What is this construction using iterated face maps of semisimplicial sets? Let $X$ be a semisimplicial set (face maps but no degeneracy maps). Fix a positive integer $k$. Let $Y_n$ be $X_{(n+1)k}$ and then define $\partial^Y_i:Y_n\to Y_{n-1}$ by 
$$\partial^Y_i = (\partial^X_{ik})^k.$$
These maps give $Y$ the structure of a semisimplicial set.
There are many variations possible on this theme, changing the indexing numbers around. I'm not only interested in this particular example, but in any similar semisimplicial set created using iterated face maps of a semisimplicial set. 
Has this construction or similar ones been studied? Does it have a name? Any obvious properties? Is there a variation with a straightforward or canonical extension to simplicial sets? Are there interesting variations where $\partial_0$ and $\partial_n$ are perturbed or different in some way? If there is something interesting that can only be said in a linear category that would also be of interest.
 A: This is just an extended comment.
The simplicial category $\mathbf{\Delta}$ can be identified with the category of nonempty, finite, totally ordered sets.  Given $I,J\in\mathbf{\Delta}$ we can equip $I\times J$ with the lexicographic order to get a new object of $\mathbf{\Delta}$.  If $f\colon I\to I'$ and $g\colon J\to J'$ are order-preserving, then $f\times g\colon I\times J\to I'\times J'$ will be order-preserving if $f$ and $g$ are injective, but not otherwise.  We thus get a monoidal structure on the semisimplicial category $\mathbf{\Delta}'$, whose morphisms are the strictly increasing maps.  A semisimplicial set is a functor $X\colon(\mathbf{\Delta}')^{\text{op}}\to\text{Sets}$, and your construction just composes it with $(-)\times [n]$.  That seems like a more conceptual way to ask the question, but it does not answer it.  
A: I don't know if this is exactly what you're looking for, but there is the edgewise subdivision of simplicial sets. It seems to be very similar to the construction you're introducing. As far as I know it's a variant of a construction introduced by Segal, and it was introduced by Bökstedt–Hsiang–Madsen (and it was later used by McCarthy in the context of Hochschild homology, and probably also other people). What's next basically comes from [BHM] below.
The construction is as follows. Fix a positive integer $r$. There's an endofunctor $$\DeclareMathOperator{\sd}{sd}\sd_r : \Delta \to \Delta$$ given by $\sd_r[m-1] = [mr-1]$ (where $[n] = \{0 < \dots < n \}$, and on nondecreasing maps is given by $\sd_r(f)(am+b) = an + f(b)$ (where $f : [m-1] \to [n-1]$, and the Euclidean division $0 \le b < m$). Then the edgewise subdivision of a simplicial set $X : \Delta^{op} \to \mathsf{Set}$ is given by $\sd_r X = X \circ \sd_r$. More precisely, $\sd_r X_n = X_{r(n+1) - 1}$, and the face and degeneracy maps are given by:
$$\bar{d}_i = d_i \circ d_{i+(n+1)} \circ \dots \circ d_{i+(r-1)(n+1)}, \\
\bar{s}_j = s_{j+(r-1)(n+2)} \circ \dots \circ s_{j + (n+2)} \circ s_j.$$
There's a canonical homeomorphism $|\sd_r X| \to |X|$, they have very a nice picture in [BHM]:

I think it's practical to write these down as matrices to see what's happening, see [McC] below. If $X$ is actually a cyclic set it's even possible to make everything compatible with the $S^1$-action. McCarthy uses these subdivisions to define explicit Adams operations on Hochschild homology (pinch the circle using the subdivision and fold). I know Ginot also uses them to define Adams operations on higher Hochschild homology. It's probably been used in other contexts too.


*

*[BHM] M. Bökstedt, W. C. Hsiang, and I. Madsen. “The cyclotomic trace and algebraic K-theory of spaces”. In: Invent. Math. 111.3 (1993), pp. 465–539. ISSN: 0020-9910. DOI: 10.1007/BF01231296. MR1202133.

*[Seg] G. Segal. Configuration-spaces and iterated loop-spaces. Invent. Math. 21 (1973), 213–221. DOI: 10.1007/BF01390197 MR331377

*[McC] R. McCarthy. “On operations for Hochschild homology”. In: Comm. Algebra 21.8 (1993), pp. 2947–2965. ISSN: 0092-7872. DOI: 10.1080/00927879308824712. MR1222750.

*[Gin] G. Ginot. “Higher order Hochschild cohomology”. In: C. R. Math. Acad. Sci. Paris 346.1-2 (2008), pp. 5–10. ISSN: 1631-073X. DOI: 10.1016/j.crma.2007.11.010. MR2383113.
