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arXiv · 2404.15119

Normal ordered grammars

Shi-Mei Ma·Toufik Mansour·Jean Yeh·Yeong-Nan Yeh
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Abstract

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.

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Shi-Mei Ma, Toufik Mansour, Jean Yeh, Yeong-Nan Yeh. 2024-04-23. Normal ordered grammars. https://arxiv.org/abs/2404.15119

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