A365044 -id:A365044 - cp's OEIS frontend

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

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

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).

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

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.

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

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

Gus Wiseman, Aug 23 2023

nonn

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

Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.

			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}

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

The nonnegative complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A365043.
The complement is counted by A365045, first differences of A365044.
The nonnegative version is A365046, first differences of A364914.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
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.
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[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combp[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A088314(n) - 1.

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

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(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)

The nonnegative version is A364915, strict A364350.
The strict case is A365006.
For subsets instead of partitions we have A365044, complement A365043.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A237667 counts sum-free partitions, binary A236912.
A364912 counts positive linear combinations of partitions.
A365068 counts combination-full partitions, strict A364839.
Cf. A085489, A108917, A151897, A325862, A364272, A364910, A364911, 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[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

a(31)-a(49) from Chai Wah Wu, Sep 20 2023

cp's OEIS Frontend

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

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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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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.

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A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

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A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

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