A354468 Number of possible ordered pairs (n_1, S) where (n_1, n_2, ..., n_k) is a partition of n, n_1 is the largest element of the partition, and S = Sum_{j=1..k} n_j^2.

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 39, 50, 66, 83, 104, 127, 157, 188, 225, 265, 312, 359, 418, 479, 547, 620, 700, 786, 884, 987, 1094, 1214, 1348, 1479, 1627, 1779, 1945, 2122, 2313, 2505, 2719, 2934, 3161, 3412, 3666, 3932, 4218, 4511, 4820, 5140, 5477, 5825

Offset: 0

Author

Noah A Rosenberg, Jun 02 2022

Keywords

• nonn

Comments

In categorical data with a sample of size n distributed over at least 1 and at most n distinct categorical types, if a dataset is summarized by an ordered pair of two numbers -- the number of observations of the most frequent type and the sum of squares of the frequencies of all types -- then a(n) gives the number of distinguishable ordered pairs across all possible datasets.

Examples

			For n=4 the a(4)=5 ordered pairs are (4,16), (3,10), (2,8), (2,6), and (1,4).
		

References

• Noah A. Rosenberg, Mathematical Properties of Population-Genetic Statistics, Princeton University Press, 2025, page 112.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800
• Noah A. Rosenberg and Donna M. Zulman, Measures of care fragmentation: mathematical insights from population genetics, Health Services Research 55 (2020), 318-327.

Crossrefs

Bounded below by A069999. Bounded above by A000041 and by A000125(n-1).

Cf. A354800.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},
     {b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})
    end:
a:= n-> add(nops(b(n-i, min(n-i, i))), i=signum(n)..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 02 2022

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n},
     Union@Flatten@{b[n, i-1], #+i^2& /@ b[n-i, Min[n-i, i]]}];
a[n_] := Sum[Length[b[n-i, Min[n-i, i]]], {i, Sign[n], n}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 05 2022, after Alois P. Heinz *)

Extensions

a(16)-a(51) from Alois P. Heinz, Jun 02 2022