A105806 Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 3, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 0, 1, 4, 3, 3, 2, 2, 1, 1, 0, 1, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 6, 5, 6, 3, 4, 2, 2, 1, 1, 0, 1, 7, 8, 6, 6, 4, 4, 2, 2, 1, 1, 0, 1, 11, 8, 9, 7, 6, 4, 4, 2, 2, 1, 1, 0, 1, 11, 13, 10, 10, 7, 7, 4, 4, 2, 2, 1, 1, 0, 1

Offset

1,17

Comments

The array with all ranks (including negative ones) is A063995.

a(n,-r)=a(n,r) for negative rank -r with r from 1,2,...,n-1 (due to conjugation of partitions of n; see the link).

Dyson's rank of a partition of n is the maximal part minus the number of parts, i.e. the number of columns minus the number of rows of the Ferrers diagram (see the link) of the partition.

Links

Lars Blomberg, Table of n, a(n) for n = 1..5050

Wolfdieter Lang, First 16 rows.

Alexander Berkovich and Frank G. Garvan, Some observations on Dyson's new symmetries of partitions, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.

Freeman J. Dyson, Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 1.

Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180.

Eric Weisstein's World of Mathematics, Conjugation of partitions of n.

Eric Weisstein's World of Mathematics, Ferrers diagram.

Formula

a(n, r)= number of partitions of n with rank r, with r from 0, 1, ..., n-1.

G.f. of column r: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(r*k) * ( x^(k*(3*k-1)/2) - x^(k*(3*k+1)/2) ). - Seiichi Manyama, May 21 2023

Example

Triangle starts:
  1;
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 1, 1, 0, 1;
  1, 2, 1, 1, 0, 1; ...
Row 6, second entry is 2 because there are 2 partitions of n=6 with rank r=2-1=1, namely (3^2) and (1^2,4).
The table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):
n\m -6 -5  -4  -3  -2  -1   0   1   2   3   4   5   6
-----------------------------------------------------
0   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
1   0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,
2   0,  0,  0,  0,  0,  1,  0,  1,  0,  0,  0,  0,  0,
3   0,  0,  0,  0,  1,  0,  1,  0,  1,  0,  0,  0,  0,
4   0,  0,  0,  1,  0,  1,  1,  1,  0,  1,  0,  0,  0,
5   0,  0,  1,  0,  1,  1,  1,  1,  1,  0,  1,  0,  0,
6   0,  1,  0,  1,  1,  2,  1,  2,  1,  1,  0,  1,  0,
7   1,  0,  1,  1,  2,  1,  3,  1,  2,  1,  1,  0,  1,
...
The central triangle is A063995, the right-hand triangle is the present sequence. - N. J. A. Sloane, Jan 23 2020

Crossrefs

For the full triangle see A063995.

Columns for r=0..5 are given in A047993, A101198, A101199, A101200, A363213, A363214.

Row sums = A064174.

Sequence in context: A194438 A144409 A131257 * A129501 A129353 A174295

Adjacent sequences: A105803 A105804 A105805 * A105807 A105808 A105809

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 11 2005

Status

approved