A325828 -id:A325828 - cp's OEIS frontend

A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.

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

Gus Wiseman, May 30 2019

The number of submultisets of a partition is the product of its multiplicities, each plus one.

			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)

Alois P. Heinz, Rows n = 0..60, flattened

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.
Cf. A000005, A000712, A002033, A005179, A088880, A108917, A126796.
Cf. A325694, A325792, A325793, A325831, A325832, A325833, A325834.

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==k&]],{n,0,10},{k,1,Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]

Sum_{k=1..A088881(n)} k * T(n,k) = A000712(n). - Alois P. Heinz, Aug 17 2019

cp's OEIS Frontend

A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.

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