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

Author

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

Keywords

• easy
• nonn
• nice

Comments

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

References

• 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.

Links

• 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.

Crossrefs

Cf. A111212, A354468, 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-> nops(b(n$2)):
seq(a(n), n=0..56);  # Alois P. Heinz, Jun 02 2022

Mathematica

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 *)

PARI

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

Formula

No generating function is known.

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

Extensions

More terms from Robert Gerbicz, Aug 27 2002

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