For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\varphi (m+1),\dots ,\varphi (m+k)$ with $m+k\le n$. Is it true that$$F(n)=(c+o(1))\log \log \log n$$for some constant $c$? Is the first pattern which fails to appear always$$\varphi (m+1)>\varphi (m+2)>\cdots >\varphi (m+k)?$$Is it true that the 'natural' ordering which mimics what happens to $\varphi (1),\dots ,\varphi (k)$ is the most likely to appear?