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A105806 Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1. 5
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; refs; listen; history; text; internal format)
OFFSET

1,17

COMMENTS

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.

REFERENCES

F. J. Dyson: Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.

LINKS

Table of n, a(n) for n=1..103.

Eric Weisstein's World of Mathematics, Conjugation of partitions of n.

Eric Weisstein's World of Mathematics, Ferrers diagram.

W. Lang: First 16 rows.

FORMULA

a(n, r)= number of partitions of n with rank r, with r from 0, 1, ..., n-1.

EXAMPLE

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).

CROSSREFS

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

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 11 2005

STATUS

approved

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Last modified January 21 05:39 EST 2020. Contains 331104 sequences. (Running on oeis4.)