A364913 -id:A364913 - cp's OEIS frontend

A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

0, 1, 2, 3, 5, 6, 11, 12, 20, 24, 35, 38, 63, 63, 92, 112, 148, 160, 230, 244, 339, 383, 478, 533, 726, 781, 978, 1123, 1394, 1526, 1960, 2112, 2630, 2945, 3518, 3964, 4856, 5261, 6307, 7099, 8464, 9258, 11140, 12155, 14419, 16093, 18589, 20565, 24342, 26597, 30948

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

			The strict partition (6,3) cannot be linearly combined to obtain 10, so is not counted under a(10).
The strict partition (4,2) has 6 = 1*4 + 1*2 so is counted under a(6), but (4,2) cannot be linearly combined to obtain 7 so is not counted under a(7).
The a(1) = 1 through a(7) = 12 strict partitions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (2,1)  (4)    (2,1)  (3)      (2,1)
                   (2,1)  (3,1)  (6)      (3,1)
                   (3,1)  (3,2)  (2,1)    (3,2)
                          (4,1)  (3,1)    (4,1)
                                 (3,2)    (4,3)
                                 (4,1)    (5,1)
                                 (4,2)    (5,2)
                                 (5,1)    (6,1)
                                 (3,2,1)  (3,2,1)
                                          (4,2,1)

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

For positive coefficients we have A088314.
The positive complement is counted by A088528.
The version for subsets is A365073.
The complement is counted by A365312.
For non-strict partitions we have A365379.
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.
Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

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,10}]

from math import isqrt
from sympy.utilities.iterables import partitions
def A365311(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,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(26)-a(50) from Chai Wah Wu, Sep 13 2023

A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 4, 5, 0, 1, 4, 8, 7, 7, 0, 1, 6, 13, 17, 12, 11, 0, 1, 6, 18, 28, 30, 19, 15, 0, 1, 8, 24, 50, 58, 53, 30, 22

Offset: 0

Gus Wiseman, Aug 20 2023

A way of writing n as a positive 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),(2,2)) are a way of writing 8 as a positive linear combination of (1,1,2), namely 8 = 3*1 + 1*1 + 2*2.

			Triangle begins:
  1
  0  1
  0  1  2
  0  1  2  3
  0  1  4  4  5
  0  1  4  8  7  7
  0  1  6 13 17 12 11
  0  1  6 18 28 30 19 15
  0  1  8 24 50 58 53 30 22
Row n = 4 counts the following linear combinations:
  .  1*4  2*2      2*1+1*2      4*1
          1*1+1*3  1*1+1*1+1*2  3*1+1*1
          1*2+1*2  1*1+1*2+1*1  2*1+2*1
          1*3+1*1  1*2+1*1+1*1  2*1+1*1+1*1
                                1*1+1*1+1*1+1*1
Row n = 5 counts the following linear combinations:
  .  1*5  1*1+1*4  2*1+1*3      3*1+1*2          5*1
          1*2+1*3  2*2+1*1      2*1+1*1+1*2      4*1+1*1
          1*3+1*2  1*1+1*1+1*3  2*1+1*2+1*1      3*1+2*1
          1*4+1*1  1*1+1*2+1*2  1*1+1*1+1*1+1*2  3*1+1*1+1*1
                   1*1+1*3+1*1  1*1+1*1+1*2+1*1  2*1+2*1+1*1
                   1*2+1*1+1*2  1*1+1*2+1*1+1*1  2*1+1*1+1*1+1*1
                   1*2+1*2+1*1  1*2+1*1+1*1+1*1  1*1+1*1+1*1+1*1+1*1
                   1*3+1*1+1*1
Array begins:
  1   0   0   0    0    0    0     0
  1   1   1   1    1    1    1     1
  2   2   4   4    6    6    8     8
  3   4   8   13   18   24   33    40
  5   7   17  28   50   70   107   143
  7   12  30  58   108  179  286   428
  11  19  53  109  223  394  696   1108
  15  30  86  194  420  812  1512  2619

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row k = 0 is A000007.
Row k = 1 is A000012.
Column n = 0 is A000041.
Column n = 1 is A000070.
Row sums are A006951.
Row k = 2 is A052928 except initial terms.
The case of strict integer partitions is A116861.
Central column is T(2n,n) = A(n,n) = A364907(n).
With rows reversed we have the nonnegative version A365004.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A237113, A364272, A364910, A364911, A364914, A364915, A365002.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combp[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]
- or -
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n-k,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]

As an array, also the number of ways to write n-k as a nonnegative linear combination of an integer partition of k (see programs).

A365002 Number of ways to write n as a nonnegative linear combination of a strict integer partition.

Original entry on oeis.org

1, 1, 2, 4, 8, 10, 26, 32, 63, 84, 157, 207, 383, 477, 768, 1108, 1710, 2261, 3536, 4605, 6869, 9339, 13343, 17653, 25785, 33463, 46752, 61549, 85614, 110861, 153719, 197345, 268623, 346845, 463513, 593363, 797082, 1011403, 1335625, 1703143, 2232161, 2820539

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

A way of writing n as a (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).

			The a(1) = 1 through a(5) = 10 ways:
  1*1  1*2  1*3      1*4      1*5
       2*1  3*1      2*2      5*1
            0*2+3*1  4*1      0*2+5*1
            1*2+1*1  0*2+4*1  0*3+5*1
                     0*3+4*1  0*4+5*1
                     1*2+2*1  1*2+3*1
                     1*3+1*1  1*3+1*2
                     2*2+0*1  1*3+2*1
                              1*4+1*1
                              2*2+1*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums of lower-left half of A364916 as an array.
Column sums of right half of A364916 as a triangle.
For all positive coefficients we have A000041, non-strict A006951.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A066328, A116861, A364272, A364907, A364910, A364911, A364912, A365004.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Sum[Length[combs[n,y]], {y,Select[Join@@IntegerPartitions/@Range[n], UnsameQ@@#&]}],{n,0,15}]

from itertools import combinations
from collections import Counter
from sympy.utilities.iterables import partitions
def A365002(n):
    aset = Counter(tuple(sorted(set(p))) for p in partitions(n))
    return sum(sum(aset[t] for t in aset if set(t).issubset(set(q))) for l in range(1,n+1) for q in combinations(range(1,n+1),l) if sum(q)<=n) # Chai Wah Wu, Sep 20 2023

a(16)-a(34) from Chai Wah Wu, Sep 20 2023
a(35)-a(38) from Chai Wah Wu, Sep 21 2023
a(0)=1 and a(39)-a(41) from Alois P. Heinz, Jan 11 2024

A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 4, 4, 1, 0, 7, 7, 8, 4, 1, 0, 11, 12, 17, 13, 6, 1, 0, 15, 19, 30, 28, 18, 6, 1, 0, 22, 30, 53, 58, 50, 24, 8, 1, 0, 30, 45, 86, 109, 108, 70, 33, 8, 1, 0, 42, 67, 139, 194, 223, 179, 107, 40, 10, 1, 0, 56, 97, 213, 328, 420, 394, 286, 143, 50, 10, 1, 0

Offset: 0

Gus Wiseman, Aug 23 2023

A way of writing n as a (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).

			Array begins:
  1  1  2   3   5    7     11
  0  1  2   4   7    12    19
  0  1  4   8   17   30    53
  0  1  4   13  28   58    109
  0  1  6   18  50   108   223
  0  1  6   24  70   179   394
  0  1  8   33  107  286   696
  0  1  8   40  143  428   1108
  0  1  10  50  199  628   1754
  0  1  10  61  254  882   2622
  0  1  12  72  332  1215  3857
  0  1  12  84  410  1624  5457
  0  1  14  99  517  2142  7637
The A(4,2) = 6 ways:
  2*2
  0*1+4*1
  1*1+3*1
  2*1+2*1
  3*1+1*1
  4*1+0*1

Alois P. Heinz, Rows n = 0..200, flattened

Row n = 0 is A000041, strict A000009.
Row n = 1 is A000070.
Column k = 0 is A000007.
Column k = 1 is A000012.
Column k = 2 is A052928 except initial terms.
Antidiagonal sums are A006951.
The case of strict integer partitions is A116861.
Main diagonal is A364907.
The transpose is A364912, also the positive version.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A066328, A108917, A237113, A364272, A364910, A364911, A364915, A365002.

b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
    end:
A:= (n, k)-> b(k$2, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 28 2024

nn=5;
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
tabv=Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,nn},{k,0,nn}]
Table[tabv[[k+1,n-k+1]],{n,0,nn},{k,0,n}]

Also the number of ways to write n-k as a *positive* linear combination of an integer partition of k.

Antidiagonals 8-11 from Alois P. Heinz, Jan 28 2024

A364907 Number of ways to write n as a nonnegative linear combination of an integer partition of n.

Original entry on oeis.org

1, 1, 4, 13, 50, 179, 696, 2619, 10119, 38867, 150407, 582065, 2260367, 8786919, 34225256, 133471650, 521216494, 2037608462, 7974105052, 31235316275, 122457794193, 480473181271, 1886555402750, 7412471695859, 29142658077266, 114643347181003, 451237737215201

Offset: 0

Gus Wiseman, Aug 18 2023

nonn

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).

			The a(0) = 1 through a(3) = 13 ways:
  0  1*1  1*2      1*3
          0*1+2*1  0*2+3*1
          1*1+1*1  1*2+1*1
          2*1+0*1  0*1+0*1+3*1
                   0*1+1*1+2*1
                   0*1+2*1+1*1
                   0*1+3*1+0*1
                   1*1+0*1+2*1
                   1*1+1*1+1*1
                   1*1+2*1+0*1
                   2*1+0*1+1*1
                   2*1+1*1+0*1
                   3*1+0*1+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

The case with no zero coefficients is A000041.
A finer version is A364906.
The version for compositions is A364908, strict A364909.
Using just strict partitions we get A364910, main diagonal of A364916.
Main diagonal of A365004.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A116861, A364350, A364839, A364911, A364912, A364913, A364914, A364915, A365002, A365003.

b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..27);  # Alois P. Heinz, Jan 28 2024

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[n]}]],{n,0,5}]

a(n) = Sum_{m:A056239(m)=n} A364906(m).
a(n) = A364912(2n,n).
a(n) = A365004(n,n).

a(9)-a(26) from Alois P. Heinz, Jan 28 2024

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

Gus Wiseman, Aug 24 2023

nonn

Also subsets of {1..n} containing n whose greatest element cannot be written as a positive linear combination of the others.

			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}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The nonempty case is A070880.
The nonnegative version is A124506, first differences of A326083.
The binary version is A288728, first differences of A007865.
A subclass is A341507.
The complement is counted by A365042, first differences of A365043.
First differences of A365044.
The nonnegative complement is A365046, first differences of A364914.
The binary complement is A365070, first differences of A093971.
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.
Cf. 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]],MemberQ[#,n]&&And@@Table[combp[#[[k]],Union[Delete[#,k]]]=={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A070880(n) + 1 for n > 0.

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

Gus Wiseman, Sep 04 2023

			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)

The complement for subsets is A365073, positive coefficients A088314.
For strict partitions we have A365312, positive coefficients A088528.
For positive coefficients we have A365323.
The complement is counted by A365379.
The version for subsets is A365380, positive coefficients A365322.
The relatively prime case is A365382.
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.
Cf. A237668, A363225, A364272, A364345, A364914, A365320.

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

a(21)-a(45) from Chai Wah Wu, Sep 13 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

Gus Wiseman, Sep 04 2023

nonn

			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 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.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

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

a(21)-a(43) from 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

Naohiro Nomoto, Nov 16 2003

Also the number of nonempty subsets of {1..n-1} that cannot be linearly combined using all positive coefficients to obtain n. - Gus Wiseman, Sep 10 2023

			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)

Alois P. Heinz, Table of n, a(n) for n = 1..100

For sets with sum < n instead of maximum < n we have A088528.
The complement is counted by A365042, including empty set A088314.
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.
A326083 counts combination-free subsets, complement A364914.
A364350 counts combination-free strict partitions, complement A364913.
Cf. A007865, A085489, A093971, A151897, A326020, A326080, A364534, A365044.

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

a(n) = 2^(n-1) - A088314(n). - Charlie Neder, Feb 08 2019

a(n) = A365045(n) - 1. - Gus Wiseman, Sep 10 2023

Edited by N. J. A. Sloane, Sep 09 2017

a(20)-a(34) from Alois P. Heinz, Feb 08 2019

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

Gus Wiseman, Aug 31 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 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)

Chai Wah Wu, Table of n, a(n) for n = 0..109

The nonnegative version for subsets appears to be A124506.
For sums instead of combinations we have A364349, binary A364533.
The nonnegative version is A364350, complement A364839.
For subsets instead of partitions we have A365044, complement A365043.
The non-strict version is A365072, nonnegative A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364912 counts linear combinations of partitions of k.
Cf. A151897, A237113, A236912, A237667, A275972, A363226, A364272, A364913, A365004, A365068.

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

a(31)-a(56) from Chai Wah Wu, Sep 20 2023

cp's OEIS Frontend

A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

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A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

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A365002 Number of ways to write n as a nonnegative linear combination of a strict integer partition.

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A365004 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of an integer partition of k.

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A364907 Number of ways to write n as a nonnegative linear combination of an integer partition of n.

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A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

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A365378 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.

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