  For a word $\pi$ and integer $i$, we define $L^i(\pi)$ to be the length of
the longest subsequence of the form $i(i+1)\cdots j$, and we let
$L(\pi):=\max_i L^i(\pi)$. In this paper we estimate the expected values of
$L^1(\pi)$ and $L(\pi)$ when $\pi$ is chosen uniformly at random from all words
which use each of the first $n$ integers exactly $m$ times. We show that
$\mathbb{E}[L^1(\pi)]\sim m$ if $n$ is sufficiently larger in terms of $m$ as
$m$ tends towards infinity, confirming a conjecture of Diaconis, Graham, He,
and Spiro. We also show that $\mathbb{E}[L(\pi)]$ is asymptotic to the inverse
gamma function $\Gamma^{-1}(n)$ if $n$ is sufficiently large in terms of $m$ as
$m$ tends towards infinity.
