A365311 -id:A365311 - cp's OEIS frontend

A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12

Offset: 0

Gus Wiseman, Sep 16 2023

First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
Rows are palindromic.
Are there only two zeros in the whole triangle?

			Triangle begins:
  1
  1  1
  1  0  1
  2  1  1  2
  2  1  0  1  2
  3  1  1  1  1  3
  4  2  2  1  2  2  4
  5  2  2  2  2  2  2  5
  6  3  2  3  1  3  2  3  6
  8  3  3  4  3  3  4  3  3  8
Row n = 6 counts the following strict partitions:
  (6)      (5,1)    (4,2)    (3,2,1)  (4,2)    (5,1)    (6)
  (5,1)    (3,2,1)  (3,2,1)           (3,2,1)  (3,2,1)  (5,1)
  (4,2)                                                 (4,2)
  (3,2,1)                                               (3,2,1)
Row n = 10 counts the following strict partitions:
  A     91    82    73    64    532   64    73    82    91    A
  64    541   532   532   541   541   541   532   532   541   64
  73    631   721   631   631   4321  631   631   721   631   73
  82    721   4321  721   4321        4321  721   4321  721   82
  91    4321        4321                    4321        4321  91
  532                                                         532
  541                                                         541
  631                                                         631
  721                                                         721
  4321                                                        4321

Robert Price, Table of n, a(n) for n = 0..1325

Columns k = 0 and k = n are A000009.
The non-strict complement is A046663, central column A006827.
Central column n = 2k is A237258.
For subsets instead of partitions we have A365381.
The non-strict case is A365543.
The complement is A365663.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#],k]&]], {n,0,10},{k,0,n}]

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Gus Wiseman, Sep 17 2023

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)

Robert Price, Table of n, a(n) for n = 2..1226
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

A088314 Cardinality of set of sets of parts of all partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 51, 61, 79, 96, 124, 148, 186, 222, 275, 326, 400, 473, 575, 673, 811, 946, 1132, 1317, 1558, 1813, 2138, 2463, 2893, 3323, 3882, 4461, 5177, 5917, 6847, 7818, 8994, 10251, 11766, 13334, 15281, 17309, 19732, 22307

Offset: 0

Naohiro Nomoto, Nov 05 2003

Number of different values of A007947(m) when A056239(m) is equal to n.
From Gus Wiseman, Sep 11 2023: (Start)
Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:
  {1}  {1}  {1}    {1}    {1}    {1}      {1}
       {2}  {3}    {2}    {5}    {2}      {7}
            {1,2}  {4}    {1,2}  {3}      {1,2}
                   {1,2}  {1,3}  {6}      {1,3}
                   {1,3}  {1,4}  {1,2}    {1,4}
                          {2,3}  {1,3}    {1,5}
                                 {1,4}    {1,6}
                                 {1,5}    {2,3}
                                 {2,4}    {2,5}
                                 {1,2,3}  {3,4}
                                          {1,2,3}
                                          {1,2,4}
Cf. A365073, A365311, A365312, A365322, A365380.
(End)

			The 7 partitions of 5 and their sets of parts are
[ #]  partition      set of parts
[ 1]  [ 1 1 1 1 1 ]  {1}
[ 2]  [ 2 1 1 1 ]    {1, 2}
[ 3]  [ 2 2 1 ]      {1, 2}  (same as before)
[ 4]  [ 3 1 1 ]      {1, 3}
[ 5]  [ 3 2 ]        {2, 3}
[ 6]  [ 4 1 ]        {1, 4}
[ 7]  [ 5 ]          {5}
so we have a(5) = |{{1}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {5}}| = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A182410.
The complement in subsets of {1..n-1} is A070880(n) = A365045(n) - 1.
The case of pairs is A365315, see also A365314, A365320, A365321.
A116861 and A364916 count linear combinations of strict partitions.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364914 and A365046 count combination-full subsets.
Cf. A000009, A007865, A088528, A088809, A093971, A326020, A364272, A364534, A365043, A364350.

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

list2set := L -> {op(L)};
a:= N -> list2set(map( list2set, combinat[partition](N) ));
seq(nops(a(n)), n=0..30);
#  Yogy Namara (yogy.namara(AT)gmail.com), Jan 13 2010
b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[],i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..40);
# Alois P. Heinz, Aug 09 2012

Table[Length[Union[Map[Union,IntegerPartitions[n]]]],{n,1,30}] (* Geoffrey Critzer, Feb 19 2013 *)
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i < 1, {},
     Union@Flatten@{b[n, i - 1], Table[If[Head[#] == List,
     Append[#, i]]& /@ b[n - i*j, i - 1], {j, 1, n/i}]}]];
a[n_] := Length[b[n, n]];
a /@ Range[0, 40] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
combp[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,1,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&&combp[n,#]!={}&]], {n,0,15}] (* Gus Wiseman, Sep 11 2023 *)

from sympy.utilities.iterables import partitions
def A088314(n): return len({tuple(sorted(set(p))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023

a(n) = 2^(n-1) - A070880(n). - Alois P. Heinz, Feb 08 2019

a(n) = A365042(n) + 1. - Gus Wiseman, Sep 13 2023

More terms and clearer definition from Vladeta Jovovic, Apr 21 2005

A365073 Number of subsets of {1..n} that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 60, 112, 244, 480, 992, 1944, 4048, 7936, 16176, 32320, 65088, 129504, 261248, 520448, 1046208, 2090240, 4186624, 8365696, 16766464, 33503744, 67064064, 134113280, 268347392, 536546816, 1073575936, 2146703360, 4294425600, 8588476416, 17178349568

Offset: 0

Gus Wiseman, Sep 01 2023

nonn

			The subset {2,3,6} has 7 = 2*2 + 1*3 + 0*6 so is counted under a(7).
The a(1) = 1 through a(4) = 14 subsets:
  {1}  {1}    {1}      {1}
       {2}    {3}      {2}
       {1,2}  {1,2}    {4}
              {1,3}    {1,2}
              {2,3}    {1,3}
              {1,2,3}  {1,4}
                       {2,3}
                       {2,4}
                       {3,4}
                       {1,2,3}
                       {1,2,4}
                       {1,3,4}
                       {2,3,4}
                       {1,2,3,4}

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

The case of positive coefficients is A088314.
The case of subsets containing n is A131577.
The binary version is A365314, positive A365315.
The binary complement is A365320, positive A365321.
The positive complement is counted by A365322.
A version for partitions is A365379, strict A365311.
The complement is counted by A365380.
The case of subsets without n is A365542.
A326083 and A124506 appear to count combination-free subsets.
A179822 and A326080 count sum-closed subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A007865, A088809, A093971, A151897, A237668, A308546, A326020, A364534, A364839, A365043, A365381.

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]],combs[n,#]!={}&]],{n,0,5}]

a(n)={
  my(comb(k,b)=while(b>>k, b=bitor(b, b>>k); k*=2); b);
  my(recurse(k,b)=
    if(bittest(b,0), 2^(n+1-k),
    if(2*k>n, 2^(n+1-k) - 2^sum(j=k, n, !bittest(b,j)),
    self()(k+1, b) + self()(k+1, comb(k,b)) )));
  recurse(1, 1<Andrew Howroyd, Sep 04 2023

Terms a(12) and beyond from Andrew Howroyd, Sep 04 2023

A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 51, 60, 71, 83, 97, 113, 132, 153, 178, 205, 238, 272, 315, 360, 413, 471, 539, 613, 698, 792, 899, 1018, 1153, 1302, 1470, 1658, 1867, 2100, 2362, 2652, 2974, 3335, 3734, 4178, 4672

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367224 /\ A005117.

			The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
The a(1) = 1 through a(10) = 5 strict partitions:
  (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                           (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                             (5,2,1)  (5,3,1)  (6,3,1)
                                                      (6,2,1)  (7,2,1)
                                                               (4,3,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214*   A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A275972 counts strict knapsack partitions, non-strict A108917.
A364272 counts sum-full strict partitions, sum-free A364349.
A365925 counts subset-sums of strict partitions, non-strict A304792.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]

A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 8, 13, 18, 26, 35, 50, 66, 92, 119, 160, 208, 275, 350, 457, 579, 742, 933, 1185, 1476, 1859, 2300, 2868, 3531, 4371, 5343, 6575, 8003, 9776, 11842, 14394, 17351, 20987, 25191, 30315, 36257, 43448, 51753, 61776, 73342, 87192, 103184, 122253, 144211

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The Heinz numbers of these partitions are given by A367226.

			The partition (4,2,1) has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(7).
The a(1) = 1 through a(7) = 13 partitions:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)
             (111)  (31)    (41)     (51)      (61)
                    (211)   (221)    (321)     (322)
                    (1111)  (311)    (411)     (331)
                            (2111)   (2211)    (421)
                            (11111)  (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218*   A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A008284 counts partitions by length, strict A008289.
A240855 counts strict partitions whose length is a part, complement A240861.
A365046 counts combination-full subsets, differences of A364914.
Cf. A088314, A363225, A364350, A365073, A365311, A365918.

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], combs[Length[#], Union[#]]!={}&]], {n,0,15}]

a(31)-a(48) from Chai Wah Wu, Nov 15 2023

A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 7, 10, 11, 15, 17, 22, 25, 32, 37, 46, 53, 65, 75, 90, 105, 124, 143, 168, 193, 224, 258, 297, 340, 390, 446, 509, 580, 660, 751, 852, 967, 1095, 1240, 1401, 1584, 1786, 2015, 2269, 2554, 2869, 3226, 3617, 4056, 4541, 5084

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367218.

			The a(3) = 1 through a(10) = 7 strict partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                (4,1)  (5,1)    (6,1)    (7,1)    (8,1)    (9,1)
                       (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                         (5,2,1)  (5,3,1)  (5,4,1)
                                                  (6,2,1)  (6,3,1)
                                                           (7,2,1)
                                                           (4,3,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220*   A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
A365046 counts combination-full subsets, differences of A364914.
Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.

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@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]

A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

1, 2, 3, 6, 12, 24, 49, 101, 207, 422, 859, 1747, 3548, 7194, 14565, 29452, 59496, 120086, 242185, 488035, 982672, 1977166, 3975508, 7989147, 16047464, 32221270, 64674453, 129775774, 260337978, 522124197, 1046911594, 2098709858, 4206361369, 8429033614, 16887728757, 33829251009, 67755866536, 135687781793, 271693909435

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			The set {1,2,4} has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(4).
The a(0) = 1 through a(4) = 12 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {1,3}    {1,3}
                  {2,3}    {1,4}
                  {1,2,3}  {2,3}
                           {2,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222*   A367223
  ranks:       A367224    A367225    A367226    A367227
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A326020 counts complete subsets.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts subsets containing two distinct elements summing to k.
Cf. A068911, A088314, A103580, A116861, A326080, A364350, A365073, A365311, A365376, A365380, A365544.

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]], combs[Length[#], Union[#]]!={}&]], {n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A367222(n):
    c, mlist = 1, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(1,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for s in mlist[k-1]:
                if s <= ws:
                    c += 1
                    break
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367223(n).

a(13)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 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

A365320 Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 7, 5, 12, 12, 27, 14, 42, 36, 47, 47, 83, 58, 109, 80, 116, 126, 172, 111, 195, 192, 219, 202, 294, 210, 342, 286, 354, 369, 409, 324, 509, 480, 523, 452, 640, 507, 711, 622, 675, 747, 865, 654, 916, 842, 964, 922, 1124, 940, 1147, 1029

Offset: 0

Gus Wiseman, Sep 06 2023

nonn

Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?

			The pair p = (3,6) cannot be linearly combined to obtain 8 or 10, so p is counted under a(8) and a(10), but we have 9 = 1*3 + 1*6 or 9 = 3*3 + 0*6, so p not counted under a(9).
The a(5) = 2 through a(10) = 12 pairs:
  (2,4)  (4,5)  (2,4)  (3,6)  (2,4)  (3,6)
  (3,4)         (2,6)  (3,7)  (2,6)  (3,8)
                (3,5)  (5,6)  (2,8)  (3,9)
                (3,6)  (5,7)  (4,6)  (4,7)
                (4,5)  (6,7)  (4,7)  (4,8)
                (4,6)         (4,8)  (4,9)
                (5,6)         (5,6)  (6,7)
                              (5,7)  (6,8)
                              (5,8)  (6,9)
                              (6,7)  (7,8)
                              (6,8)  (7,9)
                              (7,8)  (8,9)

The unrestricted version is A000217, ranks A001358.
For strict partitions we have A365312, complement A365311.
The (binary) complement is A365314, positive A365315.
The case of positive coefficients is A365321, for all subsets A365322.
For partitions we have A365378, complement A365379.
For all subsets instead of just pairs we have A365380, complement A365073.
A004526 counts partitions of length 2, shift right for strict.
A007865 counts sum-free subsets, complement A093971.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A070880, A088314, A088809, A151897, A326020, A364839.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}],combs[n,#]=={}&]],{n,0,30}]

from itertools import count
from sympy import divisors
def A365320(n):
    a = set()
    for i in range(1,n+1):
        if not n%i:
            a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
        else:
            for j in count(0,i):
                if j > n:
                    break
                k = n-j
                for d in divisors(k):
                    if d>=i:
                        break
                    a.add((d,i))
    return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 13 2023

cp's OEIS Frontend

A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

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A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

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A088314 Cardinality of set of sets of parts of all partitions of n.

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A365073 Number of subsets of {1..n} that can be linearly combined using nonnegative coefficients to obtain n.

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A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.

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A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

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A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

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