Abstract
A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid an increasing pattern. In particular, we count the Grassmannian permutations of size m avoiding the identity permutation of size k, thus solving a conjecture made by Weiner. We also refine our counts to special classes such as odd Grassmannian permutations and Grassmannian involutions. We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words.
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Acknowledgements
We would like to thank Michael Weiner for his comments on an earlier version of this paper. We would also like to thank the referees for their careful reading and helpful suggestions. The computer algebra system SageMath [25] provided valuable assistance in studying examples. This work was done during the first author’s visit to the Indian Institute of Technology Bhilai, funded by SERB, India, via the project AV/VRI/2022/0140. Krishna Menon is partially supported by a grant from the Infosys Foundation, and Anurag Singh is partially supported by the Research Initiation Grant from the Indian Institute of Technology Bhilai (No. 2009301).
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Menon, K., Singh, A. Dyck Paths, Binary Words, and Grassmannian Permutations Avoiding an Increasing Pattern. Ann. Comb. (2023). https://doi.org/10.1007/s00026-023-00667-x
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DOI: https://doi.org/10.1007/s00026-023-00667-x