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?

  • $\begingroup$ is there a typo in the question: do you mean "... condition that a $\mathfrak{sl}_3$-web only has faces ... not divisible by 4 [and hence is a Kreweras web]" rather than "... condition that a Kreweras web ..." ? $\endgroup$ – Noam Zeilberger Jun 3 at 14:54
  • $\begingroup$ (In any case, your preprint is interesting.) $\endgroup$ – Noam Zeilberger Jun 3 at 14:56
  • 1
    $\begingroup$ Well, it's a language problem. But I mean precisely what you are saying. $\endgroup$ – Martin Rubey Jun 3 at 18:06
  • $\begingroup$ @NoamZeilberger: we updated to include some explicit discussion of OEIS sequences you might be interested in- see what is now Section 6.2 of arxiv.org/abs/2005.14031. $\endgroup$ – Sam Hopkins Jun 11 at 11:35

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