Published July 2020
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Journal Article
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Path Counting and Rank Gaps in Differential Posets
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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.
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.Attached Files
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Additional details
Identifiers
- Eprint ID
- 105006
- Resolver ID
- CaltechAUTHORS:20200818-144505108
Related works
- Describes
- https://arxiv.org/abs/1806.03509 (URL)
Funding
- NSF Graduate Research Fellowship
- DGE-1122374
Dates
- Created
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2020-08-18Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field