%I A105805
%S A105805 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,
%T A105805 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,
%U A105805 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
%V A105805 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,
%W A105805 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,
%X A105805 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
%N A105805 Dyson's rank of partitions listed in the Abramowitz-Stegun order.
%C A105805 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).
%C A105805 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 Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 17 2006]
%D A105805 F. J. Dyson: Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 
               418.
%H A105805 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A105805 A. M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 
               1972, pp. 831-2.
%H A105805 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A105805.text">
               First 15 rows.</a>
%F A105805 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).
%e A105805 [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]; ...
%e A105805 Row 3 for partitions of 3 in the mentioned order: 3,(1,2),1^3 with ranks 
               2,0,-2.
%Y A105805 Sequence in context: A160706 A087509 A089596 this_sequence A049581 A114327 
               A073450
%Y A105805 Adjacent sequences: A105802 A105803 A105804 this_sequence A105806 A105807 
               A105808
%K A105805 sign,easy,tabf
%O A105805 1,4
%A A105805 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 28 2005