A325592 - OEIS

A325592

Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

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

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OFFSET

0,13

COMMENTS

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

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010

EXAMPLE

Triangle begins:

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):

(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: A035698 A230204 A372646 * A161502 A279628 A241914

Adjacent sequences: A325589 A325590 A325591 * A325593 A325594 A325595

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, May 15 2019

STATUS

approved