A364913 -id:A364913 - cp's OEIS frontend

A116861 Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 1, 3, 1, 1, 3, 1, 1, 4, 1, 0, 3, 2, 2, 2, 5, 1, 1, 3, 3, 2, 4, 2, 6, 1, 0, 5, 2, 3, 4, 4, 3, 8, 1, 1, 4, 3, 4, 7, 4, 5, 3, 10, 1, 0, 5, 3, 4, 7, 7, 6, 6, 5, 12, 1, 1, 6, 4, 3, 12, 6, 8, 7, 9, 5, 15, 1, 0, 6, 4, 5, 10, 10, 9, 10, 11, 10, 7, 18, 1, 1, 6, 4, 5, 15, 11, 13, 9, 16, 11, 13, 8, 22

Offset: 1

Emeric Deutsch, Feb 27 2006

Conjecture: Reverse the rows of the table to get an infinite lower-triangular matrix b with 1's on the main diagonal. The third diagonal of the inverse of b is minus A137719. - George Beck, Oct 26 2019
Proof: The reversed rows yield the matrix I+N where N is strictly lower triangular, N[i,j] = 0 for j >= i, having its 2nd diagonal equal to the 2nd column (1, 0, 1, 0, 1, ...): N[i+1,i] = A000035(i), i >= 1, and 3rd diagonal equal to the 3rd column of this triangle, (2, 1, 2, 3, 3, 3, ...): N[i+2,i] = A137719(i), i >= 1. It is known that (I+N)^-1 = 1 - N + N^2 - N^3 +- .... Here N^2 has not only the second but also the 3rd diagonal zero, because N^2[i+2,i] = N[i+2,i+1]*N[i+1,i] = A000035(i+1)*A000035(i) = 0. Therefore the 3rd diagonal of (I+N)^-1 is equal to -A137719 without leading 0. - M. F. Hasler, Oct 27 2019
From Gus Wiseman, Aug 27 2023: (Start)
Also the number of ways to write n-k as a nonnegative linear combination of a strict integer partition of k. Also the number of ways to write n as a (strictly) positive linear combination of a strict integer partition of k. Row n=7 counts the following:
  7*1  .  1*2+5*1  1*3+4*1  1*3+2*2  1*5+2*1      1*7
          2*2+3*1  2*3+1*1  1*4+3*1  1*3+1*2+2*1  1*4+1*3
          3*2+1*1                                 1*5+1*2
                                                  1*6+1*1
                                                  1*4+1*2+1*1
(End)

			T(10,7) = 4 because we have [6,1,1,1,1], [4,3,3], [4,2,2,1,1] and [4,2,1,1,1,1] (6+1=4+3=4+2+1=7).
Triangle starts:
  1;
  1, 1;
  1, 0, 2;
  1, 1, 1, 2;
  1, 0, 2, 1, 3;
  1, 1, 3, 1, 1,  4;
  1, 0, 3, 2, 2,  2, 5;
  1, 1, 3, 3, 2,  4, 2, 6;
  1, 0, 5, 2, 3,  4, 4, 3, 8;
  1, 1, 4, 3, 4,  7, 4, 5, 3, 10;
  1, 0, 5, 3, 4,  7, 7, 6, 6,  5, 12;
  1, 1, 6, 4, 3, 12, 6, 8, 7,  9,  5, 15;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (16)(1).

Cf. A000041 (row sums), A000009 (diagonal), A014153.
Cf. A114638 (count partitions with #parts = sum(distinct parts)).
Column 1: A000012, column 2: A000035(1..), column 3: A137719(1..).
For subsets instead of partitions we have A026820.
This statistic is ranked by A066328.
The central diagonal is T(2n,n) = A364910(n), non-strict A364907.
Partial sums of columns are columns of A364911.
Same as A364916 (offset 0) with rows reversed.
A008284 counts partitions by length, strict A008289.
A364912 counts linear combinations of partitions.
A364913 counts combination-full partitions, strict A364839.
Cf. A237667, A364272, A364915, A365002, A365004, A365006.

g:= -1+product(1+t^j*x^j/(1-x^j), j=1..40): gser:= simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local f, g, j;
      if n=0 then [1] elif i<1 then [ ] else f:= b(n, i-1);
         for j to n/i do
           f:= zip((x, y)->x+y, f, [0$i, b(n-i*j, i-1)[]], 0)
         od; f
      fi
    end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 27 2013

max = 14; s = Series[-1+Product[1+t^j*x^j/(1-x^j), {j, 1, max}], {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[Length[Select[IntegerPartitions[n],Total[Union[#]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Aug 29 2023 *)

A116861(n,k,s=0)={forpart(X=n,vecsum(Set(X))==k&&s++,k);s} \\ M. F. Hasler, Oct 27 2019

G.f.: -1 + Product_{j>=1} (1 + t^j*x^j/(1-x^j)).
Sum_{k=1..n} T(n,k) = A000041(n).
T(n,n) = A000009(n).
Sum_{k=1..n} k*T(n,k) = A014153(n-1).
T(n,1) = 1. T(n,2) = A000035(n+1). T(n,3) = A137719(n-2). - R. J. Mathar, Oct 27 2019
T(n,4) = A002264(n-1) + A121262(n). - R. J. Mathar, Oct 28 2019

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 15 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(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)

For sums of subsets instead of combinations of partitions we have A151897.
For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The complement in strict partitions is A364839, non-strict A364913.
A more strict variation is A364915.
The case of all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A116861 and A364916 count linear combinations of strict partitions.
A323092 (ranks A320340) and A120641 count double-free partitions.
A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.

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}]

from sympy.utilities.iterables import partitions
def A364350(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):
        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
    return c # Chai Wah Wu, Sep 23 2023

More terms and offset corrected by Martin Fuller, Sep 11 2023

A364914 Number of subsets of {1..n} such that some element can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 9, 20, 48, 101, 219, 454, 944, 1917, 3925, 7915, 16004, 32188, 64751, 129822, 260489, 521672, 1045060, 2091808, 4187047, 8377255, 16762285, 33531228, 67077485, 134170217, 268371678, 536772231, 1073611321, 2147282291, 4294697258, 8589527163, 17179321094

Offset: 0

Gus Wiseman, Aug 17 2023

nonn

A variation of non-binary combination-full sets where parts can be re-used. The complement is counted by A326083. The binary version is A093971. For non-re-usable parts we have A364534. First differences are A365046.

			The set {3,4,5,17} has 17 = 1*3 + 1*4 + 2*5, so is counted under a(17).
The a(0) = 0 through a(5) = 20 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,3}    {1,3}      {1,3}
               {1,2,3}  {1,4}      {1,4}
                        {2,4}      {1,5}
                        {1,2,3}    {2,4}
                        {1,2,4}    {1,2,3}
                        {1,3,4}    {1,2,4}
                        {2,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,3,5}
                                   {1,4,5}
                                   {2,3,4}
                                   {2,3,5}
                                   {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}

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

The binary complement is A007865.
The binary version without re-usable parts is A088809.
The binary version is A093971.
The complement without re-usable parts is A151897.
The complement is counted by A326083.
The version without re-usable parts is A364534.
The version for strict partitions is A364839, complement A364350.
The version for partitions is A364913.
The version for positive combinations is A365043, complement A365044.
First differences are A365046.
Cf. A011782, A085489, A103580, A116861, A124506, A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364756.

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]],Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A364914(n):
    c, mlist = 0, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m,k=m-1):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(2,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for d in w:
                for s in mlist[d-1]:
                    if s <= ws:
                        c += 1
                        break
                else:
                    continue
                break
    return c # Chai Wah Wu, Nov 17 2023

a(12)-a(34) from Chai Wah Wu, Nov 17 2023

A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 7, 12, 12, 17, 20, 26, 29, 39, 43, 54, 62, 77, 88, 107, 122, 148, 168, 200, 229, 267, 308, 360, 407, 476, 536, 623, 710, 812, 917, 1050, 1190, 1349, 1530, 1733, 1944, 2206, 2483, 2794, 3138, 3524

Offset: 0

Gus Wiseman, Aug 19 2023

nonn

			For y = (4,3,2) we can write 4 = 0*3 + 2*2, so y is counted under a(9).
For y = (11,5,3) we can write 11 = 1*5 + 2*3, so y is counted under a(19).
For y = (17,5,4,3) we can write 17 = 1*3 + 1*4 + 2*5, so y is counted under a(29).
The a(1) = 0 through a(12) = 12 strict partitions (A = 10, B = 11):
  .  .  (21)  (31)  (41)  (42)   (61)   (62)   (63)   (82)    (A1)    (84)
                          (51)   (421)  (71)   (81)   (91)    (542)   (93)
                          (321)         (431)  (432)  (532)   (632)   (A2)
                                        (521)  (531)  (541)   (641)   (B1)
                                               (621)  (631)   (731)   (642)
                                                      (721)   (821)   (651)
                                                      (4321)  (5321)  (732)
                                                                      (741)
                                                                      (831)
                                                                      (921)
                                                                      (5421)
                                                                      (6321)

For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
Non-strict versions are A364913 and the complement of A364915.
For subsets instead of partitions we have A364914, complement A326083.
The case of no all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A363226, A365002.

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}]

from sympy.utilities.iterables import partitions
def A364839(n):
    if n <= 1: return 0
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if 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

A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 2, 0, 1, 0, 4, 1, 1, 3, 1, 1, 0, 5, 2, 2, 2, 3, 0, 1, 0, 6, 2, 4, 2, 3, 3, 1, 1, 0, 8, 3, 4, 4, 3, 2, 5, 0, 1, 0, 10, 3, 5, 4, 7, 4, 3, 4, 1, 1, 0, 12, 5, 6, 6, 7, 7, 4, 3, 5, 0, 1, 0, 15, 5, 9, 7, 8, 6, 12, 3, 4, 6, 1, 1, 0

Offset: 0

Gus Wiseman, Aug 17 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).

As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.

			Array begins:
  1  1  1  2  2  3  4   5   6   8   10   12  15   18   22   27
  0  1  0  1  1  1  2   2   3   3   5    5   7    8    10   12
  0  1  1  2  1  2  4   4   5   6   9    10  13   15   19   23
  0  1  0  3  2  2  4   4   6   7   11   11  15   17   22   27
  0  1  1  3  3  3  7   7   8   10  16   17  23   27   33   42
  0  1  0  3  2  4  7   6   9   9   17   17  23   26   33   43
  0  1  1  5  3  4  12  10  13  16  26   27  36   42   52   68
  0  1  0  4  3  3  10  11  13  13  27   25  35   40   51   67
  0  1  1  5  4  5  15  13  19  20  36   37  51   58   72   97
  0  1  0  6  4  5  14  13  18  23  42   39  54   61   78   105
  0  1  1  6  4  6  20  17  23  25  54   50  69   80   98   138
  0  1  0  6  4  5  19  16  23  24  54   55  71   80   103  144
  0  1  1  8  6  7  27  23  30  35  72   70  103  113  139  199
  0  1  0  7  5  6  24  21  29  31  75   68  95   115  139  201
  0  1  1  8  5  7  31  27  36  39  90   86  122  137  178  255
  0  1  0  9  6  8  31  27  38  42  100  93  129  148  187  289
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   3  1  2  0  1  0
   4  1  1  3  1  1  0
   5  2  2  2  3  0  1  0
   6  2  4  2  3  3  1  1  0
   8  3  4  4  3  2  5  0  1  0
  10  3  5  4  7  4  3  4  1  1  0
  12  5  6  6  7  7  4  3  5  0  1  0
  15  5  9  7  8  6 12  3  4  6  1  1  0
  18  7 10 11 10  9 10 10  5  4  6  0  1  0
  22  8 13 11 16  9 13 11 15  5  4  6  1  1  0
  27 10 15 15 17 17 16 13 13 14  6  4  8  0  1  0

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

Same as A116861 with offset 0 and rows reversed, non-strict version A364912.
Row n = 0 is A000009.
Row n = 1 is A096765.
Row n = 2 is A365005.
Column k = 0 is A000007.
Column k = 1 is A000012.
Column k = 2 is A000035.
Column k = 3 is A137719.
The main diagonal is A364910.
Left half has row sums A365002.
For not just strict partitions we have A365004, diagonal A364907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A066328 adds up distinct prime indices.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A007865, A085489, A108917, A237113, A323092, A364272, A364533, A364670, A364911, A364913.

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

A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 1, 2, 6, 11, 28, 53, 118, 235, 490, 973, 2008, 3990, 8089, 16184, 32563, 65071, 130667, 261183, 523388, 1046748, 2095239, 4190208, 8385030, 16768943, 33546257, 67092732, 134201461, 268400553, 536839090, 1073670970, 2147414967, 4294829905, 8589793931

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

Includes all subsets containing both 1 and n.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,4}      {1,2,5}
                        {1,2,4}    {1,3,5}
                        {1,3,4}    {1,4,5}
                        {2,3,4}    {2,3,5}
                        {1,2,3,4}  {2,4,5}
                                   {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 complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A364914.
The positive version is A365042, first differences of A365043.
The positive complement is counted by A365045, first differences of A365044.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
A364350 counts combination-free strict partitions, complement A364839.
A085489 and A364755 count subsets without the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

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]],MemberQ[#,n]&&Or@@Table[combs[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n+1) = 2^n - A124506(n).

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

Gus Wiseman, Aug 22 2023

nonn

			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 sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The strict case is A364350.
The complement is A365068, strict A364839.
The positive case is A365072, strict A365006.
A000041 counts integer partitions, strict A000009.
A007865 counts binary sum-free sets w/ re-usable parts, complement A093971.
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. A085489, A108917, A151897, A237113, A323092, A364272, A364533, A364910, A364911, 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[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

a(n) = A000041(n) - A365068(n).

a(37)-a(59) from Chai Wah Wu, Sep 25 2023

A364911 Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 5, 2, 5, 3, 3, 1, 6, 3, 8, 4, 4, 4, 1, 7, 3, 11, 6, 6, 6, 5, 1, 8, 4, 14, 9, 8, 10, 7, 6, 1, 9, 4, 19, 11, 11, 14, 11, 9, 8, 1, 10, 5, 23, 14, 15, 21, 15, 14, 11, 10, 1, 11, 5, 28, 17, 19, 28, 22, 20, 17, 15, 12

Offset: 0

Gus Wiseman, Aug 27 2023

Also the number of ways to write any number up to n as a positive linear combination of a strict integer partition of k.

			Triangle begins:
  1
  1  1
  1  2  1
  1  3  1  2
  1  4  2  3  2
  1  5  2  5  3  3
  1  6  3  8  4  4  4
  1  7  3 11  6  6  6  5
  1  8  4 14  9  8 10  7  6
  1  9  4 19 11 11 14 11  9  8
  1 10  5 23 14 15 21 15 14 11 10
  1 11  5 28 17 19 28 22 20 17 15 12
  1 12  6 34 21 22 40 28 28 24 24 17 15
  1 13  6 40 25 27 50 38 37 34 35 27 22 18
  1 14  7 46 29 32 65 49 50 43 51 38 35 26 22
  1 15  7 54 33 38 79 62 63 59 68 55 50 41 32 27
Row n = 5 counts the following partitions:
    .    1           2     3         4       5
         1+1         2+2   1+2       1+3     1+4
         1+1+1             1+1+2     1+1+3   2+3
         1+1+1+1           1+1+1+2
         1+1+1+1+1         1+2+2
Row n = 5 counts the following positive linear combinations:
  .  1*1  1*2  1*3      1*4      1*5
     2*1  2*2  1*2+1*1  1*3+1*1  1*3+1*2
     3*1       1*2+2*1  1*3+2*1  1*4+1*1
     4*1       1*2+3*1
     5*1       2*2+1*1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column n = k is A000009.
Column k = 0 is A000012.
Column k = 1 is A000027.
Row sums are A000070.
Column k = 2 is A008619.
Columns are partial sums of columns of A116861.
Column k = 3 appears to be the partial sums of A137719.
Diagonal n = 2k is A364910.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A114638 counts partitions where (length) = (sum of distinct parts).
A116608 counts partitions by number of distinct parts.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002865, A066328, A179009, A236912, A237113, A237667, A364912, A364913, A364915, A364916, A365002, A365004.

Table[Length[Select[Array[IntegerPartitions,n+1,0,Join],Total[Union[#]]==k&]],{n,0,9},{k,0,n}]

T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

G.f.: A(x,y) = (1/(1 - x)) * Product_{k>=1} (1 - y^k + y^k/(1 - x^k)). - Andrew Howroyd, Jan 11 2024

A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 21, 32, 49, 70, 99, 135, 185, 245, 323, 418, 541, 688, 873, 1094, 1368, 1693, 2092, 2564, 3138, 3810, 4620, 5565, 6696, 8012, 9569, 11381, 13518, 15980, 18872, 22194, 26075, 30535, 35711, 41627, 48473, 56290, 65283, 75533, 87298, 100631, 115911, 133219

Offset: 0

Gus Wiseman, Aug 25 2023

nonn

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} such that some element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,4,9} has 9 = 3*3 + 0*4, but this is not strictly positive, so S is not counted under a(9).
The subset S = {3,4,10} has 10 = 2*3 + 1*4, so S is counted under a(10).
The a(0) = 0 through a(5) = 12 subsets:
  .  .  {1,2}  {1,2}    {1,2}    {1,2}
               {1,3}    {1,3}    {1,3}
               {1,2,3}  {1,4}    {1,4}
                        {2,4}    {1,5}
                        {1,2,3}  {2,4}
                        {1,2,4}  {1,2,3}
                        {1,3,4}  {1,2,4}
                                 {1,2,5}
                                 {1,3,4}
                                 {1,3,5}
                                 {1,4,5}
                                 {2,3,5}

Bert Dobbelaere, Table of n, a(n) for n = 0..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary complement is A007865, first differences A288728.
The binary version is A093971, first differences A365070.
The nonnegative complement is A326083, first differences A124506.
The nonnegative version is A364914, first differences A365046.
First differences are A365042.
The complement is counted by A365044, first differences A365045.
Without re-usable parts we have A364534, first differences A365069.
A085489 and A364755 count subsets with no sum of two distinct elements.
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.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

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]]],combp[Last[#],Union[Most[#]]]!={}&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365043(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365044(n).

a(15)-a(35) from Chai Wah Wu, Nov 20 2023

More terms from Bert Dobbelaere, Apr 28 2025

A365312 Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 05 2023

nonn

			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)

The complement for positive coefficients is counted by A088314.
For positive coefficients we have A088528.
The complement is counted by A365311.
For non-strict partitions we have A365378, complement A365379.
The version for subsets is A365380, complement A365073.
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,0,10}]

from math import isqrt
from sympy.utilities.iterables import partitions
def A365312(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 not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(26)-a(58) from Chai Wah Wu, Sep 13 2023

cp's OEIS Frontend

A116861 Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).

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A364350 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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A364914 Number of subsets of {1..n} such that some element can be written as a nonnegative linear combination of the others.

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A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

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A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.

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A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

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A364915 Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.

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A364911 Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.

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