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proposed

A000041 counts integer partitions, strict A000009.

Cf. ~A008967, A046663, A117855, `A122768, A238628, ~A299701, A365543, `A365544, A366753, A367095.

Row sums are A000041.

Column k = 1 is A088922.

allocated for Gus WisemanTriangle 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

0,5

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.

Triangle begins:

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

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)

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

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

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

The corresponding rank statistic is A366739.

A000041 counts integer partitions, strict A000009.

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. ~A008967, A046663, A117855, `A122768, A238628, ~A299701, A365543, `A365544, A366753, A367095.

allocated

nonn,tabf

Gus Wiseman, Nov 19 2023

approved

editing

allocated for Gus Wiseman

allocated

approved