Combinatorial study of the q-Catalan triangle and its generalizations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913128832303104 |
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| author | Wirdane, Youssouf |
| author_facet | Wirdane, Youssouf |
| contents | We announce a series of results on the combinatorial study of the q-Catalan triangle (C_{n,k}(q)), defined by C_{n,0}(q)=q^{n(n-1)/2} and C_{n,k}(q)=C_{n,k-1}(q)+q^{n-k-1}C_{n-1,k}(q). We establish combinatorial interpretations via a universal combinatorial family of seven components: four families of pattern-avoiding permutations weighted by inversion or co-inversion statistics, Dyck paths, binary words and triangulations. We introduce the mirror polynomial C-tilde_{n,k}(q)=q^{n(n-1)/2}C_{n,k}(q^{-1}), prove its dual recurrence and co-inversion interpretation. The q,p-Catalan triangle and a multivariate generalization opening the way to cyclotomic q-analogues are introduced. Theorems on the q-Catalan triangle via 312-avoiding permutations and the mirror recurrence are proved completely here. This is the first paper of series W0-W5 on classical and q-deformed interpretations of the Catalan triangle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14682 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Combinatorial study of the q-Catalan triangle and its generalizations Wirdane, Youssouf Combinatorics 05A15, 05A30, 05A05, 05C30 We announce a series of results on the combinatorial study of the q-Catalan triangle (C_{n,k}(q)), defined by C_{n,0}(q)=q^{n(n-1)/2} and C_{n,k}(q)=C_{n,k-1}(q)+q^{n-k-1}C_{n-1,k}(q). We establish combinatorial interpretations via a universal combinatorial family of seven components: four families of pattern-avoiding permutations weighted by inversion or co-inversion statistics, Dyck paths, binary words and triangulations. We introduce the mirror polynomial C-tilde_{n,k}(q)=q^{n(n-1)/2}C_{n,k}(q^{-1}), prove its dual recurrence and co-inversion interpretation. The q,p-Catalan triangle and a multivariate generalization opening the way to cyclotomic q-analogues are introduced. Theorems on the q-Catalan triangle via 312-avoiding permutations and the mirror recurrence are proved completely here. This is the first paper of series W0-W5 on classical and q-deformed interpretations of the Catalan triangle. |
| title | Combinatorial study of the q-Catalan triangle and its generalizations |
| topic | Combinatorics 05A15, 05A30, 05A05, 05C30 |
| url | https://arxiv.org/abs/2605.14682 |