of main theorem}
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In this section we prove the main theorem. In Section \[sec:the partition\] we describe the De Bruijn-Tengbergen-Kruyswijk partition. In Section \[sec:f\] we define the mapping ${\psi}$ and prove basic facts about it. In Section \[sec:finally the proof\] we give the proof for Theorem \[thm:main\], omitting some technical details that can be found in Section \[sec:missing claims\].

The De Bruijn-Tengbergen-Kruyswijk Partition {#sec:the partition}
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Let $n$ be an even integer. A *symmetric chain* in ${\{0,1\}^{n}}$ is a sequence of vertices $C = \{ c_k, c_{k+1}, \dots, c_{n-k} \}$ such that $|c_i| = i$ for $i=k,k+1,\ldots,n-k$, and ${{\sf distance}}(c_i,c_{i+1}) = 1$ for $i=k,k+1,\ldots,n-k-1$. We say that a symmetric chain is *monotone* if it satisfies the following property: if $c_{i-1}$ and $c_i$ differ in the $j^{\text{th}}$ coordinate, and $c_i$ and $c_{i+1}$ differ in the $(j')^{\text{th}}$ coordinate, then $j < j'$.

We shall represent a monotone symmetric chain as follows. Let $y \in {\{0,1,{\sqcup}\}^{n}}$ be such that $m = |\{ i \colon y_i = {\sqcup}\}|$ satisfies $m \equiv n \pmod 2$, and let $k = (n-m)/2$. The monotone symmetric chain $C_y = \{ c_k, c_{k+1}, \ldots, c_{n-k} \}$ is specified by $y$ as follows. For $i = k,k+1,\ldots,n-k$, the string 