A367412 - OEIS

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A367412

Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15

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OFFSET

0,5

COMMENTS

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

LINKS

Table of n, a(n) for n=0..74.

EXAMPLE

Triangle begins:

1 1

1 2

1 3 1

1 3 3

1 5 3 2

1 4 7 2 1

1 6 7 6 2

1 6 10 6 7

1 7 12 11 8 3

1 6 16 11 17 3 2

1 10 14 20 19 10 2 1

1 7 22 17 31 14 7 2

1 9 22 27 37 22 11 6

1 10 24 27 51 32 16 15

1 11 27 39 57 43 27 22 4

1 9 33 34 79 57 36 39 7 2

1 13 31 51 86 77 45 62 14 4 1

Row n = 9 counts the following partitions:

(9) (81) (711) (621) (5211)

(72) (6111) (531) (4311)

(63) (522) (432) (4221)

(54) (51111) (33111) (42111)

(333) (441) (222111) (3321)

(111111111) (411111) (2211111) (32211)

(3222) (321111)

(3111111)

(22221)

(21111111)

MATHEMATICA

DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n, 10}, {k, 0, n}], 0, 2]

CROSSREFS

Row sums are A000041.

Column k = 1 is A088922.

The non-binary version (with zeros) is A365658.

The strict non-binary version (with zeros) is A365832.

The corresponding rank statistic is A366739.

A001358 lists semiprimes, squarefree A006881, conjugate A065119.

A126796 counts complete partitions, ranks A325781, strict A188431.

A276024 counts positive subset-sums of partitions, strict A284640.

A365924 counts incomplete partitions, ranks A365830, strict A365831.

A366738 counts semi-sums of partitions, non-binary A304792.

A366741 counts semi-sums of strict partitions, non-binary A365925.

Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

Sequence in context: A244740 A088742 A256435 * A279945 A342724 A347046

Adjacent sequences: A367409 A367410 A367411 * A367413 A367414 A367415

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Nov 19 2023

STATUS

approved