A365045
Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.
Original entry on oeis.org
0, 1, 1, 2, 4, 11, 23, 53, 111, 235, 483, 988, 1998, 4036, 8114, 16289, 32645, 65389, 130887, 261923, 524014, 1048251, 2096753, 4193832, 8388034, 16776544, 33553622, 67107919, 134216597, 268434140, 536869355, 1073740012, 2147481511, 4294964834, 8589931700
Offset: 0
The subset {3,4,10} has 10 = 2*3 + 1*4 so is not counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
. {1} {2} {3} {4} {5}
{2,3} {3,4} {2,5}
{2,3,4} {3,5}
{1,2,3,4} {4,5}
{2,4,5}
{3,4,5}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
Without re-usable parts we have
A365071, first differences of
A151897.
A085489 and
A364755 count subsets w/o the sum of two distinct elements.
A088809 and
A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement
A364839.
A364913 counts combination-full 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[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[combp[#[[k]],Union[Delete[#,k]]]=={},{k,Length[#]}]&]],{n,0,10}]
A365314
Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 3, 6, 8, 14, 14, 23, 24, 33, 28, 52, 36, 55, 58, 73, 53, 95, 62, 110, 94, 105, 81, 165, 105, 133, 132, 176, 112, 225, 123, 210, 174, 192, 186, 306, 157, 223, 218, 328, 180, 354, 192, 324, 315, 288, 216, 474, 260, 383, 311, 404, 254, 491, 338, 511, 360
Offset: 0
We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
The a(2) = 1 through a(7) = 14 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3)
(2,3) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5)
(2,4) (2,3) (1,6) (1,6)
(3,4) (2,5) (2,3) (1,7)
(3,5) (2,4) (2,3)
(4,5) (2,5) (2,5)
(2,6) (2,7)
(3,4) (3,4)
(3,5) (3,7)
(3,6) (4,7)
(4,6) (5,7)
(5,6) (6,7)
For all subsets instead of just pairs we have
A365073, complement
A365380.
The case of positive coefficients is
A365315, for all subsets
A088314.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}], combs[n,#]!={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365314(n):
a = set()
for i in range(1,n+1):
if not n%i:
a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
else:
for j in count(0,i):
if j > n:
break
k = n-j
for d in divisors(k):
if d>=i:
break
a.add((d,i))
return len(a) # Chai Wah Wu, Sep 12 2023
A365378
Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 1, 4, 2, 9, 5, 13, 10, 28, 7, 45, 25, 51, 32, 101, 31, 148, 50, 166, 106, 291, 47, 374, 176, 450, 179, 721, 121, 963, 285, 1080, 474, 1534, 200, 2140, 712, 2407, 599, 3539, 481, 4546, 1014, 4885
Offset: 0
The partition (5,2,2) has distinct parts {2,5} and has 11 = 3*2 + 1*5, so is not counted under a(11).
The partition (4,2,2) cannot be linearly combined to obtain 9, so is counted under a(9).
The partition (4,2,2) has distinct parts {2,4} and has 10 = 5*2 + 0*4, so is not counted under a(10).
The a(3) = 1 through a(10) = 10 partitions:
(2) (3) (2) (4) (2) (3) (2) (3)
(3) (5) (3) (5) (4) (4)
(4) (4) (6) (5) (6)
(22) (5) (7) (6) (7)
(6) (33) (7) (8)
(22) (8) (9)
(33) (22) (33)
(42) (42) (44)
(222) (44) (63)
(62) (333)
(222)
(422)
(2222)
For positive coefficients we have
A365323.
The complement is counted by
A365379.
The relatively prime case is
A365382.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
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@@IntegerPartitions/@Range[n-1],combs[n,Union[#]]=={}&]],{n,0,10}]
-
from sympy.utilities.iterables import partitions
def A365378(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n) for b in partitions(m) if not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365321
Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 2, 4, 6, 10, 13, 18, 24, 30, 37, 46, 54, 63, 77, 85, 99, 111, 127, 141, 161, 171, 194, 210, 235, 246, 277, 293, 322, 342, 372, 389, 428, 441, 491, 504, 545, 561, 612, 635, 680, 701, 753, 773, 836, 846, 911, 932, 1000, 1017, 1082, 1103, 1176, 1193
Offset: 0
For the pair p = (2,3) we have 4 = 2*2 + 0*3, so p is not counted under A365320(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is counted under a(4).
The a(2) = 1 through a(7) = 13 pairs:
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7)
(2,3) (2,3) (2,4) (2,3) (2,4)
(2,4) (2,5) (2,5) (2,6)
(3,4) (3,4) (2,6) (2,7)
(3,5) (3,4) (3,5)
(4,5) (3,5) (3,6)
(3,6) (3,7)
(4,5) (4,5)
(4,6) (4,6)
(5,6) (4,7)
(5,6)
(5,7)
(6,7)
For all subsets instead of just pairs we have
A365322, complement
A088314.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf.
A070880,
A088571,
A088809,
A151897,
A326020,
A365043,
A365073,
A365311,
A365312,
A365378,
A365380.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}], combp[n,#]=={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365321(n):
a = set()
for i in range(1,n+1):
for j in count(i,i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d,i))
return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 12 2023
A365379
Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
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
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)
For subsets with positive coefficients we have
A088314, complement
A088528.
The case of strict partitions with positive coefficients is also
A088314.
The complement is counted by
A365378.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
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}]
-
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
A070880
Consider the 2^(n-1)-1 nonempty subsets S of {1, 2, ..., n-1}; a(n) gives number of such S for which it is impossible to partition n into parts from S such that each s in S is used at least once.
Original entry on oeis.org
0, 0, 1, 3, 10, 22, 52, 110, 234, 482, 987, 1997, 4035, 8113, 16288, 32644, 65388, 130886, 261922, 524013, 1048250, 2096752, 4193831, 8388033, 16776543, 33553621, 67107918, 134216596, 268434139, 536869354, 1073740011, 2147481510, 4294964833, 8589931699
Offset: 1
a(4)=3 because there are three different subsets S of {1,2,3} satisfying the condition: {3}, {2,3} & {1,2,3}. For the other subsets S, such as {1,2}, there is a partition of 4 which uses them all (such as 4 = 1+1+2).
From _Gus Wiseman_, Sep 10 2023: (Start)
The a(6) = 22 subsets:
{4} {2,3} {1,2,4} {1,2,3,4} {1,2,3,4,5}
{5} {2,5} {1,2,5} {1,2,3,5}
{3,4} {1,3,4} {1,2,4,5}
{3,5} {1,3,5} {1,3,4,5}
{4,5} {1,4,5} {2,3,4,5}
{2,3,4}
{2,3,5}
{2,4,5}
{3,4,5}
(End)
For sets with sum < n instead of maximum < n we have
A088528.
Allowing empty sets gives
A365045, nonnegative version apparently
A124506.
Without re-usable parts we have
A365377(n) - 1.
For nonnegative (instead of positive) coefficients we have
A365380(n) - 1.
A364350 counts combination-free strict partitions, complement
A364913.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Rest[Subsets[Range[n-1]]], combp[n,#]=={}&]],{n,7}] (* Gus Wiseman, Sep 10 2023 *)
-
from sympy.utilities.iterables import partitions
def A070880(n): return (1<Chai Wah Wu, Sep 10 2023
A365044
Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.
Original entry on oeis.org
1, 2, 3, 5, 9, 20, 43, 96, 207, 442, 925, 1913, 3911, 7947, 16061, 32350, 64995, 130384, 261271, 523194, 1047208, 2095459, 4192212, 8386044, 16774078, 33550622, 67104244, 134212163, 268428760, 536862900, 1073732255, 2147472267, 4294953778, 8589918612, 17179850312
Offset: 0
The subset S = {3,5,6,8} has 6 = 2*3 + 0*5 + 0*8 and 8 = 1*3 + 1*5 + 0*6 but neither of these is strictly positive, so S is counted under a(8).
The a(0) = 1 through a(5) = 20 subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{3} {3} {3}
{2,3} {4} {4}
{2,3} {5}
{3,4} {2,3}
{2,3,4} {2,5}
{1,2,3,4} {3,4}
{3,5}
{4,5}
{2,3,4}
{2,4,5}
{3,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
A085489 and
A364755 count subsets w/o the sum of two distinct elements.
A088809 and
A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement
A364839.
A364913 counts combination-full partitions.
Cf.
A006951,
A237113,
A237668,
A308546,
A324736,
A326020,
A326080,
A364272,
A364349,
A364534,
A365069.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],And@@Table[combp[Last[#],Union[Most[#]]]=={},{k,Length[#]}]&]],{n,0,10}]
-
from itertools import combinations
from sympy.utilities.iterables import partitions
def A365044(n):
mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
return n+1+sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] not in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023
A365315
Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 2, 4, 5, 8, 10, 12, 15, 18, 20, 24, 28, 28, 35, 37, 42, 44, 49, 49, 60, 59, 66, 65, 79, 74, 85, 84, 93, 93, 107, 100, 120, 104, 126, 121, 142, 129, 145, 140, 160, 150, 173, 154, 189, 170, 196, 176, 208, 193, 223, 202, 238, 203, 241, 227, 267, 235
Offset: 0
We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
For the pair p = (2,3), we have 4 = 2*2 + 0*3, so p is counted under A365314(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is not counted under a(4).
The a(3) = 1 through a(10) = 15 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3) (1,3) (1,3)
(1,4) (1,4) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5) (1,5) (1,5)
(2,4) (1,6) (1,6) (1,6) (1,6)
(2,3) (1,7) (1,7) (1,7)
(2,5) (2,3) (1,8) (1,8)
(3,4) (2,4) (2,3) (1,9)
(2,6) (2,5) (2,3)
(3,5) (2,7) (2,4)
(3,6) (2,6)
(4,5) (2,8)
(3,4)
(3,7)
(4,6)
For all subsets instead of just pairs we have
A088314, complement
A365322.
The case of nonnegative coefficients is
A365314, for all subsets
A365073.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf.
A070880,
A088809,
A326020,
A364534,
A365043,
A365311,
A365312,
A365378,
A365379,
A365380,
A365383.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}],combp[n,#]!={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365315(n):
a = set()
for i in range(1,n+1):
for j in count(i,i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d,i))
return len(a) # Chai Wah Wu, Sep 13 2023
A365042
Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.
Original entry on oeis.org
0, 0, 1, 2, 4, 5, 9, 11, 17, 21, 29, 36, 50, 60, 78, 95, 123, 147, 185, 221, 274, 325, 399, 472, 574, 672, 810, 945, 1131, 1316, 1557, 1812, 2137, 2462, 2892, 3322, 3881, 4460, 5176, 5916, 6846, 7817, 8993, 10250, 11765, 13333, 15280, 17308, 19731, 22306
Offset: 0
The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(7) = 11 subsets:
. . {1,2} {1,3} {1,4} {1,5} {1,6} {1,7}
{1,2,3} {2,4} {1,2,5} {2,6} {1,2,7}
{1,2,4} {1,3,5} {3,6} {1,3,7}
{1,3,4} {1,4,5} {1,2,6} {1,4,7}
{2,3,5} {1,3,6} {1,5,7}
{1,4,6} {1,6,7}
{1,5,6} {2,3,7}
{2,4,6} {2,5,7}
{1,2,3,6} {3,4,7}
{1,2,3,7}
{1,2,4,7}
Without re-usable parts we have
A365069, first differences of
A364534.
A085489 and
A364755 count subsets with no sum of two distinct elements.
A088314 counts sets that can be linearly combined to obtain n.
A088809 and
A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement
A364839.
A364913 counts combination-full 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[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combp[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]
A365070
Number of subsets of {1..n} containing n and some element equal to the sum of two other (possibly equal) elements.
Original entry on oeis.org
0, 0, 1, 1, 5, 9, 24, 46, 109, 209, 469, 922, 1932, 3858, 7952, 15831, 32214, 64351, 129813, 259566, 521681, 1042703, 2091626, 4182470, 8376007, 16752524, 33530042, 67055129, 134165194, 268328011, 536763582, 1073523097, 2147268041, 4294505929, 8589506814, 17178978145
Offset: 0
The subset {1,3} has no element equal to the sum of two others, so is not counted under a(3).
The subset {3,4,5} has no element equal to the sum of two others, so is not counted under a(5).
The subset {1,3,4} has 4 = 1 + 3, so is counted under a(4).
The subset {2,4,5} has 4 = 2 + 2, so is counted under a(5).
The a(0) = 0 through a(5) = 9 subsets:
. . {1,2} {1,2,3} {2,4} {1,2,5}
{1,2,4} {1,4,5}
{1,3,4} {2,3,5}
{2,3,4} {2,4,5}
{1,2,3,4} {1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
The complement w/o re-usable parts is
A085489, first differences of
A364755.
The case without re-usable parts is
A364756, firsts differences of
A088809.
A364350 counts combination-free strict partitions, complement
A364839.
A364913 counts combination-full partitions.
A365006 counts no positive combination-full strict ptns.
-
Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#,Total /@ Tuples[#,2]]!={}&]], {n,0,10}]
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