Thron-type continued fractions (T-fractions) for some classes of increasing trees
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915062993649664 |
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| author | Bitonti, Veronica Deb, Bishal Sokal, Alan D. |
| author_facet | Bitonti, Veronica Deb, Bishal Sokal, Alan D. |
| contents | We introduce some classes of increasing labeled and multilabeled trees, and we show that these trees provide combinatorial interpretations for certain Thron-type continued fractions with coefficients that are quasi-affine of period 2. Our proofs are based on bijections from trees to labeled Motzkin or Schröder paths; these bijections extend the well-known bijection of Françon--Viennot (1979) interpreted in terms of increasing binary trees. This work can also be viewed as a sequel to the recent work of Elvey Price and Sokal (2020), where they provide combinatorial interpretations for Thron-type continued fractions with coefficients that are affine. Towards the end of the paper, we conjecture an equidistribution of vincular patterns on permutations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thron-type continued fractions (T-fractions) for some classes of increasing trees Bitonti, Veronica Deb, Bishal Sokal, Alan D. Combinatorics 05A19 (Primary), 05A05, 05A15, 05A30, 05C05, 05C30, 30B70 (Secondary) We introduce some classes of increasing labeled and multilabeled trees, and we show that these trees provide combinatorial interpretations for certain Thron-type continued fractions with coefficients that are quasi-affine of period 2. Our proofs are based on bijections from trees to labeled Motzkin or Schröder paths; these bijections extend the well-known bijection of Françon--Viennot (1979) interpreted in terms of increasing binary trees. This work can also be viewed as a sequel to the recent work of Elvey Price and Sokal (2020), where they provide combinatorial interpretations for Thron-type continued fractions with coefficients that are affine. Towards the end of the paper, we conjecture an equidistribution of vincular patterns on permutations. |
| title | Thron-type continued fractions (T-fractions) for some classes of increasing trees |
| topic | Combinatorics 05A19 (Primary), 05A05, 05A15, 05A30, 05C05, 05C30, 30B70 (Secondary) |
| url | https://arxiv.org/abs/2412.10214 |