A326034 -id:A326034 - cp's OEIS frontend

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 1, 1, 0, 1, 1, 2, 2, 4, 3, 2, 1, 1, 0, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 0, 1, 1, 3, 3, 4, 6, 4, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 6, 4

Offset: 0

Gus Wiseman, Jun 03 2019

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

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  1  2  1  1
  0  1  1  1  2  1  1
  0  1  1  2  3  2  1  1
  0  1  1  2  1  3  2  1  1
  0  1  1  2  2  4  3  2  1  1
  0  1  1  2  3  1  4  3  2  1  1
  0  1  1  3  3  4  6  4  3  2  1  1
  0  1  1  1  1  3  1  6  4  3  2  1  1
  0  1  1  3  3  5  4  7  6  4  3  2  1  1
  0  1  1  2  3  5  4  1  7  6  4  3  2  1  1
  0  1  1  2  3  4  6  6 11  7  6  4  3  2  1  1
Row n = 9 counts the following partitions:
  (111111111)  (22221)  (333)   (432)  (54)     (63)    (72)   (81)  (9)
                        (3222)  (441)  (522)    (621)   (711)
                                       (531)    (6111)
                                       (51111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.

Row sums are A108917.
Column k = 3 is A326034.
Cf. A002033, A196723, A275972, A276024.
Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

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.

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.

A342684 Number of knapsack partitions of n with largest part 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 11, 1, 8, 6, 10, 7, 13, 9, 15, 6, 12, 10, 15, 8, 18, 10, 17, 6, 17, 12, 17, 9, 18, 13, 22, 7, 19, 10, 19, 13, 20, 14, 24, 4, 20, 12, 19, 13, 23, 15, 21, 4, 20, 13, 23, 11, 23, 15, 20, 7, 20, 12, 22, 15, 24, 12, 22

Offset: 0

Fausto A. C. Cariboni, May 18 2021

nonn

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

I computed terms a(n) for n = 0..40000 and the subsequence a(98)-a(937) of length 840 is repeated continuously.

			The initial nonzero values count the following partitions:
   8: (8)
   9: (8,1)
  10: (8,1,1), (8,2)
  11: (8,1,1,1), (8,2,1), (8,3)

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

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

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

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

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

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A342684 Number of knapsack partitions of n with largest part 8.

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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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