On the number of permutations with bounded run lengths

In this work we obtain recurrent formulae for the number of permutations with either increasing or monotonic (i.e., both increasing and decreasing) runs of bounded length. Our formulae allow one to efficiently compute the number of such permutations. In particular, we use the formulae to find and correct a few miscalculations in the classic 1966 book by David, Kendall, and Barton. We further use our formulae to derive differential equations for the corresponding exponential generating functions. In the case of increasing runs, we solve these equations and obtain closed-form expressions for the generating functions.