Continued fractions for cycle-alternating permutations
Saved in:
| Title: | Continued fractions for cycle-alternating permutations |
|---|---|
| Authors: | Deb, Bishal, Sokal, Alan |
| Contributors: | Deb, Bishal |
| Source: | The Ramanujan Journal. 65:1013-1060 |
| Publication Status: | Preprint |
| Publisher Information: | Springer Science and Business Media LLC, 2024. |
| Publication Year: | 2024 |
| Subject Terms: | Permutation, S-fraction, 05A19 (Primary), 05A05, 05A15, 05A30, 11B68, 30B70 (Secondary), cycle-alternating permutation, [MATH] Mathematics [math], alternating Laguerre digraph, alternating cycle, continued fraction, secant numbers, FOS: Mathematics, Mathematics - Combinatorics, tangent numbers, Dyck path, Combinatorics (math.CO), Laguerre digraph |
| Description: | A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index i is either a cycle valley ($$\sigma ^{-1}(i)>i σ - 1 ( i ) > i < σ ( i ) ) or a cycle peak ($$\sigma ^{-1}(i)\sigma (i)$$ σ - 1 ( i ) < i > σ ( i ) ). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes–Rogers and Jacobi–Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs. |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1572-9303 1382-4090 |
| DOI: | 10.1007/s11139-024-00905-7 |
| DOI: | 10.48550/arxiv.2304.06545 |
| Access URL: | http://arxiv.org/abs/2304.06545 https://hal.science/hal-04717223v1 https://hal.science/hal-04717223v1/document https://doi.org/10.1007/s11139-024-00905-7 |
| Rights: | CC BY |
| Accession Number: | edsair.doi.dedup.....8dd18428f0495c60ab302d940e204453 |
| Database: | OpenAIRE |
Full Text Finder 
Nájsť tento článok vo Web of Science | Abstract: | A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index i is either a cycle valley ($$\sigma ^{-1}(i)>i σ - 1 ( i ) > i < σ ( i ) ) or a cycle peak ($$\sigma ^{-1}(i)\sigma (i)$$ σ - 1 ( i ) < i > σ ( i ) ). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes–Rogers and Jacobi–Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs. |
|---|---|
| ISSN: | 15729303 13824090 |
| DOI: | 10.1007/s11139-024-00905-7 |