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A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n. 7
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 0, 1, 3, 4, 2, 0, 0, 1, 0, 1, 4, 3, 3, 0, 0, 0, 1, 0, 1, 4, 7, 2, 2, 0, 0, 0, 1, 0, 1, 5, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 5, 10, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 6, 9, 5, 1, 2, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.

LINKS

Table of n, a(n) for n=0..90.

EXAMPLE

Triangle begins:

  1

  0  1

  0  1  1

  0  1  1  1

  0  1  2  0  1

  0  1  2  2  0  1

  0  1  3  2  0  0  1

  0  1  3  4  2  0  0  1

  0  1  4  3  3  0  0  0  1

  0  1  4  7  2  2  0  0  0  1

  0  1  5  6  4  2  0  0  0  0  1

  0  1  5 10  6  4  2  0  0  0  0  1

  0  1  6  9  5  1  2  0  0  0  0  0  1

  0  1  6 14 10  5  2  2  0  0  0  0  0  1

  0  1  7 13 11  3  3  2  0  0  0  0  0  0  1

  0  1  7 19 16  7  3  2  2  0  0  0  0  0  0  1

Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):

   (C)  (66)   (444)   (3333)  (81111)  (222222)  (111111111111)

        (75)   (543)   (5511)           (711111)

        (84)   (552)   (7221)

        (93)   (732)   (7311)

        (A2)   (741)   (9111)

        (B1)   (822)

               (831)

               (921)

               (A11)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n, {k}], UnsameQ@@Total/@Union[Subsets[#]]&]], {n, 0, 15}, {k, 0, n}]

CROSSREFS

Row sums are A000041.

Column k = 2 is A004526.

Column k = 3 is A325690.

Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.

Sequence in context: A165276 A035698 A230204 * A161502 A279628 A241914

Adjacent sequences:  A325589 A325590 A325591 * A325593 A325594 A325595

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, May 15 2019

STATUS

approved

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Last modified May 27 21:07 EDT 2020. Contains 334671 sequences. (Running on oeis4.)