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A325987
Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.
1
1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 1, 1, 2, 1, 1, 0, 1, 0, 3, 0, 3, 0, 4, 0, 1, 0, 3, 0, 1, 1, 3, 1, 3, 0, 3, 2, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 5, 0, 3, 0, 5, 0, 3, 0, 6, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 4, 0
OFFSET
0,10
COMMENTS
The number of submultisets of a partition is the product of its multiplicities, each plus one.
LINKS
FORMULA
Sum_{k=1..A088881(n)} k * T(n,k) = A000712(n). - Alois P. Heinz, Aug 17 2019
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 0 2
0 1 1 1 1 1
0 1 0 2 0 3 0 1
0 1 1 3 0 1 1 2 1 1
0 1 0 3 0 3 0 4 0 1 0 3
0 1 1 3 1 3 0 3 2 1 0 4 0 1 1 1
0 1 0 5 0 3 0 5 0 3 0 6 0 1 0 3 0 2 0 1
0 1 1 4 0 5 0 7 2 1 1 4 0 1 2 5 0 3 0 2 1 0 0 2
Row n = 7 counts the following partitions (empty columns not shown):
(7) (43) (322) (421) (31111) (3211)
(52) (331) (2221) (22111)
(61) (511) (4111) (211111)
(1111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==k&]], {n, 0, 10}, {k, 1, Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]
CROSSREFS
Row lengths are A088881.
Row sums are A000041.
Diagonal n = k is A325830 interspersed with zeros.
Diagonal n + 1 = k is A325828.
Diagonal n - 1 = k is A325836.
Column k = 3 appears to be A137719.
Sequence in context: A114591 A161849 A056175 * A359324 A353421 A105241
KEYWORD
nonn,look,tabf
AUTHOR
Gus Wiseman, May 30 2019
STATUS
approved