Not HoTT-specific, some of the recent type/proof-theory work of Noam
Zeilberger may pique the interest of combinatorists. For example,

  A sequent calculus for the Tamari order
  Noam Zeilberger, 2019
  https://arxiv.org/abs/1803.10080

uses a sequent calculus to describe the structure of an order relation on
trees that combinatorist have studied, and re-derive counting results on
the intervals of that order.

(Reasoning on the structure of an order relation is not too far from the
"higher" concerns of HoTT and higher category-theory.
And the idea of using adequate proof structures to prove a coherence result
has been used in many other more categorical settings.)

On Mon, Jul 1, 2019 at 1:36 PM Andrej Bauer <andrej.bauer@andrej.com> wrote:

> In a couple of days I am gonig to give a talk about HoTT in front of
> 250 combinatorialists at http://fpsac2019.fmf.uni-lj.si
>
> I have some ideas about how to explain that HoTT is relevant to a
> mathematician who studies "simple finite objects", but I'd be
> interested to hear if anyone has anything else to say. I'll gladly
> acknowledge good ideas.
>
> My current plan is to discuss, after a suitable introduction:
>
> 1. The difference between Σ and ∃ is the difference between "explicit
> construction" and "abstract proof of existence".
>
> 2. Discuss univalence and how we get "isomorphic structures are equal".
>
> 3. I will advertise Brent Yorgey's PhD thesis about combinatorial
> spieces, and probably cite some gems from it
> (
> https://homotopytypetheory.org/2016/07/20/combinatorial-species-and-finite-sets-in-hott/
> )
>
> I don't have a good feeling for what might pique a combinatorialist's
> interest. Does anyone here?
>
> With kind regards,
>
> Andrej
>
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