A365545 - OEIS

A365545

Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 2, 0, 0, 0, 0, 5, 2, 0, 0, 5, 0, 1, 0

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OFFSET

0,18

COMMENTS

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Is column k = n - 7 given by A325695?

LINKS

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

EXAMPLE

Triangle begins:

1 0

0 1 0

1 0 1 0

0 1 0 1 0

0 0 2 0 1 0

1 0 0 2 0 1 0

1 0 0 0 3 0 1 0

0 1 1 0 0 3 0 1 0

0 0 3 0 0 0 4 0 1 0

1 0 0 2 2 0 0 4 0 1 0

1 0 0 0 5 0 0 0 5 0 1 0

2 0 0 0 0 5 2 0 0 5 0 1 0

2 0 1 0 0 0 8 0 0 0 6 0 1 0

1 1 3 0 0 0 0 7 3 0 0 6 0 1 0

2 0 4 0 1 0 0 0 12 0 0 0 7 0 1 0

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

2 0 3 0 7 0 1 0 0 0 16 0 0 0 8 0 1 0

3 0 0 2 6 3 3 1 0 0 0 15 4 0 0 8 0 1 0

Row n = 12: counts the following partitions:

(6,3,2,1) . . . . (9,2,1) (6,5,1) . . (11,1) . (12) .

(5,4,2,1) (8,3,1) (6,4,2) (10,2)

(7,4,1) (9,3)

(7,3,2) (8,4)

(5,4,3) (7,5)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n, 0, 10}, {k, 0, n}]

CROSSREFS

Row sums are A000009, non-strict A000041.

The complement (positive subset-sums) is also A365545 with rows reversed.

Weighted row sums are A365922, non-strict A365918.

The non-strict version is A365923, complement A365658, rank stat A325799.

A046663 counts partitions without a subset summing to k, strict A365663.

A126796 counts complete partitions, ranks A325781, strict A188431.

A364350 counts combination-free strict partitions, complement A364839.

A365543 counts partitions with a submultiset summing to k, strict A365661.

A365924 counts incomplete partitions, ranks A365830, strict A365831.

Cf. A006827, A284640, A304792, A364272, A365921.

Sequence in context: A241910 A065717 A070092 * A342770 A341519 A307837

Adjacent sequences: A365542 A365543 A365544 * A365546 A365547 A365548

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Sep 24 2023

STATUS

approved