\big) \big\}.$$* ]{}

[ *If $n=4$ and $N=6$, then $\varphi_k\colon \mathbb{L}_{6}^{5} \to 2^{L_{4}^6}\times 2^{L_{4}^6}$, for $k =1,2$. Furthermore, if $w=(1,2,2,3,3,4)$, then $$\begin{aligned}
\varphi_1(w) & = \big\{\big((1,2,4,5),(1,3,4,5) \big),\big( (1,2,4,6),(1,3,4,6) \big),\big( (1,2,5,6),(1,3,5,6) \big) \big\} \\
\varphi_2(w) & = \big\{\big((1,2,3,5),(1,2,4,5) \big),\big( (1,2,3,6),(1,2,4,6) \big)  \big\}\end{aligned}$$* ]{}

The action of all the $\varphi_k$ restricted to ordered multi-indices $\mathcal{L}_{2(n-1)}^{N-1}$ is relatively easy to describe. Define $\rho = \rho_{N,n} \colon \mathcal{L}_{2(n-1)}^{N-1} \to 2^{L_{n}^N}$ by $$\label{rhoo}
\rho(x) = \left\{ y \in L_{n}^N  : y_1 \leq x_1,\ x_{2(k-1)} <  y_k \leq x_{2(k-1)+1},\ x_{2(n-1)} < y_n,\  k=2,\ldots, n-1 \right\}$$

Let $n=4$, $N=6$. Then $\rho\colon {{\mathcal{L}}}_6^5 \to L_4^6$,

$$\rho(112244) = \left\{y\in L_4^6\colon y_1\leq 1,\ 1<y_2\leq 2,\ 2<y_3\leq 4,\ 4<y_4\right\} = \left\{ 1235, 1236, 1245, 1246 \right\}.$$

\[lem:rho\] Let $x\in \mathcal{L}_{2(n-1)}^{N-1}$. If $(y,y')\in \varphi_k(x)$, then $y'=y$ and $$\{(y,y) : y\in \rho(x)\} = \bigcap_{k=1}^{n-2} \varphi_k(x).$$

If $x\in \mathcal{L}_{2(n-1)}^{N-1}$, then $x_{2k}<x_{2k+1}$ and hence $$\varphi_k(x) = \{(y,y) : y_k\leq x_{2k-1},\ x_{2k}<y_{k+1} \leq x_{2k+1},\ x_{2(k+1)}< y_{k+2}\}.$$ Therefore, $$\bigcap_{k=1}^{n-2} \varphi_k(x) = \{(y,y) : y_1\leq x_{1},\ x_{2(k-1)} <  y_k \leq x_{2(k-1)+1},\ x_{2(n-1)} < y_n,\  k=2,\ldots, n-1\}$$ which by definition of $\rho(x)$ is equal to the set $\{(y,y): y\in\rho(x)\}$.

For $n < N$, set $$\mathcal{P}_{2n}^{N}:={\left\{ { (x_1,\dots,x_{2n} ) \in \{1,\dots,N\}^{2n} : x_{2k-1}=x_{2k} < x_{2k+1}=x_{2k+2},\ k= 1,\ldots, n-1}\right\}} \subset \mathcal{L}_{2n}^{N}.$$ 