The symmetric functions catalog

An overview of symmetric functions and related topics

2020-05-05

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

Matchings and crossings

Polygon dissections with $k$ edges

In [RSW04], the authors consider dissections of polygons with $k$ non-crossing edges, where each edge connects two vertices, $X_{n,k}.$ Let $\grpc_{n}$ act by rotation. This gives the CSP-triple

\[ \left( X_{n,k}, \grpc_{n}, \frac{1}{[n+k]_q}\qbinom{n+k}{k+1}_q \qbinom{n-3}{k}_q \right). \]

This can also be turned into a CSP on the set $\SYT((k+1)^2 1^{n-k-3}),$ but it is not explained what the corresponding group action is.

See also [BR14] for some related results.

Matchings with $k$ crossings

In [LB17], matchings on $[2n]$ with $k$ crossings is studied. Let $X_{n,k}$ be the set of such matchings, and let $\grpc_{2n}$ act on such matchings via rotation. The case $k=0$ correspond to the Catalan case. For $k=1,2,3$ they prove the following instances of cyclic sieving:

\[ \left( X_{n,1}, \grpc_{2n}, \qbinom{2n}{n-2}_q \right) , \quad \left( X_{n,2}, \grpc_{2n}, \frac{[n+3]_q}{[2]_q} \qbinom{2n}{n-3}_q \right) \]

and

\[ \left( X_{n,3}, \grpc_{2n}, \frac{1}{[3]_q} \qbinom{n+5}{2}_q \qbinom{2n}{n-4}_q + \qbinom{2n}{n-3}_q \right). \]
Problem.

Find a cyclic sieving phenomena on matchings with more than $3$ crossings.

Annular non-crossing permutations

Let $\pi$ be a permutation on $[n]$ and draw the directed edges $i\to \pi(i)$ on a circle with vertices $1,\dotsc,n$ on the boundary. If all edges are non-crossing, and all cycles are oriented clockwise, we say that $\pi$ is a non-crossing permutation. Such permutations are in natural bijection with Catalan objects.

In [Kim13], a generalization of non-crossing permutations are studied. Instead of a circle, one chooses $n$ and $m$ and places exterior vertices $1,\dotsc,n$ on a circle clockwise, and interior vertices $n+1,\dotsc,n+m$ counter-clockwise on an interior circle.

A permutation is $(n,m)$-non-crossing if one can draw the edges as before in a non-crossing manner, such that every cycle is oriented clockwise. A non-crossing permutation is called connected if it contains at least one cycle with both exterior and interior vertices. The paper [Kim13] only consider non-crossing permutations with at least one connected cycle. Let $ANC(n,m)$ denote the set of connected $(n,m)$-non-crossing permutations.

Let $C$ act on the annulus by rotation, where the order of $C$ is $\gcd(m,n).$ Then

\[ \left( ANC(n,m), C, \frac{[2nm]_q}{[n+m]_q}\qbinom{2n-1}{n}_q\qbinom{2m-1}{m}_q \right) \]

is a CSP-triple.

Several refinements, such as the $q$-Narayana and $q$-Kreweras analogue are also considered in [Kim13] and proved to have CSP. The type $B$ variants, where all non-crossing partitions are symmetric with respect to rotation by half a turn, are also covered. For example,

\[ \left( ANC_B(n,m), C, \frac{[2nm]_q}{[n+m]_q}\qbinom{2n}{n}_q\qbinom{2m}{m}_q \right) \]

is a CSP-triple.

Non-crossing trees, forests and graphs

In [Klu12], we have the following CSP. Let $X_{n,k}$ be the set of non-crossing forests with $n$ vertices and $k$ components. These are forests drawn with vertices on a circle and edges inside, such that edges do not cross.

Let

\[ f_{n,k}(q) \coloneqq \frac{1}{[2n-k]_q} \qbinom{n}{k-1}_q \qbinom{3n-2k-1}{n-k}_q, \]

and let $\grpc_n$ act on $X_{n,k}$ by $2\pi/n$ rotation. Then $(X_{n,k},\grpc_n,f_{n,k}(q))$ is a CSP triple. This is also proved in [Poz11].

We can also count graphs according to edges. In [Poz11], the following $q$-analogue is considered, which enumerates the number of non-crossing graphs with $n$ vertices and $k$ edges, $Y_{n,k}$:

\[ g_{n,k}(q) \coloneqq \frac{1}{[n-1]_q} \sum_{j=0}^{n-2} q^{j(j+n-k+2)}\qbinom{n-1}{k-j}_q \qbinom{n-1}{j+1}_q \qbinom{n-2+j}{n-2}_q. \]

Then $(Y_{n,k},\grpc_n,g_{n,k}(q))$ is a CSP-triple, under rotation.

References