A326017 -id:A326017 - cp's OEIS frontend

A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 0, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 2, 5, 0, 29, 2, 9, 8, 20, 2

Offset: 1

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

The Heinz numbers of these partitions are given by A326018.

			The initial terms count the following partitions:
  15: (5,4,3,3)
  21: (7,6,5,3)
  21: (7,5,3,3,3)
  24: (8,7,6,3)
  25: (7,5,5,4,4)
  27: (9,8,7,3)
  27: (9,7,6,5)
  27: (8,7,3,3,3,3)
  31: (10,8,6,6,1)
  33: (11,9,7,3,3)
  33: (11,8,5,5,4)
  33: (11,7,6,6,3)
  33: (11,7,3,3,3,3,3)
  33: (11,5,5,4,4,4)
  33: (10,9,8,3,3)
  33: (10,8,6,6,3)
  33: (10,8,3,3,3,3,3)

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326015, A326017, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Range[Max@@#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]

A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 4, 4, 5, 3, 3, 4, 6, 2, 7, 6, 13, 9, 19, 16, 27, 21, 40, 33, 47, 37, 54, 48, 66, 51, 65, 65, 77, 64, 80, 71, 96, 60, 106, 95, 112, 93, 152, 114, 191, 131, 242, 192, 303, 210, 366, 300, 482, 352, 581, 450, 713, 539, 882, 689, 995

Offset: 1

Gus Wiseman, Jun 03 2019

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

These are the subsets counted by A325867, ordered by sum rather than maximum.

			The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):
  {1}  {2,1}  {3,1}  {3,2}  {4,2,1}  {5,2,1}  {4,3,2}  {6,3,1}  {5,4,2}
                                              {5,3,1}  {7,2,1}  {6,3,2}
                                              {6,2,1}           {6,4,1}
                                                                {7,3,1}
.
  {5,4,3}  {6,4,3}  {6,5,3}  {6,5,4}    {7,5,4}    {7,6,4}
  {7,3,2}  {6,5,2}  {8,5,1}  {7,6,2}    {9,4,3}    {9,5,3}
  {7,4,1}  {7,4,2}  {9,3,2}  {8,4,2,1}  {9,6,1}    {9,6,2}
  {8,3,1}  {7,5,1}                      {9,4,2,1}  {8,4,3,2}
           {9,3,1}                                 {9,5,2,1}
                                                   {10,4,2,1}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..600

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]]
maxsks[n_]:=Select[Select[IntegerPartitions[n],UnsameQ@@#&&ksQ[#]&],Select[Table[Append[#,i],{i,Complement[Range[Max@@#],#]}],ksQ]=={}&];
Table[Length[maxsks[n]],{n,30}]

A326034 Number of knapsack partitions of n with largest part 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.
Appears to repeat the terms (2,2,2,3,1,3) ad infinitum.
I computed terms a(n) for n = 0..5000 and (2,2,2,3,1,3) is repeated continuously starting at a(8). - Fausto A. C. Cariboni, May 14 2021

			The initial values count the following partitions:
   3: (3)
   4: (3,1)
   5: (3,2)
   5: (3,1,1)
   6: (3,3)
   7: (3,3,1)
   7: (3,2,2)
   8: (3,3,2)
   8: (3,3,1,1)
   9: (3,3,3)
   9: (3,2,2,2)
  10: (3,3,3,1)
  10: (3,3,2,2)
  11: (3,3,3,2)
  11: (3,3,3,1,1)
  11: (3,2,2,2,2)
  12: (3,3,3,3)
  13: (3,3,3,3,1)
  13: (3,3,3,2,2)
  13: (3,2,2,2,2,2)
  14: (3,3,3,3,2)
  14: (3,3,3,3,1,1)
  15: (3,3,3,3,3)
  15: (3,2,2,2,2,2,2)

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325592, A326015, A326016, A326017, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
kst[n_]:=Select[IntegerPartitions[n,All,{1,2,3}],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[kst[n],Max@@#==3&]],{n,0,30}]

A343321 Number of knapsack partitions of n with largest part 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 1, 4, 3, 5, 5, 4, 4, 6, 5, 7, 2, 6, 5, 8, 5, 4, 6, 7, 6, 8, 2, 8, 6, 7, 7, 5, 5, 8, 7, 8, 2, 8, 6, 9, 6, 3, 7, 9, 5, 8, 3, 8, 6, 8, 6, 5, 6, 7, 7, 9, 1, 8, 7, 8, 6, 4, 6, 9, 6, 7, 3, 9, 5, 8, 7, 4, 6, 8, 6, 9, 2, 7, 7, 9, 5, 4, 7

Offset: 0

Fausto A. C. Cariboni, May 14 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..10000 and (6,7,7,5,5,8,7,8,2,8,6,9,6,3,7,9,5,8,3,8,6,8,6,5,6,7,7,9,1,8,7,8,6,4,6,9,6,7,3,9,5,8,7,4,6,8,6,9,2,7,7,9,5,4,7,8,6,8,2,9) is repeated continuously starting at a(32).

			The initial values count the following partitions:
   5: (5)
   6: (5,1)
   7: (5,1,1)
   7: (5,2)
   8: (5,1,1,1)
   8: (5,2,1)
   8: (5,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326017, A326034, A344310.

A344310 Number of knapsack partitions of n with largest part 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 1, 2, 3, 3, 1, 3, 3, 3, 2, 3, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4

Offset: 0

Fausto A. C. Cariboni, May 14 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..10000 and (3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4) is repeated continuously starting at a(18).

			The initial values count the following partitions:
   4: (4)
   5: (4,1)
   6: (4,1,1)
   6: (4,2)
   7: (4,1,1,1)
   7: (4,2,1)
   7: (4,3)
   8: (4,4)

Cf. A108917, A275972, A326017, A326034, A343321.

A344340 Number of knapsack partitions of n with largest part 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 1, 4, 4, 6, 5, 7, 3, 7, 4, 8, 6, 10, 2, 7, 6, 9, 6, 9, 2, 9, 5, 9, 7, 9, 2, 8, 7, 10, 5, 9, 3, 10, 6, 8, 7, 10, 3, 9, 6, 10, 6, 10, 4, 9, 6, 9, 8, 11, 1, 9, 7, 11, 7, 8, 3, 10, 7, 10, 6, 10, 2, 10, 8, 9, 6, 9, 4, 11, 5, 9, 7

Offset: 0

Fausto A. C. Cariboni, May 15 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..10000 and (6,10,6,10,4,9,6,9,8,11,1,9,7,11,7,8,3,10,7,10,6,10,2,10,8,9,6,9,4,11,5,9,7,11,3,8,7,10,7,10,2,10,6,10,8,9,2,9,8,11,5,9,3,11,7,8,7,10,3,10) is repeated continuously starting at a(50).

			The initial values count the following partitions:
   6: (6)
   7: (6,1)
   8: (6,1,1)
   8: (6,2)
   9: (6,1,1,1)
   9: (6,2,1)
   9: (6,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326017, A326034, A343321, A344310.

A326035 Number of uniform knapsack partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 9, 10, 12, 12, 17, 16, 20, 25, 27, 29, 35, 39, 44, 57, 53, 66, 75, 84, 84, 114, 112, 131, 133, 162, 167, 209, 192, 242, 250, 289, 279, 363, 348, 417, 404, 502, 487, 608, 557, 706, 682, 835, 773, 1004, 922, 1149, 1059, 1344, 1257, 1595

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition is uniform if all parts appear with the same multiplicity, and knapsack if every distinct submultiset has a different sum.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..650

Cf. A002033, A047966, A072774, A108917, A275972, A276024, A299702.

Cf. A325592, A325858, A326015, A326016, A326017, A326036, A326037.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ks[n_]:=Select[IntegerPartitions[n],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[ks[n],SameQ@@Length/@Split[#]&]],{n,30}]

A344412 Number of knapsack partitions of n with largest part 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 1, 6, 5, 8, 7, 10, 8, 8, 9, 11, 8, 13, 11, 13, 5, 14, 8, 13, 10, 17, 12, 8, 10, 14, 13, 14, 12, 18, 3, 15, 11, 15, 14, 17, 12, 8, 12, 15, 13, 20, 12, 14, 5, 17, 15, 17, 10, 18, 14, 9, 13, 18, 13, 15, 15, 18, 5, 18, 11

Offset: 0

Fausto A. C. Cariboni, May 17 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..25000 and the subsequence a(72)-a(491) of length 420 is repeated continuously.

			The initial nonzero values count the following partitions:
   7: (7)
   8: (7,1)
   9: (7,1,1), (7,2)
  10: (7,1,1,1), (7,2,1), (7,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326017, A326034, A343321, A344310, A344340.

A344625 Number of knapsack partitions of n with largest part 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 11, 12, 1, 10, 7, 11, 10, 17, 12, 18, 16, 12, 15, 19, 13, 25, 20, 17, 22, 29, 6, 25, 20, 22, 20, 28, 16, 31, 21, 14, 23, 33, 15, 24, 22, 25, 28, 30, 8, 31, 20, 22, 22, 36, 16, 34, 26, 14, 23, 26, 22, 33, 25, 24

Offset: 0

Fausto A. C. Cariboni, May 25 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..50000 and the subsequence a(128)-a(2647) of length 2520 is repeated continuously.

			The initial nonzero values count the following partitions:
   9: (9)
  10: (9,1)
  11: (9,1,1), (9,2)
  12: (9,1,1,1), (9,2,1), (9,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..3000

Cf. A108917, A275972, A326017, A326034, A342684, A343321, A344310, A344340, A344412.

A344635 Number of knapsack partitions of n with largest part 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 1, 13, 9, 16, 11, 20, 14, 24, 16, 25, 9, 27, 14, 29, 19, 32, 16, 34, 19, 37, 11, 32, 17, 38, 19, 32, 22, 41, 19, 40, 14, 38, 22, 41, 22, 39, 18, 44, 26, 46, 8, 46, 24, 38, 23, 40, 21, 48, 28, 42, 12

Offset: 0

Fausto A. C. Cariboni, May 25 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..50000 and the subsequence a(162)-a(2681) of length 2520 is repeated continuously.

			The initial nonzero values count the following partitions:
  10: (10)
  11: (10,1)
  12: (10,1,1), (10,2)
  13: (10,1,1,1), (10,2,1), (10,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..3000

Cf. A108917, A275972, A326017, A326034, A342684, A343321, A344310, A344340, A344412, A344625.

cp's OEIS Frontend

A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

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A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

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A326034 Number of knapsack partitions of n with largest part 3.

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A343321 Number of knapsack partitions of n with largest part 5.

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A344310 Number of knapsack partitions of n with largest part 4.

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A344340 Number of knapsack partitions of n with largest part 6.

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A326035 Number of uniform knapsack partitions of n.

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A344412 Number of knapsack partitions of n with largest part 7.

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A344625 Number of knapsack partitions of n with largest part 9.

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A344635 Number of knapsack partitions of n with largest part 10.

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