[seqfan] combinations of partitions of n

jnthn stdhr jstdhr at gmail.com
Sat Oct 7 20:33:32 CEST 2017


Let P(n) be the number of partitions of n, and T(n,k) be the triangle of
partitions of n into k parts.  For each row in the triangle, how many ways
are there to combine values of k so that the sum equals m times n, where
the factor m ranges from 1,...,floor( P(n) / n )?

Floor(P(n)/n) = A086740. By hand, the sequence begins:  1, 1, 1, 1, 2, 3,
6, 12, 24.  Using a(9) as an example, which is not necessarily correct:

   Row 9 of the triangle = {1,4,7,6,5,3,2,1,1}.
   Maximum of factor m is floor(P(9) / 9) = floor(30/9) = 3.
   The combinations I get are:

      m = 1:  (7,2), (7,1,1), (6,3), (6,2,1), (6,1,1,1), (5,4), (5,3,1),
(5,2,1,1), (4,3,2), (4,3,1,1), (4,2,1,1,1)

      m = 2:  (7,6,3,2), (7,6,3,1,1), (7,6,2,1,1,1), (7,5,4,2),
(7,5,3,2,1), (7,4,3,2,1,1), (6,5,4,3), (6,5,4,2,1), (6,5,4,1,1,1),
(6,5,3,2,1,1), (6,4,3,2,1,1,1)

      m = 3: (7,6,5,4,3,2), (7,6,5,4,3,1,1)

Since there are always repeated values of k > 1 at every n = 2P(n) and n =
2P(n) + 1, the combinations may have "bumps" at these values of n.  And
since values of k grow rather quickly, m grows fairly quickly (e.g at n =
35, m = 425).

-Jonathan


More information about the SeqFan mailing list