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 A105805 Dyson's rank of partitions listed in the Abramowitz-Stegun order. 9
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The sequence of row lengths of this array is [1,2,3,5,7,11,15,22,30,42,56,77,...] from A000041(n), n>=1 (partition numbers). 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] REFERENCES F. J. Dyson: Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418. LINKS 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]. A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2. Wolfdieter Lang, First 16 rows. FORMULA a(n, k)= rank of the k-th partition of n in Abramowitz-Stegun order (see reference). The rank of a partition is the maximal part minus the number of parts (m in the table of Abramowitz-Stegun). a(n,k) = A049085(n,k) - A036043(n,k). Alford Arnold, Aug 02 2010 EXAMPLE [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. From Wolfdieter Lang, Jul 18 2013  (Start) 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. Th row n=8 (see the link) is no longer antisymmetric. See the Franklin T. Adams-Watters correction above. (End) MAPLE # ASPrts is implemented in A119441 A105805 := proc(n, k)     local pi;     pi := ASPrts(n)[k] ;     max(op(pi))-nops(pi) ; end proc: for n from 1 do     for k from 1 to A000041(n) do         printf("%d, ", A105805(n, k)) ;     end do:     printf("\n") ; end do: # R. J. Mathar, Jul 17 2013 CROSSREFS A209616 (sum of positive ranks). Sequence in context: A262878 A317239 A089596 * A194547 A257570 A220417 Adjacent sequences:  A105802 A105803 A105804 * A105806 A105807 A105808 KEYWORD sign,easy,tabf AUTHOR Wolfdieter Lang, Apr 28 2005 STATUS approved

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Last modified November 19 08:40 EST 2018. Contains 317347 sequences. (Running on oeis4.)