A365923 - OEIS

A365923

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

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 4, 0, 2, 0, 1, 0, 5, 1, 0, 3, 1, 1, 0, 8, 0, 3, 0, 3, 0, 1, 0, 10, 2, 1, 2, 2, 3, 1, 1, 0, 16, 0, 5, 0, 3, 0, 5, 0, 1, 0, 20, 2, 2, 4, 2, 6, 0, 4, 1, 1, 0, 31, 0, 6, 0, 8, 0, 5, 0, 5, 0, 1, 0, 39, 4, 4, 4, 1, 6, 6, 3, 2, 6, 1, 1, 0

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OFFSET

0,7

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.

LINKS

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

EXAMPLE

The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).

Triangle begins:

1 0

1 1 0

2 0 1 0

2 1 1 1 0

4 0 2 0 1 0

5 1 0 3 1 1 0

8 0 3 0 3 0 1 0

10 2 1 2 2 3 1 1 0

16 0 5 0 3 0 5 0 1 0

20 2 2 4 2 6 0 4 1 1 0

31 0 6 0 8 0 5 0 5 0 1 0

39 4 4 4 1 6 6 3 2 6 1 1 0

55 0 13 0 8 0 12 0 6 0 6 0 1 0

71 5 8 7 3 5 3 16 3 6 0 6 1 1 0

Row n = 6 counts the following partitions:

(321) (411) . (51) (33) (6) .

(3111) (42)

(2211) (222)

(21111)

(111111)

MATHEMATICA

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

CROSSREFS

Row sums are A000041.

The rank statistic counted by this triangle is A325799.

The strict case is A365545, weighted row sums A365922.

The complement (positive subset-sum) is A365658.

Weighted row sums are A365918, for positive subset-sums A304792.

A046663 counts partitions w/o a submultiset 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. A000009, A006827, A122768, A364272, A365919, A365921.

Sequence in context: A342003 A339737 A364916 * A089650 A085513 A362913

Adjacent sequences: A365920 A365921 A365922 * A365924 A365925 A365926

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Sep 24 2023

STATUS

approved