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
Examples
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)
Crossrefs
Programs
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Mathematica
Table[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{n,0,10},{k,0,n}]