Ron,
> T(n,m)=binomial(m!,n) seems not to be in OEIS?
No doubt you've noticed the diagonal in such an array:
A086687: Binomial(n!,n).
A related sequence is A109892:
a(n) = least integer of the form (n!+1)(n!+2)...(n!+k)/n!.
Is A109892 equivalent to binomial(n!+n,n)?