A363213 -id:A363213 - cp's OEIS frontend

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

Wolfdieter Lang, Mar 11 2005

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.

			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

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.

For the full triangle see A063995.
Columns for r=0..5 are given in A047993, A101198, A101199, A101200, A363213, A363214.
Row sums = A064174.

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

A363231 Number of partitions of n with rank 4 or higher (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 209, 267, 343, 435, 551, 693, 870, 1084, 1351, 1672, 2066, 2542, 3121, 3815, 4658, 5664, 6875, 8319, 10049, 12102, 14553, 17452, 20894, 24959, 29766, 35420, 42089, 49911, 59100, 69856, 82452, 97152, 114324, 134315

Offset: 1

Seiichi Manyama, May 22 2023

nonn

In general, for r>=0, Sum_{k>=1} (-1)^(k-1) * p(n - k*(3*k + 2*r - 1)/2) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + (12*r-5)*Pi/144) / sqrt(n/6)), where p() is the partition function. - Vaclav Kotesovec, May 26 2023

			a(7) = 2 counts these partitions: 7, 6+1.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), A363230 (r=3), this sequence (r=4).

Cf. A000041, A140090, A363213.

Table[Count[IntegerPartitions[n],?(#[[1]]-Length[#]>3&)],{n,60}] (* _Harvey P. Dale, Jul 29 2024 *)

a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+7)/2));

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+7)/2).
a(n) = p(n-5) - p(n-13) + p(n-24) - ... + (-1)^(k-1) * p(n-k*(3*k+7)/2) + ..., where p() is A000041().
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 43*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

cp's OEIS Frontend

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

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