A272660 Number of distinct characteristic polynomials of n X n matrices with elements {t, 1, 2} where t is an indeterminate.

1, 3, 36, 1782, 760678

Offset: 0

Author

N. J. A. Sloane, May 15 2016

Keywords

• nonn
• more

Examples

			From _Robin Visser_, May 01 2025: (Start)
For n = 1, the a(1) = 3 possible characteristic polynomials are x - 2, x - 1, and x - t.
For n = 2, the a(2) = 36 possible characteristic polynomials are x^2 - 2x - 1, x^2 - 2x, x^2 + (-t-2)x, x^2 + (-t-2)x - t^2+2t, x^2 + (-t-1)x, x^2 + (-t-1)x - t^2+t, x^2 - 4x, x^2 - 4x + 2, x^2 - 4x + 3, x^2 - 3x - 2, x^2 - 3x, x^2 - 3x + 1, x^2 - 2tx + t^2-4, x^2 - 2tx + t^2-2, x^2 - 2tx + t^2-1, x^2 + (-t-2)x + t, x^2 - 4x - t+4, x^2 - 4x - t^2+4, x^2 - 2x - 3, x^2 - 2tx, x^2 - 2tx + t^2-t, x^2 - 2tx + t^2-2t, x^2 - 4x - 2t+4, x^2 - 3x - t^2+2, x^2 - 3x - t+2, x^2 - 3x - 2t+2, x^2 - 2x - t^2+1, x^2 - 2x - t+1, x^2 - 2x - 2t+1, x^2 + (-t-2)x + 2t-4, x^2 + (-t-2)x + 2t-2, x^2 + (-t-2)x + 2t-1, x^2 + (-t-1)x + t-4, x^2 + (-t-1)x + t-2, x^2 + (-t-1)x + t-1, and x^2 + (-t-1)x - t. (End)
		

References

• Robert M. Corless, Bohemian Eigenvalues, Talk Presented at Computational Discovery in Mathematics (ACMES 2), University of Western Ontario, May 12 2016. (Talk based on joint work with Steven E. Thornton, Sonia Gupta, Jonathan Brino-Tarasoff, Venkat Balasubramanian.)

Crossrefs

Six classes of matrices mentioned in Rob Corless's talk: A272658, A272659, A272660, A272661, A272662, A272663.

Programs

Sage

import itertools
def a(n):
    ans, t = set(), SR('t')
    W = itertools.product([t, 1, 2], repeat=n*n)
    for w in W: ans.add(Matrix(SR, n, n, w).charpoly())
    return len(ans)  # Robin Visser, May 01 2025

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 28 2023

a(4) from Robin Visser, May 01 2025