The symmetric functions catalog

See the cyclic sieving phenomenon page for the definition and related theorems.

#Lattice paths

#Circular Dyck paths

In [ALP19], we consider the family \(\CDP(n,w)\) of circular Dyck paths (area sequences of circular unit interval digraphs) with bounded width. These are all vectors of integers \(\avec=(a_1,\dotsc,a_n)\) which satisfy

• \(0 \leq a_i \leq w-1\) for \(1 \leq i \leq n,\)

• \(a_{i+1} \leq a_{i} + 1\) for \(1 \leq i \leq n,\) (index mod \(n\)).

We consider the following \(q\)-analog of \(|\CDP(n,w)|:\) \[\begin{aligned} |\CDP(n,w)|_q = \sum_{s \in \setZ} \sum_{j=1}^w q^{s^2\delta + s(j+1)} \left( \qbinom{2n-1}{n-1-\delta s}_q - \qbinom{2n-1}{n+j+\delta s}_q \right), \end{aligned}\] where \(\delta = w+2.\)

Let \(\alpha\) act on circular Dyck paths via cyclic shift of the area sequence. We show that \[\left\{ \left( \CDP(n,w), \langle \alpha \rangle, |\CDP(n,w)|_q \right) \right\}_{n=1}^{\infty}\] is an instance of a Lyndon-like CSP family.

#Fans of Dyck paths

J. Pappe, S. Pfannerer, A. Schilling, and Mary Claire Simone study promotion on \(r\)-fans of Dyck paths and on vacillating tableaux [PPSS24]. They construct an injection into chord diagrams that intertwines promotion with rotation, using promotion-evacuation diagrams and Fomin growth diagrams. This gives cyclic sieving phenomena for \(r\)-fans of Dyck paths and for the corresponding vacillating tableaux.

#Lattice walks in the plane

In [MOP15], the authors consider a \(q\)-analog of lattice walks in the plane, parametrized by the total number of steps right, left, up and down \((r,l,u,d).\) Suppose we use non-commuting variables with the relation \(xy = qyx,\) and define \(Z_{r,l,u,d}(q)\) via the relation \[(x+y+x^{-1}+y^{-1})^n = \sum_{\substack{r,l,u,d \\ u+d+l+r = n}} Z_{r,l,u,d}(q) y^{-u} y^{d} x^{r} x^{-l}.\] In particular, \(Z_{r,0,u,0}(q) = \qbinom{r+u}{u}_q.\) Evaluating \(Z_{r,l,u,d}(q)\) at roots of unity is interesting, since these values are related to the Hofstadter Hamiltonian.

The authors evaluate special cases of \(Z_{r,l,u,d}(q)\) when \(r-l\) and \(u-d\) are multiples of \(n,\) and this seems to suggest a cyclic sieving under \(\grpc_n.\)

Bibliography

1. [ALP19]Per Alexandersson, Svante Linusson and Samu Potka. The cyclic sieving phenomenon on circular Dyck paths. The Electronic Journal of Combinatorics, 26:1–32, October 2019.
   .bib

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   @article{AlexanderssonLinussonPotka2019,
     doi = {10.37236/8720},
     url2 = {https://doi.org/10.37236/8720},
     year = {2019},
     month = oct,
     publisher = {The Electronic Journal of Combinatorics},
     author = {Per Alexandersson and Svante Linusson and Samu Potka},
     title = {The Cyclic Sieving Phenomenon on Circular {D}yck Paths},
     journal = {The Electronic Journal of Combinatorics},
     volume = {26},
     paper = {P4.16},
     pages = {1--32}
   }
   

2. [MOP15]Stefan Mashkevich, Stéphane Ouvry and Alexios Polychronakos. Statistics of two-dimensional random walks, the cyclic sieving phenomenon and the Hofstadter model. Journal of Physics A: Mathematical and Theoretical, 48(40):405001, September 2015.
   .bib

   Copy Download

   @article{MashkevichOuvryPolychronakos2015,
     doi = {10.1088/1751-8113/48/40/405001},
     url2 = {https://doi.org/10.1088/1751-8113/48/40/405001},
     year  = {2015},
     month = sep,
     publisher = {{IOP} Publishing},
     volume = {48},
     number = {40},
     pages = {405001},
     author = {Stefan Mashkevich and  St{\'{e}}phane Ouvry and  Alexios Polychronakos},
     title = {Statistics of two-dimensional random walks,  the cyclic sieving phenomenon and the {H}ofstadter model},
     journal = {Journal of Physics A: Mathematical and Theoretical}
   }
   

3. [PPSS24]Joseph Pappe, Stephan Pfannerer, Anne Schilling and Mary Claire Simone. Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux. Journal of Algebra, 655:794–842, 2024.
   .bib

   Copy Download

   @article{PappePfannererSchillingSimone2024,
     author = {Joseph Pappe and Stephan Pfannerer and Anne Schilling and Mary Claire Simone},
     title = {Promotion and growth diagrams for fans of {D}yck paths and vacillating tableaux},
     year = {2024},
     journal = {Journal of Algebra},
     volume = {655},
     pages = {794--842},
     doi = {10.1016/j.jalgebra.2023.07.038},
     url = {https://doi.org/10.1016/j.jalgebra.2023.07.038},
     eprint = {2212.13588}
   }