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Number of distinct characteristic polynomials for 4 X 4 matrices with entries from {0, 1, ..., n}.
0

%I #7 Jan 12 2024 16:06:58

%S 1,333,58335,2875405

%N Number of distinct characteristic polynomials for 4 X 4 matrices with entries from {0, 1, ..., n}.

%t mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}];

%t flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]];

%t charPolyMat[n_Integer?Positive] := charPolyMat[n] = FullSimplify[CoefficientList[Expand[CharacteristicPolynomial[mat[n], x]], x]];

%t a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[Table[Evaluate[charPolyMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}], d^2 - 1]]];

%t Table[a[4, n], {n, 0, 2}]

%Y Cf. A366448 (2 X 2 matrices), A366551 (3 X 3 matrices).

%Y Cf. A272659.

%K nonn,more,hard

%O 0,2

%A _Robert P. P. McKone_, Dec 07 2023