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A365661
Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.
46
1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12
OFFSET
0,7
COMMENTS
First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
Rows are palindromic.
Are there only two zeros in the whole triangle?
EXAMPLE
Triangle begins:
1
1 1
1 0 1
2 1 1 2
2 1 0 1 2
3 1 1 1 1 3
4 2 2 1 2 2 4
5 2 2 2 2 2 2 5
6 3 2 3 1 3 2 3 6
8 3 3 4 3 3 4 3 3 8
Row n = 6 counts the following strict partitions:
(6) (5,1) (4,2) (3,2,1) (4,2) (5,1) (6)
(5,1) (3,2,1) (3,2,1) (3,2,1) (3,2,1) (5,1)
(4,2) (4,2)
(3,2,1) (3,2,1)
Row n = 10 counts the following strict partitions:
A 91 82 73 64 532 64 73 82 91 A
64 541 532 532 541 541 541 532 532 541 64
73 631 721 631 631 4321 631 631 721 631 73
82 721 4321 721 4321 4321 721 4321 721 82
91 4321 4321 4321 4321 91
532 532
541 541
631 631
721 721
4321 4321
MATHEMATICA
Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#], k]&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Columns k = 0 and k = n are A000009.
The non-strict complement is A046663, central column A006827.
Central column n = 2k is A237258.
For subsets instead of partitions we have A365381.
The non-strict case is A365543.
The complement is A365663.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Sequence in context: A171533 A115236 A307777 * A284593 A190672 A327910
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 16 2023
STATUS
approved