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%I #70 Jun 22 2022 12:50:51
%S 1,1,2,3,5,7,9,13,18,21,27,34,39,46,54,61,72,83,92,106,118,130,145,
%T 162,176,193,209,226,246,265,284,308,330,352,375,402,426,453,480,508,
%U 538,570,598,631,661,694,730,765,800,835,872,911,951,992,1030,1071,1115
%N 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.
%C 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.
%C 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]
%C 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]
%D Zachary Albertson and Evan Willett, "Possible Dimensions of Commutators of Matrices", Senior Thesis, Wake Forest University, May 09, 2002.
%H Joerg Arndt, <a href="/A069999/b069999.txt">Table of n, a(n) for n = 0..1000</a>
%H Christian Brouder, William J. Keith, and Ângela Mestre, <a href="http://arxiv.org/abs/1301.0874">Closed forms for a multigraph enumeration</a>, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
%H Phillip Tomas Heikoop, <a href="https://digitalcommons.wpi.edu/mqp-all/6822">Dimensions of Matrix Subalgebras</a>, Bachelor's Thesis, Worcester Polytechnic Institute (Massachusetts, 2019).
%H Hideki Innan, Kangyu Zhang, Paul Marjoram, Simon Tavare, and Noah A. Rosenberg, <a href="https://doi.org/10.1534/genetics.104.032219">Statistical tests of the coalescent model based on the haplotype frequency distribution and the number of segregating sites</a>, Genetics 169 (2005), 1763-1777.
%H Noah A. Rosenberg and Donna M. Zulman, <a href="https://doi.org/10.1111/1475-6773.13263">Measures of care fragmentation: mathematical insights from population genetics</a>, Health Services Research 55 (2020), 318-327.
%H David Savitt and R. P. Stanley, <a href="https://doi.org/10.37236/1484">A Note on the Symmetric Powers of the Standard Representation of S_n</a>, Electronic J. Combinat, 7 (2000) #R6.
%F No generating function is known.
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},
%p {b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})
%p end:
%p a:= n-> nops(b(n$2)):
%p seq(a(n), n=0..56); # _Alois P. Heinz_, Jun 02 2022
%t 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 *)
%o (PARI)
%o a069999(N)= \\ terms up to a(N), b-file format
%o {
%o my( V = vector(N) );
%o V[1] = 'x;
%o print(1," ", 1 );
%o for (j=2, N,
%o my( t = x^(j*j) );
%o for (a=1, j-1,
%o my( b = j - a );
%o if ( a > b, break() );
%o t += V[a] * V[b];
%o );
%o t = Pol( apply( x->x!=0, Vec(t) ) );
%o print(j," ", vecsum( Vec(t) ) );
%o V[j] = t;
%o );
%o } \\ _Joerg Arndt_, Apr 19 2019
%Y Cf. A111212, A354468, A354800.
%K easy,nonn,nice
%O 0,3
%A Jim Kuzmanovich (kuz(AT)wfu.edu), Apr 26 2002
%E More terms from _Robert Gerbicz_, Aug 27 2002
%E a(0)=1 prepended by _Alois P. Heinz_, Jun 02 2022
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