Joint equidistributions of mesh patterns 123 and 132 with minus antipodal shadings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911305807429632 |
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| author | Lv, Shuzhen Zhang, Philip B. |
| author_facet | Lv, Shuzhen Zhang, Philip B. |
| contents | The study of joint equidistributions of mesh patterns 123 and 132 with the same symmetric shadings was recently initiated by Kitaev and Lv, where 75 of 80 potential joint equidistributions were proven. In this paper, we prove 112 out of 126 potential joint equidistributions of mesh patterns 123 and 132 with the same minus antipodal shadings. As a byproduct, we present 562 joint equidistribution results for non-symmetric and non-minus-antipodal shadings. To achieve this, we construct bijections, find recurrence relations, and obtain generating functions. Moreover, we demonstrate that the joint distributions of several pairs of mesh patterns are related to the unsigned Stirling numbers of the first kind. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Joint equidistributions of mesh patterns 123 and 132 with minus antipodal shadings Lv, Shuzhen Zhang, Philip B. Combinatorics 05A05, 05A15 The study of joint equidistributions of mesh patterns 123 and 132 with the same symmetric shadings was recently initiated by Kitaev and Lv, where 75 of 80 potential joint equidistributions were proven. In this paper, we prove 112 out of 126 potential joint equidistributions of mesh patterns 123 and 132 with the same minus antipodal shadings. As a byproduct, we present 562 joint equidistribution results for non-symmetric and non-minus-antipodal shadings. To achieve this, we construct bijections, find recurrence relations, and obtain generating functions. Moreover, we demonstrate that the joint distributions of several pairs of mesh patterns are related to the unsigned Stirling numbers of the first kind. |
| title | Joint equidistributions of mesh patterns 123 and 132 with minus antipodal shadings |
| topic | Combinatorics 05A05, 05A15 |
| url | https://arxiv.org/abs/2506.23148 |