The number of triangulations of Delta^2 x Delta^(k) is between alpha^(k^2) and beta*(k^2) where alpha = (27/16)^(1/4) ~ 1.13975 and beta = 6^(1/6) ~ 1.34800 [p. 10 of Santos's handwritten notes about "The Cayley trick"].
There are arithmetic errors in Santos's lecture notes "The Cayley trick". The same table gives lozenge tilings of k*Delta^2.
The first column (indexed by k) of the table on p. 9 in Santos' handwritten notes "The Cayley trick" is actually the sequence (
A273464(k, k*(k-1)/2 + 1): k >= 1).
In later published papers, Santos (2004, 2005) mentions that the number of triangulations of Delta^2 x Delta^k grows as exp(
A244996*k^2/2 + o(k^2)) as k -> infinity. Notice that exp(
A244996 * k^2/2) =
A242710^(k^2/2). [See Theorem 1 and Theorem 4.9. Probably Theorem 1, part (2), in Santos (2004) has a typo.]
Note that alpha = (27/16)^(1/4) ~ 1.13975 <
A242710^(k^2/2) ~ 1.175311 < beta = 6^(1/6) ~ 1.34800 (where alpha and beta are given on the first paragraph of these comments).
The reason the name of the sequence has "Delta^2 x Delta^(k-1)" rather than "Delta^2 x Delta^k" is because (according to Santos) the number of triangulations of Delta^2 x Delta^(k-1) equals k! times the number of lozenge tilings of k*Delta^2. (End)
