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proposed

allocated for Gus WisemanRegular 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

0,13

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)

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

Column sums are A000041. Row sums are A325193.

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

allocated

nonn,tabl

Gus Wiseman, Apr 13 2019

approved

editing

allocated for Gus Wiseman

allocated

approved