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Path Counting and Rank Gaps in Differential Posets

Gaetz, Christian and Venkataramana, Praveen (2020) Path Counting and Rank Gaps in Differential Posets. Order, 37 (2). pp. 279-286. ISSN 0167-8094. https://resolver.caltech.edu/CaltechAUTHORS:20200818-144505108

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Abstract

We study the gaps Δp_n between consecutive rank sizes in r-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller’s result that Δp_n ≥ 1, which resolved a longstanding conjecture of Stanley, by showing that Δp_n ≥ 2r. We also obtain stronger bounds in the case that the poset has many substructures called threads.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s11083-019-09504-4DOIArticle
https://arxiv.org/abs/1806.03509arXivDiscussion Paper
ORCID:
AuthorORCID
Gaetz, Christian0000-0002-3748-4008
Additional Information:© 2020 Springer Verlag. Received 06 August 2018; Accepted 07 August 2019; Published 21 September 2019. The authors wish to thank Fabrizio Zanello for helping to initiate this joint project and Richard Stanley for his helpful conversations. We are also grateful to Patrick Byrnes for making his computer code available and Alexander Miller for providing useful references. C.G. was supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.
Funders:
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1122374
Subject Keywords:Differential poset; Rank growth; Thread element; Hasse diagram; Path counting
Issue or Number:2
Record Number:CaltechAUTHORS:20200818-144505108
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200818-144505108
Official Citation:Gaetz, C., Venkataramana, P. Path Counting and Rank Gaps in Differential Posets. Order 37, 279–286 (2020). https://doi.org/10.1007/s11083-019-09504-4
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105006
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:18 Aug 2020 21:52
Last Modified:18 Aug 2020 21:52

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