Chain Decompositions of qt-Catalan Numbers via Local Chains

Abstract

The qt-Catalan number \({{\,\mathrm{Cat}\,}}_n(q,t)\) enumerates integer partitions contained in an \(n\times n\) triangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018) proposed a new approach to understanding the symmetry property \({{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)\) based on decomposing the set of all integer partitions into infinite chains. Each such global chain \(\mathcal {C}_{\mu }\) has an opposite chain \(\mathcal {C}_{\mu ^*}\); these combine to give a new small slice of \({{\,\mathrm{Cat}\,}}_n(q,t)\) that is symmetric in q and t. Here, we advance the agenda of Lee et al. (SIAM J Discr Math 32:191–232, 2018) by developing a new general method for building the global chains \(\mathcal {C}_{\mu }\) from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most \(11\). This proves that for all n, the terms in \({{\,\mathrm{Cat}\,}}_n(q,t)\) of degree at least \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -11\) are symmetric in q and t.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments.

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Correspondence to Li Li.

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Kyungyong Lee was supported by NSF Grant DMS 1800207, 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 Nicholas A. Loehr).

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Han, S., Lee, K., Li, L. et al. Chain Decompositions of qt-Catalan Numbers via Local Chains. Ann. Comb. (2020). https://doi.org/10.1007/s00026-020-00512-5

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Keywords

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

Mathematics Subject Classification

  • 05A19
  • 05A17
  • 05E05