In settling the main conjecture of Cyclic action on Kreweras walks, see https://arxiv.org/abs/2005.14031, a rather interesting object popped up.

Recall from

*Kuperberg, Greg*, **Spiders for
rank 2 Lie algebras**,
Commun. Math. Phys. 180, No. 1, 109-151
(1996). ZBL0870.17005.

that an $\mathfrak{sl}_3$-web $W$ is a planar graph, embedded in a
disk, with *boundary vertices* labeled $1,2,\ldots,m$ arranged on
the rim of the disk in counterclockwise order, and any number of
(unlabeled) *internal vertices* such that

- $W$ is
*trivalent*: all the boundary vertices have degree one, while all the internal vertices have degree three; - $W$ is
*bipartite*: the vertices (both boundary and internal) are colored white and black, with edges only between oppositely colored vertices.

An $\mathfrak{sl}_3$-web is *non-elliptic* or *irreducible* if all of
its internal faces have at least $6$ sides. Non-elliptic
$\mathfrak{sl}_3$-webs with all boundary vertices white are in
bijection with rectangular standard Young tableaux having $3$ rows:

*Khovanov, Mikhail; Kuperberg, Greg*, **Web bases for $\text{sl}(3)$ are not dual canonical**, Pac. J. Math. 188, No. 1, 129-153 (1999). ZBL0929.17012.

*Tymoczko, Julianna*, **A simple bijection between standard $3\times n$ tableaux and irreducible webs for $\mathfrak{sl}_{3}$**, J. Algebr. Comb. 35, No. 4, 611-632 (2012). ZBL1242.05277.

We call a non-elliptic $\mathfrak{sl}_3$-web *Kreweras*, if it has no faces with a multiple of $4$ sides.

One reason why these objects are interesting is the fact that $$ \sum_W 2^{\kappa(W)}=\frac{4^n}{(n+1)(2n+1)}\binom{3n}{n}, $$ where the sum is over all Kreweras webs $W$ with $3n$ boundary vertices and $\kappa(W)$ connected components. This is the number of Kreweras words, see http://oeis.org/A006335.

The sequence of numbers of connected Kreweras webs, with $3,6,9,12,15,18,\dots$ boundary vertices, is $1,2,12,104,1088,12768,\dots$, unknown to the OEIS. However, Lagrange inversion shows that this equals $2^{n+1}\cdot\frac{(4n+1)!}{(3n+2)!(n+1)!}$. Note that this is $2^n$ times http://oeis.org/A000260.

The sequence of numbers of connected Kreweras webs up to rotation, with $3,6,9,12,15,18,\dots$ boundary vertices, is $1,1,2,10,76,714,\dots$, also unknown to the OEIS.

Question:Can the condition, that a Kreweras web only has faces whose number of sides is not divisible by $4$, be given a representation theoretic meaning?