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A272663
Number of distinct characteristic polynomials of n X n matrices with elements {t, 1}, where t is an indeterminate.
8
1, 2, 9, 68, 1161, 65348
OFFSET
0,2
LINKS
Robert M. Corless, Bohemian Eigenvalues, Talk Presented at Computational Discovery in Mathematics (ACMES 2), University of Western Ontario, May 12 2016. Rotman Institute of Philosophy, Youtube video. (Talk based on joint work with Steven E. Thornton, Sonia Gupta, Jonathan Brino-Tarasoff, Venkat Balasubramanian.)
EXAMPLE
From Robin Visser, May 01 2025: (Start)
For n = 1, the a(1) = 2 possible characteristic polynomials are x - 1 and x - t.
For n = 2, the a(2) = 9 possible characteristic polynomials are x^2 - 2*x, x^2 - 2*t*x, x^2 - 2*t*x + t^2 - t, x^2 + (-t - 1)*x, x^2 + (-t - 1)*x - t^2 + t, x^2 - 2*x - t^2 + 1, x^2 - 2*t*x + t^2 - 1, x^2 - 2*x - t + 1, and x^2 + (-t - 1)*x + t - 1. (End)
PROG
(SageMath)
import itertools
def a(n):
ans, t = set(), SR('t')
W = itertools.product([t, 1], repeat=n*n)
for w in W: ans.add(Matrix(SR, n, n, w).charpoly())
return len(ans) # Robin Visser, May 01 2025
CROSSREFS
Six classes of matrices mentioned in Rob Corless's talk: A272658, A272659, A272660, A272661, A272662, A272663.
Sequence in context: A354730 A193160 A255537 * A006849 A319285 A316652
KEYWORD
nonn,hard,more
AUTHOR
N. J. A. Sloane, May 15 2016
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Sep 28 2023
a(5) from Robin Visser, May 04 2025
STATUS
approved