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A105805
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Irregular triangle read by rows: T(n,k) is the Dyson's rank of the k-th partition of n in Abramowitz-Stegun order.
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11
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0, 1, -1, 2, 0, -2, 3, 1, 0, -1, -3, 4, 2, 1, 0, -1, -2, -4, 5, 3, 2, 1, 1, 0, -1, -1, -2, -3, -5, 6, 4, 3, 2, 2, 1, 0, 0, 0, -1, -2, -2, -3, -4, -6, 7, 5, 4, 3, 2, 3, 2, 1, 1, 0, 1, 0, -1, -1, -2, -1, -2, -3, -3, -4, -5, -7, 8, 6, 5, 4, 3, 4, 3, 2, 1, 2, 1, 0, 2, 1, 0, 0, -1, -1, 0, -1, -2, -2, -3, -2, -3, -4, -4, -5, -6, -8, 9, 7, 6, 5, 4, 3, 5, 4, 3
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OFFSET
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1,4
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COMMENTS
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The rank of a partition is the largest part minus the number of parts.
Just for n <= 6, row n is antisymmetric due to conjugation of partitions (see links under A105806): a(n, p(n)-(k-1)) = a(n,k), k = 1..floor(p(n)/2). [Comment corrected by Franklin T. Adams-Watters, Jan 17 2006]
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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EXAMPLE
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Triangle begins:
[0];
[1, -1];
[2, 0, -2];
[3, 1, 0, -1, -3];
[4, 2, 1, 0, -1, -2, -4];
[5, 3, 2, 1, 1, 0, -1, -1, -2, -3, -5];
...
Row 3 for partitions of 3 in the mentioned order: 3,(1,2),1^3 with ranks 2,0,-2.
Row n = 7 is [6, 4, 3, 2, 2, 1, 0 , 0, 0, -1, -2, -2, -3, -4, -6].
This is also antisymmetric, but by accident, because a(7,7) = 0 for the partition (1,3^2), conjugate to (2^2,3) with a(7,8) = 0, and a(7,9) = 0 for (1^3,4) which is self-conjugate.
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MAPLE
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local pi;
pi := ASPrts(n)[k] ;
max(op(pi))-nops(pi) ;
end proc:
for n from 1 do
end do:
printf("\n") ;
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CROSSREFS
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KEYWORD
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sign,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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