  This is an etude in experimental semi-rigorous (rigorizable!) mathematics.
The leading asymptotics was brilliantly derived by Allan Berele and Amitai
Regev for general hooks H(k,l) and general powers z, but what about more
refined asymptotics? For small k and l, one can "guess" a linear recurrence
(since we live in the holonomic ansatz) and using the Birkhoff-Trjitzinsky
method, beautifully implemented in Doron Zeilberger's Maple package AsyRec
(that has been incorporated into the present Maple package), we computed
amazing refined asymptotics, that confirm, with a vengeance, the Berele-Regev
asymptotic formula, and especially the impressive constant in front!
