A364349 -id:A364349 - cp's OEIS frontend

A365070 Number of subsets of {1..n} containing n and some element equal to the sum of two other (possibly equal) elements.

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

Gus Wiseman, Aug 24 2023

nonn

These are binary sum-full sets where elements can be re-used. The complement is counted by A288728. The non-binary version is A365046, complement A124506. For non-re-usable parts we have A364756, complement A085489.

			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}

Andrew Howroyd, Table of n, a(n) for n = 0..88
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement w/o re-usable parts is A085489, first differences of A364755.
First differences of A093971.
The non-binary complement is A124506, first differences of A326083.
The complement is counted by A288728, first differences of A007865.
For partitions (not requiring n) we have A363225, strict A363226.
The case without re-usable parts is A364756, firsts differences of A088809.
The non-binary version is A365046, first differences of A364914.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
A365006 counts no positive combination-full strict ptns.
Cf. A050291, A051026, A151897, A236912, A237113, A237668, A326080, A364349, A364533, A364670.

Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#,Total /@ Tuples[#,2]]!={}&]], {n,0,10}]

First differences of A093971.

a(21) onwards added (using A093971) by Andrew Howroyd, Jan 13 2024

A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 17, 41, 88, 201, 418, 892, 1838, 3798, 7716, 15740

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

The complement is counted by A365071. The binary case is A364756. Allowing elements to be re-used gives A365070. A version for partitions (but not requiring n) is A237668.

			The subset {2,4,6} has 6 = 4 + 2 so is counted under a(6).
The subset {1,2,4,7} has 7 = 4 + 2 + 1 so is counted under a(7).
The subset {1,4,5,8} has 5 = 4 + 1 so is counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                    {1,2,3,4}  {2,3,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

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

The complement w/ re-usable parts is A288728, first differences of A007865.
First differences of A364534.
The binary complement is A364755, first differences of A085489.
The binary version is A364756, first differences of A088809.
The version with re-usable parts is A365070, first differences of A093971.
The complement is counted by A365071, first differences of A151897.
A124506 counts nonnegative combination-free subsets, differences of A326083.
A365046 counts nonnegative combination-full subsets, differences of A364914.
For partitions: A108917, A236912, A237113, A237668, A364532, A364913.
Strict partitions: A116861, A364272, A364349, A364350, A364839, A364916.
Cf. A050291, A326080, A363226, A364346, A364348, A364670.

Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#, Total/@Subsets[#, {2,Length[#]}]]!={}&]],{n,0,10}]

a(n) = 2^(n-1) - A365070(n).

First differences of A364534.

A365071 Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 15, 23, 40, 55, 94, 132, 210, 298, 476, 644, 1038, 1406, 2149, 2965, 4584, 6077, 9426, 12648, 19067, 25739, 38958, 51514, 78459, 104265, 155436, 208329, 312791, 411886, 620780, 823785, 1224414, 1631815, 2437015, 3217077, 4822991

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

The complement is counted by A365069. The binary version is A364755, complement A364756. For re-usable parts we have A288728, complement A365070.

			The subset {1,3,4,6} has 4 = 1 + 3 so is not counted under a(6).
The subset {2,3,4,5,6} has 6 = 2 + 4 and 4 = 1 + 3 so is not counted under a(6).
The a(0) = 0 through a(6) = 15 subsets:
  .  {1}  {2}    {3}    {4}      {5}      {6}
          {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
                 {2,3}  {2,4}    {2,5}    {2,6}
                        {3,4}    {3,5}    {3,6}
                        {1,2,4}  {4,5}    {4,6}
                        {2,3,4}  {1,2,5}  {5,6}
                                 {1,3,5}  {1,2,6}
                                 {2,4,5}  {1,3,6}
                                 {3,4,5}  {1,4,6}
                                          {2,3,6}
                                          {2,5,6}
                                          {3,4,6}
                                          {3,5,6}
                                          {4,5,6}
                                          {3,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..85
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

First differences of A151897.
The version with re-usable parts is A288728 first differences of A007865.
The binary version is A364755, first differences of A085489.
The binary complement is A364756, first differences of A088809.
The complement is counted by A365069, first differences of A364534.
The complement w/ re-usable parts is A365070, first differences of A093971.
A108917 counts knapsack partitions, strict A275972.
A124506 counts combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
Partitions: A236912, A237113, A237668, A364532, A364272, A364349, A364913.
Cf. A050291, A095944, A103580, A324741, A326080, A326117, A341507, A364533.

Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#, Total/@Subsets[#,{2,Length[#]}]]=={}&]], {n,0,10}]

a(n) + A365069(n) = 2^(n-1).

First differences of A151897.

a(14) onwards added (using A151897) by Andrew Howroyd, Jan 13 2024

A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037

Offset: 0

Gus Wiseman, Sep 16 2023

Are n = 1, 2, 4 the only n such that none of these partitions has 1?

Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?

			The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
            (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
            (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
                                (8,1,1)      (6,2,2,2)      (9,3,2)
                                (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
                                (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
                                                            (11,2,1)
                                                            (4,4,4,2)
                                                            (5,3,3,3)
                                                            (6,4,2,2)
                                                            (8,2,2,2)
                                                            (11,1,1,1)
                                                            (4,4,2,2,2)
                                                            (6,2,2,2,2)
                                                            (4,2,2,2,2,2)
                                                            (2,2,2,2,2,2,2)

For n instead of 2n we have A126796.
Central column n = 2k of A365658.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002219 counts partitions of 2n with a submultiset summing to n.
A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
A122768 counts distinct nonempty submultisets of partitions.
A299701 counts sums of submultisets of prime indices, of partitions A304792.
A364272 counts sum-full strict partitions, sum-free A364349.
A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
Cf. A000041, A008967, A095944, A364911, A365376, A365377.

msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
    c = 0
    for p in partitions(n<<1):
        q, s = list(Counter(p).elements()), set()
        for l in range(1,len(q)+1):
            for k in multiset_combinations(q,l):
                s.add(sum(k))
                if len(s) > n:
                    break
            else:
                continue
            break
        if len(s)==n:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(21)-a(38) from Chai Wah Wu, Sep 20 2023

a(39)-a(43) from Chai Wah Wu, Sep 21 2023

A366317 Number of unordered pairs of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 15, 21, 36, 55, 78, 120, 171, 253, 378, 528, 741, 1081, 1485, 2080, 2926, 4005, 5460, 7503, 10153, 13695, 18528, 24753, 32896, 43956, 57970, 76245, 100576, 131328, 171405, 223446, 289180, 373680, 482653, 619941, 794430, 1017451, 1296855

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(1) = 1 through a(7) = 15 unordered pairs of strict partitions:
  {1,1}  {2,2}  {3,3}    {4,4}    {5,5}    {6,6}      {7,7}
                {3,21}   {4,31}   {5,32}   {6,42}     {7,43}
                {21,21}  {31,31}  {5,41}   {6,51}     {7,52}
                                  {32,32}  {42,42}    {7,61}
                                  {32,41}  {42,51}    {43,43}
                                  {41,41}  {51,51}    {43,52}
                                           {6,321}    {43,61}
                                           {42,321}   {52,52}
                                           {51,321}   {52,61}
                                           {321,321}  {61,61}
                                                      {7,421}
                                                      {43,421}
                                                      {52,421}
                                                      {61,421}
                                                      {421,421}

For non-strict partitions we have A086737.
The disjoint case is A108796, non-strict A260669.
The ordered version is A304990, disjoint A032302.
The ordered disjoint case is A365662.
Excluding constant pairs gives A366132.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000712, A007582, A054440, A064914, A260664.

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2],OrderedQ]],{n,0,30}]

a(n) = A000217(A000009(n)).

Composition of A000009 and A000217.

A364841 Number of subsets S of {1..n} containing no element equal to the sum of a k-multiset of elements of S, for any 2 <= k <= |S|.

Original entry on oeis.org

1, 2, 3, 6, 9, 15, 21, 34, 49, 75, 105

Offset: 0

Gus Wiseman, Aug 15 2023

			The a(0) = 1 through a(5) = 15 subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {1,3}  {4}    {4}
                {2,3}  {1,3}  {5}
                       {1,4}  {1,3}
                       {2,3}  {1,4}
                       {3,4}  {1,5}
                              {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {3,4,5}

Cf. A007865, A085489, A103580, A151897, A236912, A326020, A326080, A326083, A364349, A364534.

Table[Length[Select[Subsets[Range[n]], Intersection[#,Join@@Table[Total/@Tuples[#,k], {k,2,Length[#]}]]=={}&]],{n,0,10}]

cp's OEIS Frontend

A365070 Number of subsets of {1..n} containing n and some element equal to the sum of two other (possibly equal) elements.

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A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

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A365071 Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

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A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

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A366317 Number of unordered pairs of strict integer partitions of n.

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A364841 Number of subsets S of {1..n} containing no element equal to the sum of a k-multiset of elements of S, for any 2 <= k <= |S|.

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