A367978 Number of distinct characteristic polynomials for 4 X 4 matrices with entries from {0, 1, ..., n}.

1, 333, 58335, 2875405, 47125558

Offset: 0

Author

Robert P. P. McKone, Dec 07 2023

Keywords

• nonn
• more
• hard

Crossrefs

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

Cf. A272659.

Programs

Mathematica

mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}];
flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]];
charPolyMat[n_Integer?Positive] := charPolyMat[n] = FullSimplify[CoefficientList[Expand[CharacteristicPolynomial[mat[n], x]], x]];
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]]];
Table[a[4, n], {n, 0, 2}]

Sage

import itertools
def a(n):
    ans, W = set(), itertools.product(range(n+1), repeat=16)
    for w in W: ans.add(Matrix(ZZ, 4, 4, w).charpoly())
    return len(ans)  # Robin Visser, May 04 2025

Extensions

a(4) from Robin Visser, May 04 2025