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Chain Decompositions of qt-Catalan Numbers: Tail Extensions and Flagpole Partitions

Abstract

This article is part of an ongoing investigation of the combinatorics of qt-Catalan numbers \({{\,\mathrm{Cat}\,}}_n(q,t)\). We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit k into a disjoint union of chains \({\mathcal {C}}_{\mu }\) indexed by partitions of size k. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property \({{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)\). Previously, we introduced a map that builds the tail part of each chain \({\mathcal {C}}_{\mu }\). Our first main contribution here is to extend this map to construct larger second-order tails for each chain. Second, we introduce new classes of partitions called flagpole partitions and generalized flagpole partitions. Third, we describe a recursive construction for building the chain \({\mathcal {C}}_{\mu }\) for a (generalized) flagpole partition \(\mu \), assuming that the chains indexed by certain specific smaller partitions (depending on \(\mu \)) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.

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References

  1. Maria Monks Gillespie, Two \(q,t\)-symmetry problems in symmetric function theory, Open Problems in Algebraic Combinatorics blog (online at http://samuelfhopkins.com/OPAC/files/opacblog_master.pdf), pp. 38–41.

  2. J. Haglund, The \(q,t\)-Catalan Numbers and the Space of Diagonal Harmonics, with an Appendix on the Combinatorics of Macdonald Polynomials, AMS University Lecture Series (2008).

  3. S. Han, K. Lee, L. Li, and N. Loehr, Chain decompositions of \(q,t\)-Catalan numbers via local chains, Ann. Comb. 24 (2020), 739–765.

    MathSciNet  Article  Google Scholar 

  4. S. Han, K. Lee, L. Li, and N. Loehr, Extended Appendix of [3], online at sites.google.com/oakland.edu/li2345/code-and-data

  5. G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75–115.

    MathSciNet  Article  Google Scholar 

  6. K. Lee, L. Li, and N. Loehr, A combinatorial approach to the symmetry of \(q,t\)-Catalan numbers, SIAM J. Discrete Math. 32 (2018), 191–232.

    MathSciNet  Article  Google Scholar 

  7. K. Lee, L. Li, and N. Loehr, Limits of modified higher \(q,t\)-Catalan numbers, Electron. J. Combin. 20(3) (2013), research paper P4, 23 pages (electronic).

  8. N. Loehr and G. Warrington, A continuous family of partition statistics equidistributed with length, J. Combin. Theory Ser. A 116 (2009), 379–403.

    MathSciNet  Article  Google Scholar 

  9. A. M. Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics Vol. 2, edited by R. L. Graham, M. Groetschel, and L. Lovasz, Elsevier (1995), 1063–1229.

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Correspondence to Li Li.

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Communicated by Jang Soo Kim.

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Kyungyong Lee was supported by NSF Grant DMS 2042786, the Korea Institute for Advanced Study (KIAS), and the University of Alabama.

This work was supported by a grant from the Simons Foundation/SFARI (Grant #633564 to N.A.L.).

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Han, S., Lee, K., Li, L. et al. Chain Decompositions of qt-Catalan Numbers: Tail Extensions and Flagpole Partitions. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00590-7

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  • DOI: https://doi.org/10.1007/s00026-022-00590-7

Keywords

  • q
  •  t-Catalan numbers
  • Dyck paths
  • Dinv statistic
  • Joint symmetry
  • Integer partitions
  • Chain decompositions

Mathematics Subject Classification

  • 05A19
  • 05A17
  • 05E05