A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 4, 1, 0, 0, 0, 2, 6, 3, 0, 0, 0, 0, 2, 6, 6, 1, 0, 0, 0, 0, 2, 8, 9, 3, 0, 0, 0, 0, 0, 2, 8, 13, 6, 1, 0, 0, 0, 0, 0, 2, 10, 16, 11, 3, 0, 0, 0, 0, 0, 0, 2, 10, 20, 17, 6, 1, 0, 0, 0, 0, 0, 0

Offset

1,5

Links

Table of n, a(n) for n=1..79.

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Example

Triangle begins:
  1
  2  0
  2  1  0
  2  3  0  0
  2  4  1  0  0
  2  6  3  0  0  0
  2  6  6  1  0  0  0
  2  8  9  3  0  0  0  0
  2  8 13  6  1  0  0  0  0
  2 10 16 11  3  0  0  0  0  0
  2 10 20 17  6  1  0  0  0  0  0
  2 12 24 25 11  3  0  0  0  0  0  0
  2 12 28 33 19  6  1  0  0  0  0  0  0
  2 14 32 44 29 11  3  0  0  0  0  0  0  0
  2 14 38 53 43 19  6  1  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)          (54)        (333)      (4221)    (51111)
  (111111111)  (63)        (432)      (4311)
               (72)        (441)      (5211)
               (81)        (522)      (6111)
               (22221)     (531)      (42111)
               (222111)    (621)      (411111)
               (2211111)   (711)
               (21111111)  (3222)
                           (3321)
                           (32211)
                           (33111)
                           (321111)
                           (3111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Min[Length[#], Max[#]]==k&]], {n, 15}, {k, n}]

Crossrefs

Column k = 2 is A265283. Column k = 3 is A325228.

Cf. A051924, A096771, A115720, A263297, A325188, A325189, A325192, A325193, A325194, A325224, A325225, A325229, A325232.

Sequence in context: A025436 A170976 A134109 * A325188 A170978 A238353

Adjacent sequences: A325224 A325225 A325226 * A325228 A325229 A325230

Keyword

nonn,tabl

Author

Gus Wiseman, Apr 12 2019

Status

approved