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Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.
9

%I #6 Nov 18 2023 18:18:34

%S 1,1,1,2,1,2,3,2,2,2,5,3,4,2,3,7,5,6,4,3,3,11,7,9,6,6,3,4,15,11,13,10,

%T 9,6,4,4,22,15,20,13,15,9,8,4,5,30,22,27,21,21,15,12,8,5,5,42,30,39,

%U 28,30,21,20,12,10,5,6,56,42,53,41,42,33,28,20,15,10,6,6

%N Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.

%C 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.

%e The partition y = (3,2,1,1) has semi-sum 3 = 2+1, but no semi-sum 6, so y is counted under T(7,3) but not under T(7,6).

%e Triangle begins:

%e 1

%e 1 1

%e 2 1 2

%e 3 2 2 2

%e 5 3 4 2 3

%e 7 5 6 4 3 3

%e 11 7 9 6 6 3 4

%e 15 11 13 10 9 6 4 4

%e 22 15 20 13 15 9 8 4 5

%e 30 22 27 21 21 15 12 8 5 5

%e 42 30 39 28 30 21 20 12 10 5 6

%e 56 42 53 41 42 33 28 20 15 10 6 6

%e 77 56 73 55 60 42 44 28 25 15 12 6 7

%e Row n = 7 counts the following partitions:

%e (511) (421) (331) (421) (511) (61)

%e (4111) (3211) (322) (4111) (421) (52)

%e (3211) (2221) (3211) (322) (331) (43)

%e (31111) (22111) (31111) (3211)

%e (22111) (211111) (2221)

%e (211111) (22111)

%e (1111111)

%t Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#, {2}],k]&]], {n,2,10}, {k,2,n}]

%Y Column k = 0 is A000041.

%Y Column n = k is A004526.

%Y The complement for all submultisets is A046663, strict A365663.

%Y For subsets instead of partitions we have A365541, non-binary A365381.

%Y The non-binary version is A365543, strict A365661.

%Y Row sums are A366738.

%Y The strict case is A367405.

%Y Cf. A122768, A108917, A299701, A304792, A364272, A364911, A365658.

%K nonn,tabl

%O 2,4

%A _Gus Wiseman_, Nov 17 2023