A354800 -id:A354800 - cp's OEIS frontend

A069999 Number of possible dimensions for commutators of n X n matrices; it is independent of the field. Or, given a partition P = (p_1, p_2, ..., p_m) of n with p_1 >= p_2 >= ... >= p_m, let S(P) = sum_j (2j-1)p_j; then a(n) = number of integers that are an S(P) for some partition.

1, 1, 2, 3, 5, 7, 9, 13, 18, 21, 27, 34, 39, 46, 54, 61, 72, 83, 92, 106, 118, 130, 145, 162, 176, 193, 209, 226, 246, 265, 284, 308, 330, 352, 375, 402, 426, 453, 480, 508, 538, 570, 598, 631, 661, 694, 730, 765, 800, 835, 872, 911, 951, 992, 1030, 1071, 1115

Offset: 0

Jim Kuzmanovich (kuz(AT)wfu.edu), Apr 26 2002

Or, given such a partition P of n, let T(P) = sum_i p_i^2; then a(n) = number of integers that are a T(P) for some P. While T(P) need not equal S(P) for a given partition, the two sets of integers are equal. Or, expand the infinite product prod_k 1/(1-x^{k^2}y^k) as a power series; then a(n) = number of terms of the form x^my^n having a nonzero coefficient.
The least m for which there are distinct partitions x(1)+...+x(k) of n for which the sums of squares {x(i)^2} are not distinct is 6. - Clark Kimberling, Mar 06 2012
a(n) is also the number of possible counts of intersection points of n lines in the plane, no three concurrent. This is because n lines, grouped into pencils of size a_1,...,a_k, meet in P=Sum_{iAlon Amit, May 20 2019

Zachary Albertson and Evan Willett, "Possible Dimensions of Commutators of Matrices", Senior Thesis, Wake Forest University, May 09, 2002.
Noah A. Rosenberg, Mathematical Properties of Population-Genetic Statistics, Princeton University Press, 2025, page 112.

Joerg Arndt, Table of n, a(n) for n = 0..1000
Christian Brouder, William J. Keith, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute (Massachusetts, 2019).
Hideki Innan, Kangyu Zhang, Paul Marjoram, Simon Tavare, and Noah A. Rosenberg, Statistical tests of the coalescent model based on the haplotype frequency distribution and the number of segregating sites, Genetics 169 (2005), 1763-1777.
Noah A. Rosenberg and Donna M. Zulman, Measures of care fragmentation: mathematical insights from population genetics, Health Services Research 55 (2020), 318-327.
David Savitt and R. P. Stanley, A Note on the Symmetric Powers of the Standard Representation of S_n, Electronic J. Combinat, 7 (2000) #R6.

Cf. A111212, A354468, A354800.

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-> nops(b(n$2)):
seq(a(n), n=0..56);  # Alois P. Heinz, Jun 02 2022

p[n_, k_] := (IntegerPartitions[n]^2)[[k]]; s[n_, k_] := Sum[p[n, k][[i]], {i, 1, Length[p[n, k]]}]; t = Table[s[n, k], {n, 1, 20}, {k, 1, Length[IntegerPartitions[n]]}]; Table[Length[Union[t[[n]]]], {n, 1, 20}] (* Clark Kimberling, Mar 06 2012 *)

a069999(N)=  \\ terms up to a(N), b-file format
{
    my( V = vector(N) );
    V[1] = 'x;
    print(1," ", 1 );
    for (j=2, N,
        my( t = x^(j*j) );
        for (a=1, j-1,
            my( b = j - a );
            if ( a > b, break() );
            t += V[a] * V[b];
        );
        t = Pol( apply( x->x!=0, Vec(t) ) );
        print(j," ", vecsum( Vec(t) ) );
        V[j] = t;
    );
}  \\ Joerg Arndt, Apr 19 2019

No generating function is known.

Asymptotic to n^2/2. - Raphael R.M. Esquivel, Dec 19 2024

More terms from Robert Gerbicz, Aug 27 2002

a(0)=1 prepended by Alois P. Heinz, Jun 02 2022

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.

Original entry on oeis.org

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

Noah A Rosenberg, Jun 02 2022

nonn

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.

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

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

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.

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

Cf. A354800.

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

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 *)

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

cp's OEIS Frontend

A069999 Number of possible dimensions for commutators of n X n matrices; it is independent of the field. Or, given a partition P = (p_1, p_2, ..., p_m) of n with p_1 >= p_2 >= ... >= p_m, let S(P) = sum_j (2j-1)p_j; then a(n) = number of integers that are an S(P) for some partition.

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