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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.

%I A325458 #9 Jan 22 2023 17:51:18
%S A325458 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,
%T A325458 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,
%U A325458 0,0,0,0,0,2,11,12,12,8,11
%N 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.
%C A325458 Conjectured to be equal to A049597.
%F A325458 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.
%e A325458 Triangle begins:
%e A325458   1
%e A325458   0  1
%e A325458   0  0  2
%e A325458   0  0  0  3
%e A325458   0  0  0  1  4
%e A325458   0  0  0  0  2  5
%e A325458   0  0  0  0  2  3  6
%e A325458   0  0  0  0  0  4  4  7
%e A325458   0  0  0  0  0  3  6  5  8
%e A325458   0  0  0  0  0  1  6  8  6  9
%e A325458   0  0  0  0  0  0  6  9 10  7 10
%e A325458   0  0  0  0  0  0  2 11 12 12  8 11
%e A325458   0  0  0  0  0  0  2  9 16 15 14  9 12
%e A325458   0  0  0  0  0  0  0  7 16 21 18 16 10 13
%e A325458   0  0  0  0  0  0  0  4 18 23 26 21 18 11 14
%e A325458   0  0  0  0  0  0  0  3 12 29 30 31 24 20 12 15
%e A325458   0  0  0  0  0  0  0  1 12 27 40 37 36 27 22 13 16
%e A325458   0  0  0  0  0  0  0  0  8 26 42 51 44 41 30 24 14 17
%e A325458   0  0  0  0  0  0  0  0  6 23 48 57 62 51 46 33 26 15 18
%e A325458   0  0  0  0  0  0  0  0  2 21 44 70 72 73 58 51 36 28 16 19
%e A325458 Row n = 9 counts the following partitions:
%e A325458   (333)  (54)     (63)      (72)       (9)
%e A325458          (432)    (522)     (621)      (81)
%e A325458          (441)    (531)     (5211)     (711)
%e A325458          (3222)   (4221)    (42111)    (6111)
%e A325458          (3321)   (4311)    (321111)   (51111)
%e A325458          (22221)  (32211)   (2211111)  (411111)
%e A325458                   (33111)              (3111111)
%e A325458                   (222111)             (21111111)
%e A325458                                        (111111111)
%t A325458 Table[Length[Select[IntegerPartitions[n],If[n==0,k==0,First[#]+Length[#]-1==k]&]],{n,0,19},{k,0,n}]
%Y A325458 Row sums are A000041.
%Y A325458 Column sums are 2^(k - 1) for k > 0.
%Y A325458 Cf. A008284, A049597, A093641, A116608, A325242, A325349, A325351, A325355, A325359, A325459.
%K A325458 nonn,tabl
%O A325458 0,6
%A A325458 _Gus Wiseman_, May 04 2019