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Search: id:A105806
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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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4
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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
(list; table; graph; listen)
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OFFSET
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1,17
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COMMENT
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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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REFERENCES
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F. J. Dyson: Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.
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LINKS
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Eric Weisstein's World of Mathematics, Conjugation of partitions of n.
Eric Weisstein's World of Mathematics, Ferrers diagram.
W. Lang: First 16 rows.
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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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Rows [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).
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CROSSREFS
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Columns for r=0..3 are given in: A047993, A101198, A101199, A101200, ...
Row sums = A064174.
Sequence in context: A027052 A144409 A131257 this_sequence A129501 A158511 A092921
Adjacent sequences: A105803 A105804 A105805 this_sequence 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 (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 11 2005
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