Dinv, area, and bounce for (k)over-right-arrow-Dyck paths

doi:10.1016/j.aam.2022.102455

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	Dinv, area, and bounce for (k)over-right-arrow-Dyck paths
	Xin, Guoce 1; Zhang, Yingrui 2
	2023-02-01
发表期刊	ADVANCES IN APPLIED MATHEMATICS
ISSN	0196-8858
卷号	143 页码:19
摘要	The well-known q, t-Catalan sequence has two combinatorial interpretations as weighted sums of ordinary Dyck paths: one is Haglund's area-bounce formula, and the other is Haiman's dinv-area formula. The zeta map was constructed to connect these two formulas: it is a bijection from ordinary Dyck paths to themselves, and it takes dinv to area, and area to bounce. Such a result was extended for k-Dyck paths by Loehr. The zeta map was extended by Armstrong-Loehr-Warrington for a very general class of paths. In this paper, weextend the dinv-area-bounce result for (k) over right arrow -Dyck paths by: i) giving a geometric construction for the bounce statistic of a k-Dyck path, which includes the k-Dyck paths and ordinary Dyck paths as special cases; ii) giving a geometric interpretation of the dinv statistic of a (k) over right arrow -Dyck path. Our bounce construction is inspired by Loehr's construction and Xin-Zhang's linear algorithm for inverting the sweep map on (k) over right arrow -Dyck paths. Our dinv interpretation is inspired by Garsia-Xin's visual proof of dinv-to-area result on rational Dyck paths. (c) 2022 Elsevier Inc. All rights reserved.
关键词	q, t-Catalan numbers Sweep map (k)over-right-arrow-Dyck paths
DOI	10.1016/j.aam.2022.102455
收录类别	SCI
语种	英语
资助项目	NSFC ; [11871067]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000882526400002
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/60578
专题	中国科学院数学与系统科学研究院
通讯作者	Zhang, Yingrui
作者单位	1.Capital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Xin, Guoce,Zhang, Yingrui. Dinv, area, and bounce for (k)over-right-arrow-Dyck paths[J]. ADVANCES IN APPLIED MATHEMATICS,2023,143:19.
APA	Xin, Guoce,&Zhang, Yingrui.(2023).Dinv, area, and bounce for (k)over-right-arrow-Dyck paths.ADVANCES IN APPLIED MATHEMATICS,143,19.
MLA	Xin, Guoce,et al."Dinv, area, and bounce for (k)over-right-arrow-Dyck paths".ADVANCES IN APPLIED MATHEMATICS 143(2023):19.