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A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0. 0
1, 0, 3, 18, 24, 39, 3456, 8190, 5928, 2109, 7619508, 17094240, 13700700, 4215120, 417153, 149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179, 26394940582090344, 59388527912287392, 47325384827973252, 15262273318168800, 1648005959253654, 74268805562952, 1233891662727 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
EXAMPLE
1;
0, 3;
18, 24, 39;
3456, 8190, 5928, 2109;
7619508, 17094240, 13700700, 4215120, 417153;
149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179;
MATHEMATICA
nn = 7; q = 3; b[p_, i_] := Count[p, i]; d[p_, i_] :=Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; g[u_, v_] :=
Total[Map[v^Length[#] u^Total[#]/aut[1, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Table[Take[(Table[ Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v]^3 Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid
CROSSREFS
Cf. A346209 (column k=0), A290516 (main diagonal), A060722 (row sums).
Sequence in context: A174029 A103715 A131860 * A263578 A048080 A306279
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jul 16 2021
STATUS
approved

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Last modified April 21 19:32 EDT 2024. Contains 371885 sequences. (Running on oeis4.)