Problem 253

Tidying Up A

A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers $1$ to $40$ in order.

Every night, the child’s father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up the pieces at random and places them in the correct order.
As the caterpillar is built up in this way, it forms distinct segments that gradually merge together.
The number of segments starts at zero (no pieces placed), generally increases up to about eleven or twelve, then tends to drop again before finishing at a single segment (all pieces placed).

For example:

Piece Placed	Segments So Far
$12$	$1$
$4$	$2$
$29$	$3$
$6$	$4$
$34$	$5$
$5$	$4$
$35$	$4$
$\ldots$	$\ldots$

Let $M$ be the maximum number of segments encountered during a random tidy-up of the caterpillar.

For a caterpillar of ten pieces, the number of possibilities for each $M$ is

$M$	Possibilities
$1$	$512$
$2$	$250912$
$3$	$1815264$
$4$	$1418112$
$5$	$144000$

so the most likely value of $M$ is $3$ and the average value is $385643/113400 = 3.400732$, rounded to six decimal places.

The most likely value of $M$ for a forty-piece caterpillar is $11$; but what is the average value of $M$?

Give your answer rounded to six decimal places.

清理（一）

小朋友有一个“数字毛毛虫”玩具，包含有$40$片拼板，分别标有编号；如果把它们都拼起来，将会组成一条直线，且按照$1$到$40$顺序排列。

每天晚上，小朋友的爸爸都要把玩具房里撒了一地的毛毛虫拼板捡起来。他捡的时候是完全随机的，捡起来之后，再按照正确的顺序拼好。
这样一来，毛毛虫拼板将会构成分离的片段，并且不断合并直到组成完整的毛毛虫。
片段数从$0$开始（没有捡起任何一块拼板），不断上升到大约$11$或$12$，然后再次下降，直到最终只有一段（所有的拼板都组合起来了）。

例如：

捡起的拼板标号	目前为止的片段数
$12$	$1$
$4$	$2$
$29$	$3$
$6$	$4$
$34$	$5$
$5$	$4$
$35$	$4$
$\ldots$	$\ldots$

记$M$为随机清理毛毛虫拼板的过程中最大的片段数。

若毛毛虫拼板共有十片，出现不同$M$的可能情况数分别是：

$M$	可能情况
$1$	$512$
$2$	$250912$
$3$	$1815264$
$4$	$1418112$
$5$	$144000$

因此，最可能出现的$M$值为$3$，$M$的平均值为$385643/113400 = 3.400732$，保留六位小数。

若毛毛虫拼板共有四十片，最可能出现的$M$值为$11$；那么$M$的平均值是多少？将你的答案保留六位小数。