Combinatorial proofs on the joint distribution of descents and inverse descents
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2022
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| _version_ | 1866913529453346816 |
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| author | Li, Frank Z. K. Liu, Xunhao |
| author_facet | Li, Frank Z. K. Liu, Xunhao |
| contents | Let $A_{n,i,j}$ be the number of permutations on $[n]$ with $(i-1)$ descents and $(j-1)$ inverse descents.Carlitz, Roselle and Scoville in 1966 first revealed some combinatorial and arithmetic properties of $A_{n,i,j}$,which contain a recurrence of $A_{n,i,j}$.Using the idea of balls in boxes,Petersen gave a combinatorial interpretation for the generating function of $A_{n,i,j}$,and obtained the same recurrence of $A_{n,i,j}$ from its generating function.Subsequently, Petersen asked whether there is a visual way to understand this recurrence.In this paper,after observing the internal structures of permutation grids,we present a combinatorial proof for the recurrence of $A_{n,i,j}$.Let $I_{n,k}$ and $J_{n,k}$ be the number of involutions and fixed-point free involutions on $[n]$ with $k$ descents,respectively.With the help of algebraic method on generating functions,Guo and Zeng derived two recurrences of $I_{n,k}$ and $J_{2n,k}$ that play an essential role in the proof of their unimodal properties.Surprisingly,the constructive approach to the recurrence of $A_{n,i,j}$ is found to fuel the combinatorial interpretations of these two recurrences of $I_{n,k}$ and $J_{2n,k}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_05307 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Combinatorial proofs on the joint distribution of descents and inverse descents Li, Frank Z. K. Liu, Xunhao Combinatorics 05A05, 05A19 Let $A_{n,i,j}$ be the number of permutations on $[n]$ with $(i-1)$ descents and $(j-1)$ inverse descents.Carlitz, Roselle and Scoville in 1966 first revealed some combinatorial and arithmetic properties of $A_{n,i,j}$,which contain a recurrence of $A_{n,i,j}$.Using the idea of balls in boxes,Petersen gave a combinatorial interpretation for the generating function of $A_{n,i,j}$,and obtained the same recurrence of $A_{n,i,j}$ from its generating function.Subsequently, Petersen asked whether there is a visual way to understand this recurrence.In this paper,after observing the internal structures of permutation grids,we present a combinatorial proof for the recurrence of $A_{n,i,j}$.Let $I_{n,k}$ and $J_{n,k}$ be the number of involutions and fixed-point free involutions on $[n]$ with $k$ descents,respectively.With the help of algebraic method on generating functions,Guo and Zeng derived two recurrences of $I_{n,k}$ and $J_{2n,k}$ that play an essential role in the proof of their unimodal properties.Surprisingly,the constructive approach to the recurrence of $A_{n,i,j}$ is found to fuel the combinatorial interpretations of these two recurrences of $I_{n,k}$ and $J_{2n,k}$. |
| title | Combinatorial proofs on the joint distribution of descents and inverse descents |
| topic | Combinatorics 05A05, 05A19 |
| url | https://arxiv.org/abs/2212.05307 |