A346384 - OEIS

%I #13 Jul 15 2021 03:41:51

%S 1,1,1,27,20,1,6291,4719,221,1,13589289,10191960,477750,2120,1,

%T 266377183929,199782888129,9364822830,41559870,19481,1,

%U 47123189360124723,35342392020078780,1656674625945339,7352106327720,3446299857,176540,1

%N 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.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%e              1;

%e              1,            1;

%e             27,           20,          1;

%e           6291,         4719,        221,        1;

%e       13589289,     10191960,     477750,     2120,     1;

%e   266377183929, 199782888129, 9364822830, 41559870, 19481, 1;

%t 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[

%t    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}];

%t 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[

%t     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

%Y Cf. A051680 (column k=0), A053290 (row sums).

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Jul 14 2021