2024/04/23 by Shi-Mei Ma, Ma, Shi-Mei, Toufik Mansour +5
Computer Science · #05A05 #05A20 #Combinatorics (math.CO) #FOS: Mathematics #Natural Language Processing Techniques #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2404.15119
openalex publication_date 2024/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the theory of normal ordered grammars, which gives a natural generalization of the normal ordering problem. To illustrate the main idea, we explore normal ordered grammars associated with the Eulerian polynomials and the second-order Eulerian polynomials. In particular, we present a normal ordered grammatical interpretation for the (cdes,cyc) (p,q)-Eulerian polynomials, where cdes and cyc are the cycle descent and cycle statistics, respectively. The exponential generating function for a family of polynomials, generated by a normal ordered grammar associated with the second-order Eulerian polynomials, reveals an interesting feature: its expression involves the generating function for Catalan numbers as its exponent. In the final part, we discuss some normal ordered grammars related to the type B Eulerian polynomials. A normal ordered grammatical interpretation of the up-down run polynomial is also established.