The symmetric functions catalog

An overview of symmetric functions and related topics

2019-10-17

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 $q$-analogue of $|\CDP(n,w)|$:

\begin{align} |\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{align}

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.

Lattice walks in the plane

In [MOP15], the authors consider a $q$-analogue 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 seem to suggest a cyclic sieving under $\grpc_n.$

References