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A325458
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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.
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2
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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
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OFFSET
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0,6
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COMMENTS
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Conjectured to be equal to A049597.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], If[n==0, k==0, First[#]+Length[#]-1==k]&]], {n, 0, 19}, {k, 0, n}]
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CROSSREFS
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Column sums are 2^(k - 1) for k > 0.
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KEYWORD
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AUTHOR
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STATUS
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approved
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