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arXiv · 2312.02830

Differential operators, grammars and Young tableaux

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

In algebraic combinatorics and formal calculation, context-free grammar is defined by a formal derivative based on a set of substitution rules. In this paper, we investigate this issue from three related viewpoints. Firstly, we introduce a differential operator method. As one of the applications, we deduce a new grammar for the Narayana polynomials. Secondly, we investigate the normal ordered grammars associated with the Eulerian polynomials. Thirdly, motivated by the theory of differential posets, we introduce a box sorting algorithm which leads to a bijection between the terms in the expansion of $(cD)^nc$ and a kind of ordered weak set partitions, where $c$ is a smooth function in the indeterminate $x$ and $D$ is the derivative with respect to $x$. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of $(cD)^nc$ in terms of standard Young tableaux. Combining this with the theory of context-free grammars, we provide a unified interpretations for the Ramanujan polynomials, Andr\'e polynomials, left peak polynomials, interior peak polynomials, Eulerian polynomials of types $A$ and $B$, $1/2$-Eulerian polynomials, second-order Eulerian polynomials, and Narayana polynomials of types $A$ and $B$ in terms of standard Young tableaux. Along the same lines, we present an expansion of the powers of $c^kD$ in terms of standard Young tableaux, where $k$ is a positive integer. In particular, we provide four interpretations for the second-order Eulerian polynomials. All of the above apply to the theory of formal differential operator rings.

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Shi-Mei Ma, Jean Yeh, Yeong-Nan Yeh. 2023-12-05. Differential operators, grammars and Young tableaux. https://arxiv.org/abs/2312.02830

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