.values()) == 1:
s = set(p)
for q in s:
if tuple(sorted(s-{q})) in alist[q]:
break
else:
c += 1
return c # Chai Wah Wu, Sep 20 2023
A365072
Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.
Original entry on oeis.org
1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 17, 15, 31, 34, 53, 65, 109, 117, 196, 224, 328, 405, 586, 673, 968, 1163, 1555, 1889, 2531, 2986, 3969, 4744, 6073, 7333, 9317, 11053, 14011, 16710, 20702, 24714, 30549, 36127, 44413, 52561, 63786, 75583, 91377, 107436, 129463
Offset: 0
The a(1) = 1 through a(8) = 6 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (43) (44)
(1111) (11111) (222) (52) (53)
(111111) (322) (332)
(1111111) (2222)
(11111111)
The a(11) = 17 partitions:
(11) (9,2) (7,2,2) (5,3,2,1) (4,3,2,1,1) (1,1,1,1,1,1,1,1,1,1,1)
(8,3) (6,3,2) (5,2,2,2) (3,2,2,2,2)
(7,4) (5,4,2) (4,3,2,2)
(6,5) (5,3,3) (3,3,3,2)
(4,4,3)
For subsets instead of partitions we have
A365044, complement
A365043.
A364912 counts positive linear combinations of partitions.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Union/@IntegerPartitions[n], Function[ptn,!Or@@Table[combp[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]],{n,0,15}]
-
from sympy.utilities.iterables import partitions
def A365072(n):
if n <= 1: return 1
alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
c = 1
for p in partitions(n,k=n-1):
s = set(p)
for q in s:
if tuple(sorted(s-{q})) in alist[q]:
break
else:
c += 1
return c # Chai Wah Wu, Sep 20 2023
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