A105805 Irregular triangle read by rows: T(n,k) is the Dyson's rank of the k-th partition of n in Abramowitz-Stegun order.

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

Offset: 1

Author

Wolfdieter Lang, Apr 28 2005

Keywords

• sign
• easy
• tabf

Comments

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

Row lengths give A000041, n >= 1.

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]

First differs from A330368 at a(49) = T(7,5). - Omar E. Pol, Dec 31 2019

Examples

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

Links

• 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.
• Freeman J. Dyson, Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.
• Wolfdieter Lang, First 16 rows.

Crossrefs

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

Programs

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

Formula

a(n,k) = A049085(n,k) - A036043(n,k). - Alford Arnold, Aug 02 2010

Extensions

Name clarified by Omar E. Pol, Dec 31 2019