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A105806
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Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.
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14
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1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 3, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 0, 1, 4, 3, 3, 2, 2, 1, 1, 0, 1, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 6, 5, 6, 3, 4, 2, 2, 1, 1, 0, 1, 7, 8, 6, 6, 4, 4, 2, 2, 1, 1, 0, 1, 11, 8, 9, 7, 6, 4, 4, 2, 2, 1, 1, 0, 1, 11, 13, 10, 10, 7, 7, 4, 4, 2, 2, 1, 1
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OFFSET
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1,17
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COMMENTS
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The array with all ranks (including negative ones) is A063995.
a(n,-r)=a(n,r) for negative rank -r with r from 1,2,...,n-1 (due to conjugation of partitions of n; see the link).
Dyson's rank of a partition of n is the maximal part minus the number of parts, i.e. the number of columns minus the number of rows of the Ferrers diagram (see the link) of the partition.
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LINKS
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Lars Blomberg, Table of n, a(n) for n = 1..5050
Wolfdieter Lang, First 16 rows.
Alexander Berkovich and Frank G. Garvan, Some observations on Dyson's new symmetries of partitions, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.
Freeman J. Dyson, Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 1.
Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180.
Eric Weisstein's World of Mathematics, Conjugation of partitions of n.
Eric Weisstein's World of Mathematics, Ferrers diagram.
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FORMULA
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a(n, r)= number of partitions of n with rank r, with r from 0, 1, ..., n-1.
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EXAMPLE
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Triangle starts:
[1];
[0,1];
[1,0,1];
[1,1,0,1];
[1,1,1,0,1];
[1,2,1,1,0,1]; ...
Row 6, second entry is 2 because there are 2 partitions of n=6 with rank r=2-1=1, namely (3^2) and (1^2,4).
The table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):
n\m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
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0 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
2 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0,
3 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0,
4 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0,
5 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0,
6 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 0,
7 1, 0, 1, l, 2, 1, 3, 1, 2, 1, 1, 0, 1,
...
The central triangle is A063995, the right-hand triangle is the present sequence. - N. J. A. Sloane, Jan 23 2020
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CROSSREFS
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For the full triangle see A063995.
Columns for r=0..3 are given in A047993, A101198, A101199, A101200, ...
Row sums = A064174.
Sequence in context: A194438 A144409 A131257 * A129501 A129353 A174295
Adjacent sequences: A105803 A105804 A105805 * A105807 A105808 A105809
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang, Mar 11 2005
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STATUS
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approved
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