On some Tutte polynomial sequences in the square lattice

Let T(Lm,n;x,y) be the Tutte polynomial of the square lattice Lm,n, for integers m,n∈Z>0. Using a family of Tutte polynomial inequalities established by the author in a previous work, we study the analytical properties of the sequences (T(Lm,n;x,y)1/mn:n∈Z>0) for a fixed m∈Z>0, and (T(Ln,n;x,y)1/n2:n∈Z>0), in the region x,y⩾1. We show that these sequences are monotonically increasing when (x−1)(y−1)>1. We also compute lower bounds for these limits when (x−1)(y−1)>1, and upper bounds when (x−1)(y−1)<1. At the point (x=2, y=1), where the Tutte polynomial is known to count the number of forests, we compute , which improves upon the previous best upper bound of 3.74101 obtained by Calkin, Merino, Noble and Noy (2003).