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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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%I #40 Jan 24 2020 03:23:12

%S 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,

%T -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,

%U -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

%N Irregular triangle read by rows: T(n,k) is the Dyson's rank of the k-th partition of n in Abramowitz-Stegun order.

%C The rank of a partition is the largest part minus the number of parts.

%C Row lengths give A000041, n >= 1.

%C 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]

%C First differs from A330368 at a(49) = T(7,5). - _Omar E. Pol_, Dec 31 2019

%H 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 [alternative scanned copy].

%H 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 Freeman J. Dyson, <a href="https://www.jstor.org/stable/2304400">Problems for solution nr. 4261</a>, Am. Math. Month. 54 (1947) 418.

%H Wolfdieter Lang, <a href="/A105805/a105805.txt"> First 16 rows.</a>

%F a(n,k) = A049085(n,k) - A036043(n,k). - _Alford Arnold_, Aug 02 2010

%e Triangle begins:

%e [0];

%e [1, -1];

%e [2, 0, -2];

%e [3, 1, 0, -1, -3];

%e [4, 2, 1, 0, -1, -2, -4];

%e [5, 3, 2, 1, 1, 0, -1, -1, -2, -3, -5];

%e ...

%e Row 3 for partitions of 3 in the mentioned order: 3,(1,2),1^3 with ranks 2,0,-2.

%e From _Wolfdieter Lang_, Jul 18 2013: (Start)

%e Row n = 7 is [6, 4, 3, 2, 2, 1, 0 , 0, 0, -1, -2, -2, -3, -4, -6].

%e 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.

%e Row n=8 (see the link) is no longer antisymmetric. See the _Franklin T. Adams-Watters_ correction above. (End)

%p # ASPrts is implemented in A119441

%p A105805 := proc(n,k)

%p local pi;

%p pi := ASPrts(n)[k] ;

%p max(op(pi))-nops(pi) ;

%p end proc:

%p for n from 1 do

%p for k from 1 to A000041(n) do

%p printf("%d,",A105805(n,k)) ;

%p end do:

%p printf("\n") ;

%p end do: # _R. J. Mathar_, Jul 17 2013

%Y Cf. A000041, A036043, A049085, A209616 (sum of positive ranks), A330368 (another version).

%K sign,easy,tabf

%O 1,4

%A _Wolfdieter Lang_, Apr 28 2005

%E Name clarified by _Omar E. Pol_, Dec 31 2019