A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 7, 1, 0, 0, 0

Offset

0,13

Links

Table of n, a(n) for n=0..90.

Example

Triangle begins:
  1
  0  0
  0  1  0
  0  0  0  0
  0  0  2  0  0
  0  0  0  1  0  0
  0  0  0  2  1  0  0
  0  0  0  0  2  0  0  0
  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  4  2  0  0  0
  0  0  0  0  0  0  2  5  1  0  0  0
  0  0  0  0  0  0  2  4  7  1  0  0  0
  0  0  0  0  0  0  0  2  6  6  0  0  0  0
  0  0  0  0  0  0  0  2  4  9  7  0  0  0  0
  0  0  0  0  0  0  0  0  2  6 11  5  0  0  0  0
  0  0  0  0  0  0  0  0  2  4 10 14  5  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  6 13 15  3  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  4 10 19 17  2  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  6 14 22 17  1  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  4 10 21 29 17  1  0  0  0  0
Row n = 16 counts the following partitions:
  (8)         (72)       (64)      (533)    (444)
  (11111111)  (711)      (622)     (542)    (3333)
              (2211111)  (631)     (551)    (4332)
              (3111111)  (6211)    (5222)   (4422)
                         (61111)   (5321)   (4431)
                         (222211)  (5411)
                         (322111)  (32222)
                         (331111)  (33221)
                         (421111)  (33311)
                         (511111)  (42221)
                                   (43211)
                                   (44111)
                                   (52211)
                                   (53111)

Mathematica

Table[Length[Select[IntegerPartitions[k], Max[Length[#], Max[#]]==n-k&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

Column sums are A000041. Row sums are A325193.

Cf. A051924, A071724, A096771, A115720, A263297, A325178, A325192.

Sequence in context: A326075 A060862 A325801 * A066087 A294927 A080224

Adjacent sequences: A325191 A325192 A325193 * A325195 A325196 A325197

Keyword

nonn,tabl

Author

Gus Wiseman, Apr 13 2019

Status

approved