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From Dyck Paths to Standard Young Tableaux

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Abstract

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions.

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Notes

  1. See [21, §3.1] or [23, §7.11] for an introduction to RSK insertion.

  2. The equivalence of this definition with (5.1) in the case \(d=1\) is shown in [4].

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Author notes

  1. Jordan O. Tirrell

    Present address: Department of Mathematics and Computer Science, Washington College, Chestertown, MD, 21620, USA

Authors and Affiliations

  1. Penn State Altoona, 3000 Ivyside Park, Altoona, PA, 16601, USA

    Juan B. Gil & Michael D. Weiner

  2. Department of Mathematics, Bucknell University, Lewisburg, PA, 17837, USA

    Peter R. W. McNamara

  3. Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA, 01075, USA

    Jordan O. Tirrell

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  1. Juan B. Gil
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  2. Peter R. W. McNamara
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  3. Jordan O. Tirrell
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  4. Michael D. Weiner
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Correspondence to Peter R. W. McNamara.

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Peter McNamara was partially supported by grant #245597 from the Simons Foundation.

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Gil, J.B., McNamara, P.R.W., Tirrell, J.O. et al. From Dyck Paths to Standard Young Tableaux. Ann. Comb. 24, 69–93 (2020). https://doi.org/10.1007/s00026-019-00482-3

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  • DOI: https://doi.org/10.1007/s00026-019-00482-3

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