Darij Grinberg's Interesting Remark "On Invariance Properties of Entries of Matrix Powers"

Posted: July 29, 2021

I enjoyed your preprint about the Larcombe-Fennessey identity that came out today (arXiv:2107.13092v1, "On Invariance Properties of Entries of Matrix Powers"). I'd like to remark that (2) holds not only if $A$ is tridiagonal, but more generally if A has the property that

(*) au,v = a(v,u) = 0, whenever u ≤ i and v > i satisfy v - u > 1.

(That is, there is a "tridiagonal bottleneck" between i and i+1 in A.)

Your proof still applies to this generalization, except that the words should not be "continuous" but rather need to pass the (i,i+1) checkpoint whenever they cross the border between "≤ i" and "> i". Instead of continuity, you thus need to make a "what goes up must come down" argument when constructing the bijection.