Recurrences and formulae in an extension of the Eulerian numbers

In this paper, the author develops a general formula for numbers which are extensions of the Eulerian numbers. A_m ^tn, r) is defined to be the number of n-permutations with r rises of magnitude at least m. Intermediate steps in the derivation include a vertical recurrence relation. recurrence relations of the diagonals both as numbers and as polynomials, and recurrences and formulae strictly on the diagonals.