A051680 -id:A051680 - cp's OEIS frontend

A346209 Number of n X n matrices over GF(3) with no eigenvalues in GF(3), i.e., neither 0 nor 1 nor 2 is an eigenvalue.

1, 0, 18, 3456, 7619508, 149200289280, 26394940582090344, 42062797470468915399168, 603463180651533072058654437264, 77927374189849689541269666899007713280, 90570450400853976077932766909301405665963077152

Offset: 0

Geoffrey Critzer, Jul 10 2021

nonn

Equivalently, a(n) is the number of n X n matrices over GF(3) whose characteristic polynomial has no linear factors.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A002820, A051680, A027376, A053290.

nn = 10; q = 3; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[Product[1/(1 - u^d/q^(r d)), {r, 1, \[Infinity]}]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], u]

Sum_{n>=0} a(n)*x^n/A053290(n) = Product_{d>=2} (Product_{r>=1} 1/(1-x^d/3^(r*d)))^A027376(d).

A053071 Number of n X n invertible binary matrices A such that A^5+I is invertible.

Original entry on oeis.org

0, 2, 48, 4480, 2887680, 5373624320, 44196975083520, 1442855588252876800, 188570467779447305011200, 98800579402758985681259724800, 207089099087390763078860569942425600

Offset: 1

Vladeta Jovovic, Mar 18 2000

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Cf. A002820, A051680, A053060.

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.

Original entry on oeis.org

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

Offset: 0

Geoffrey Critzer, Jul 14 2021

			             1;
             1,            1;
            27,           20,          1;
          6291,         4719,        221,        1;
      13589289,     10191960,     477750,     2120,     1;
  266377183929, 199782888129, 9364822830, 41559870, 19481, 1;

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

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

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

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A346209 Number of n X n matrices over GF(3) with no eigenvalues in GF(3), i.e., neither 0 nor 1 nor 2 is an eigenvalue.

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A053071 Number of n X n invertible binary matrices A such that A^5+I is invertible.

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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.

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