A325458 Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 2, 3, 6, 0, 0, 0, 0, 0, 4, 4, 7, 0, 0, 0, 0, 0, 3, 6, 5, 8, 0, 0, 0, 0, 0, 1, 6, 8, 6, 9, 0, 0, 0, 0, 0, 0, 6, 9, 10, 7, 10, 0, 0, 0, 0, 0, 0, 2, 11, 12, 12, 8, 11

Offset: 0

Author

Gus Wiseman, May 04 2019

Keywords

• nonn
• tabl

Comments

Conjectured to be equal to A049597.

Examples

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  0  3
  0  0  0  1  4
  0  0  0  0  2  5
  0  0  0  0  2  3  6
  0  0  0  0  0  4  4  7
  0  0  0  0  0  3  6  5  8
  0  0  0  0  0  1  6  8  6  9
  0  0  0  0  0  0  6  9 10  7 10
  0  0  0  0  0  0  2 11 12 12  8 11
  0  0  0  0  0  0  2  9 16 15 14  9 12
  0  0  0  0  0  0  0  7 16 21 18 16 10 13
  0  0  0  0  0  0  0  4 18 23 26 21 18 11 14
  0  0  0  0  0  0  0  3 12 29 30 31 24 20 12 15
  0  0  0  0  0  0  0  1 12 27 40 37 36 27 22 13 16
  0  0  0  0  0  0  0  0  8 26 42 51 44 41 30 24 14 17
  0  0  0  0  0  0  0  0  6 23 48 57 62 51 46 33 26 15 18
  0  0  0  0  0  0  0  0  2 21 44 70 72 73 58 51 36 28 16 19
Row n = 9 counts the following partitions:
  (333)  (54)     (63)      (72)       (9)
         (432)    (522)     (621)      (81)
         (441)    (531)     (5211)     (711)
         (3222)   (4221)    (42111)    (6111)
         (3321)   (4311)    (321111)   (51111)
         (22221)  (32211)   (2211111)  (411111)
                  (33111)              (3111111)
                  (222111)             (21111111)
                                       (111111111)
		

Crossrefs

Row sums are A000041.

Column sums are 2^(k - 1) for k > 0.

Cf. A008284, A049597, A093641, A116608, A325242, A325349, A325351, A325355, A325359, A325459.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],If[n==0,k==0,First[#]+Length[#]-1==k]&]],{n,0,19},{k,0,n}]

Formula

Franklin T. Adams-Watters has conjectured at A049597 that the k-th column gives the coefficients of the sum of Gaussian polynomials [k,m] for m = 0..k.