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

0,3

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_{i<j} a_i a_j points, and such sums P are bijective with sums of squares S=a_1^2+...+a_k^2, thanks to n^2=S+2P. For example, a(10)=27 since 10 lines can meet in 0, 9, 16, 17, 21, 23, 24, 25 or 27..45 points. [Alon Amit, May 20 2019]

References

Zachary Albertson and Evan Willett, "Possible Dimensions of Commutators of Matrices", Senior Thesis, Wake Forest University, May 09, 2002.

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.

Formula

No generating function is known.

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 *)

Prog

(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

Crossrefs

Cf. A111212, A354468, A354800.

Sequence in context: A200672 A341497 A332686 * A271661 A035563 A240063

Adjacent sequences: A069996 A069997 A069998 * A070000 A070001 A070002

Keyword

easy,nonn,nice

Author

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

Extensions

More terms from Robert Gerbicz, Aug 27 2002

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

Status

approved