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

Refined Asymptotics and Explicit Recurrences for the numbers of Young tableaux in the (k,l) hook for k+l less than six