Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.15119 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929325509443584 |
|---|---|
| author | Ma, Shi-Mei Mansour, Toufik Yeh, Jean Yeh, Yeong-Nan |
| author_facet | Ma, Shi-Mei Mansour, Toufik Yeh, Jean Yeh, Yeong-Nan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal ordered grammars Ma, Shi-Mei Mansour, Toufik Yeh, Jean Yeh, Yeong-Nan Combinatorics 05A05, 05A20 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. |
| title | Normal ordered grammars |
| topic | Combinatorics 05A05, 05A20 |
| url | https://arxiv.org/abs/2404.15119 |