Context-free Grammars for Permutations and Increasing Trees

TL;DR

This paper introduces grammatical labeling techniques to generate and analyze combinatorial objects like permutations and increasing trees, deriving generating functions and establishing new combinatorial identities without differential equations.

Contribution

It develops a grammatical framework for describing combinatorial objects, providing new derivations of generating functions and connecting permutations with increasing trees.

Findings

Derived grammars for Eulerian and Andre9 polynomials

Connected permutations with increasing trees via grammatical methods

Provided combinatorial proofs for permutation and tree correspondences

Abstract

In this paper, we introduce the notion of a grammatical labeling to describe a recursive process of generating combinatorial objects based on a context-free grammar. For example, by labeling the ascents and descents of a Stirling permutation, we obtain a grammar for the second-order Eulerian polynomials. By using the grammar for $0$-$1$-$2$ increasing trees given by Dumont, we obtain a grammatical derivation of the generating function of the Andr\'e polynomials obtained by Foata and Sch\"utzenberger, without solving a differential equation. We also find a grammar for the number $T(n,k)$ of permutations of $[n]=\{1,2,\ldots, n\}$ with $k$ exterior peaks, which was independently discovered by Ma. We demonstrate that Gessel's formula for the generating function of $T(n,k)$ can be deduced from this grammar. Moreover, by using grammars we show that the number of the permutations of $[n]$ with $k$ exterior peaks equals the number of increasing trees on $[n]$ with $2k+1$ vertices of even degree. A combinatorial proof of this fact is also presented.