A365380 -id:A365380 - cp's OEIS frontend

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

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

A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 5, 2, 2, 1, 16, 8, 8, 10, 10, 7, 5, 5, 2, 2, 1, 32, 16, 16, 20, 20, 23, 15, 15, 12, 12, 8, 5, 5, 2, 2, 1, 64, 32, 32, 40, 40, 46, 47, 38, 33, 35, 29, 28, 21, 17, 14, 13, 8, 5, 5, 2, 2, 1, 128, 64, 64, 80, 80, 92, 94, 102, 79, 82, 76, 75, 68, 64, 53, 48, 43, 34, 33, 23, 19, 15, 13, 8, 5, 5, 2, 2, 1

Offset: 0

Gus Wiseman, Sep 08 2023

Row lengths are A000124(n) = 1 + n*(n+1)/2.

			Triangle begins:
   1
   2  1
   4  2  2  1
   8  4  4  5  2  2  1
  16  8  8 10 10  7  5  5  2  2  1
  32 16 16 20 20 23 15 15 12 12  8  5  5  2  2  1
  64 32 32 40 40 46 47 38 33 35 29 28 21 17 14 13  8  5  5  2  2  1
Array begins:
     k=0   k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9
-------------------------------------------------------
n=0:  1
n=1:  2     1
n=2:  4     2    2    1
n=3:  8     4    4    5    2    2    1
n=4:  16    8    8    10   10   7    5    5    2    2
n=5:  32    16   16   20   20   23   15   15   12   12
n=6:  64    32   32   40   40   46   47   38   33   35
n=7:  128   64   64   80   80   92   94   102  79   82
n=8:  256   128  128  160  160  184  188  204  207  184
n=9:  512   256  256  320  320  368  376  408  414  440
The T(5,8) = 12 subsets are:
  {3,5}  {1,2,5}  {1,2,3,4}  {1,2,3,4,5}
         {1,3,4}  {1,2,3,5}
         {1,3,5}  {1,2,4,5}
         {2,3,5}  {1,3,4,5}
         {3,4,5}  {2,3,4,5}

Row lengths are A000124 = number of distinct sums of subsets of {1..n}.
Central column/main diagonal is A365376.
A000009 counts sets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A093971, A103580, A131577, A151897, A326080, A364272, A364534, A365073, A365377, A365380.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],k]&]],{n,0,8},{k,0,n*(n+1)/2}]

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

A367219 Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 2, 4, 4, 7, 6, 11, 9, 16, 16, 23, 22, 35, 33, 48, 50, 69, 70, 99, 99, 136, 142, 187, 194, 261, 267, 346, 367, 468, 489, 626, 650, 824, 870, 1081, 1135, 1421, 1485, 1833, 1942, 2374, 2501, 3062, 3220, 3915, 4145, 4987, 5274, 6363, 6709, 8027

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			3 cannot be written as a nonnegative linear combination of 2 and 5, so (5,2,2) is counted under a(9).
The a(2) = 1 through a(10) = 7 partitions:
  (2)  (3)  (4)  (5)  (6)      (7)    (8)      (9)      (10)
                      (3,3)    (4,3)  (4,4)    (5,4)    (5,5)
                      (2,2,2)         (5,3)    (6,3)    (6,4)
                                      (4,2,2)  (5,2,2)  (7,3)
                                                        (4,4,2)
                                                        (6,2,2)
                                                        (2,2,2,2,2)

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.
A124506 appears to count combination-free subsets, differences of A326083.
A365046 counts combination-full subsets, differences of A364914.
Cf. A068911, A088314, A116861, A364345, A364350, A365073, A365312, A365380.

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(56) from Chai Wah Wu, Nov 15 2023

A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 23, 24, 29, 32, 37, 41, 49, 54, 63, 72, 82, 93, 108, 122, 139, 159, 180, 204, 231, 261, 293, 331, 370, 415, 464, 518, 575, 641, 710, 789, 871, 965, 1064, 1177, 1294, 1428, 1569, 1729, 1897

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367219.

			The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):
  2  3  4  5  6  7   8   9   A   B    C    D    E    F    G
                 43  53  54  64  65   75   76   86   87   97
                         63  73  74   84   85   95   96   A6
                                 83   93   94   A4   A5   B5
                                 542  642  A3   B3   B4   C4
                                           652  752  C3   D3
                                           742  842  654  754
                                                     762  862
                                                     852  952
                                                     942  A42

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.
A124506 appears to count combination-free subsets, differences of A326083.
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.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365541 counts subsets containing two distinct elements summing to k.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A088314, A103580, A116861, A364346, A364350, A364533, A365312, A365380.

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,0,30}]

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

A367223 Number of subsets of {1..n} whose cardinality cannot be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 15, 27, 49, 90, 165, 301, 548, 998, 1819, 3316, 6040, 10986, 19959, 36253, 65904, 119986, 218796, 399461, 729752, 1333162, 2434411, 4441954, 8097478, 14746715, 26830230, 48773790, 88605927, 160900978, 292140427, 530487359, 963610200, 1751171679, 3183997509

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			3 cannot be written as a nonnegative linear combination of 2, 4, and 5, so {2,4,5} is counted under a(6).
The a(2) = 1 through a(6) = 15 subsets:
  {2}  {2}  {2}    {2}      {2}
       {3}  {3}    {3}      {3}
            {4}    {4}      {4}
            {3,4}  {5}      {5}
                   {3,4}    {6}
                   {3,5}    {3,4}
                   {4,5}    {3,5}
                   {2,4,5}  {3,6}
                            {4,5}
                            {4,6}
                            {5,6}
                            {2,4,5}
                            {2,4,6}
                            {2,5,6}
                            {4,5,6}

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
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.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A116861 counts positive linear combinations of strict partitions of k.
A364916 counts linear combinations of strict partitions of k.
A366320 counts subsets without a subset summing to k, with A365381.
Cf. A068911, A088314, A103580, A237667, A326020, A326080, A364350, A365073, A365312, A365377, A365380.

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 A367223(n):
    c, mlist = 0, []
    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:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367222(n).

a(14)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 2025

A365376 Number of subsets of {1..n} with a subset summing to n.

Original entry on oeis.org

1, 1, 2, 5, 10, 23, 47, 102, 207, 440, 890, 1847, 3730, 7648, 15400, 31332, 62922, 127234, 255374, 514269, 1030809, 2071344, 4148707, 8321937, 16660755, 33384685, 66812942, 133789638, 267685113, 535784667, 1071878216, 2144762139, 4290261840, 8583175092, 17168208940, 34342860713

Offset: 0

Gus Wiseman, Sep 08 2023

nonn

			The a(1) = 1 through a(4) = 10 sets:
  {1}  {2}    {3}      {4}
       {1,2}  {1,2}    {1,3}
              {1,3}    {1,4}
              {2,3}    {2,4}
              {1,2,3}  {3,4}
                       {1,2,3}
                       {1,2,4}
                       {1,3,4}
                       {2,3,4}
                       {1,2,3,4}

The case containing n is counted by A131577.
The version with re-usable parts is A365073.
The complement is counted by A365377.
The complement w/ re-usable parts is A365380.
Main diagonal of A365381.
A000009 counts sets summing to n, multisets A000041.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A088809, A093971, A103580, A151897, A236912, A237668, A326080, A364534.

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

isok(s, n) = forsubset(#s, ss, if (vecsum(vector(#ss, k, s[ss[k]])) == n, return(1)));
a(n) = my(nb=0); forsubset(n, s, if (isok(s, n), nb++)); nb; \\ Michel Marcus, Sep 09 2023

from itertools import combinations, chain
from sympy.utilities.iterables import partitions
def A365376(n):
    if n == 0: return 1
    nset = set(range(1,n+1))
    s, c = [set(p) for p in partitions(n,m=n,k=n) if max(p.values(),default=1) == 1], 1
    for a in chain.from_iterable(combinations(nset,m) for m in range(2,n+1)):
        if sum(a) >= n:
            aset = set(a)
            for p in s:
                if p.issubset(aset):
                    c += 1
                    break
    return c # Chai Wah Wu, Sep 09 2023

a(n) = 2^n-A365377(n). - Chai Wah Wu, Sep 09 2023

a(16)-a(25) from Michel Marcus, Sep 09 2023
a(26)-a(32) from Chai Wah Wu, Sep 09 2023
a(33)-a(35) from Chai Wah Wu, Sep 10 2023

A366320 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} without a subset summing to k.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 6, 6, 7, 8, 8, 6, 6, 9, 11, 11, 14, 14, 15, 16, 16, 12, 12, 9, 17, 17, 20, 20, 24, 27, 27, 30, 30, 31, 32, 32, 24, 24, 18, 17, 26, 31, 29, 35, 36, 43, 47, 50, 51, 56, 59, 59, 62, 62, 63

Offset: 1

Gus Wiseman, Oct 12 2023

			Triangle begins:
   1
   2  2  3
   4  4  3  6  6  7
   8  8  6  6  9 11 11 14 14 15
  16 16 12 12  9 17 17 20 20 24 27 27 30 30 31
  32 32 24 24 18 17 26 31 29 35 36 43 47 50 51 56 59 59 62 62 63
Row n = 3 counts the following subsets:
  {}     {}     {}   {}     {}     {}
  {2}    {1}    {1}  {1}    {1}    {1}
  {3}    {3}    {2}  {2}    {2}    {2}
  {2,3}  {1,3}       {3}    {3}    {3}
                     {1,2}  {1,2}  {1,2}
                     {2,3}  {1,3}  {1,3}
                                   {2,3}

Row lengths are A000217.
The diagonal T(n,n) is A365377, complement A365376.
The complement is counted by A365381.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 counts combination-free subsets, differences of A326083.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A093971, A103580, A151897, A326080, A364534, A365073, A365380.

Table[Length[Select[Subsets[Range[n]],FreeQ[Total/@Subsets[#],k]&]],{n,8},{k,n*(n+1)/2}]

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A367219 Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A367223 Number of subsets of {1..n} whose cardinality cannot be written as a nonnegative linear combination of the elements.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A365376 Number of subsets of {1..n} with a subset summing to n.

Views

Author

Keywords