# Slice

For $n\in{\Bbb Z}_+$ and $(a,b,c)$ in the triangular grid

$\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},$

the slice $\Gamma_{a,b,c}$ is the set of all strings in $[3]^n$ with $a$ occurences of $1$, $b$ occurences of $2$, and $c$ occurences of $3$. Thus, for instance,

$\Gamma_{1,2,0} = \{ 011, 101, 101 \}.$

The cube $[3]^n$ has $\frac{(n+1)(n+2)}{2}$ slices, and each slice $\Gamma_{a,b,c}$ has cardinality $\frac{n!}{a!b!c!}$.

The slice $\Gamma_{a,b,c}$ is incident to $2^a+2^b+2^c-3$ combinatorial lines.

The equal-slices measure on $[3]^n$ gives each slice a total measure of $1$.

The $k=2$ analogue of a slice, sometimes called a layer, plays a role in the proof of Sperner's theorem.

A combinatorial line must touch three slices $\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}$ corresponding to an equilateral triangle in $\Delta_n$. Conversely, one might hope that any sufficiently "rich" subsets of three slices $\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}$ in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of $\Gamma_{a+r,b,c}$ of strings whose first digit is $1$, and the subsets of $\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}$ of strings whose first digit is $2$.