On the number of permutations with bounded run lengths

TL;DR

This paper develops recursive formulas to count permutations with bounded run lengths, enabling efficient computation, correction of historical errors, and derivation of generating functions with closed-form solutions for increasing runs.

Contribution

It introduces new recursive formulas for counting permutations with bounded runs, corrects previous miscalculations, and derives explicit generating functions.

Findings

Recursive formulas for permutations with bounded runs

Corrected errors in classic combinatorial literature

Closed-form generating functions for increasing runs

Abstract

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.