login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=1..105.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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