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A346384 Triangle read by rows. T(n,k) is the number of invertible n X n matrices over GF(3) such that the dimension of the eigenspace corresponding to the eigenvalue 1 is k, 0 <= k <= n, n >= 0. 0
1, 1, 1, 27, 20, 1, 6291, 4719, 221, 1, 13589289, 10191960, 477750, 2120, 1, 266377183929, 199782888129, 9364822830, 41559870, 19481, 1, 47123189360124723, 35342392020078780, 1656674625945339, 7352106327720, 3446299857, 176540, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
EXAMPLE
1;
1, 1;
27, 20, 1;
6291, 4719, 221, 1;
13589289, 10191960, 477750, 2120, 1;
266377183929, 199782888129, 9364822830, 41559870, 19481, 1;
MATHEMATICA
nn = 6; 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]}]; A027376 = 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}]]]; Map[Select[#, # > 0 &] &, Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
Series[(g[u, v] /. v -> 1)*g[u, v]* Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A027376[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}]] // Grid
CROSSREFS
Cf. A051680 (column k=0), A053290 (row sums).
Sequence in context: A040704 A366673 A182141 * A247434 A022983 A023469
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jul 14 2021
STATUS
approved

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Last modified July 24 20:30 EDT 2024. Contains 374585 sequences. (Running on oeis4.)