occurence of periodical fragments
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#### An application of Dilworth theorem.

For tails $u$ and $v$ put $u<v$ if $u \prec v$ and $u$ is to the left from $v$. Then by Dilworth theorem, $\Omega$ can be divided into $p_{n, d}$ chains such that in each chain $u \prec v$ if $u$ is to the left from $v$. Paint the initial positions of the tails into $p_{n, d}$ colors according to their occurence in chains. Fix a positive integer $p$. To each positive integer $i$ from 1 to $\left|\Omega\right|$, attach $B^p(i)$, an ordered set of $p_{n, d}$ words $\{f(i,j)\}$ constructed as follows:

*For each $j = 1, 2,\ldots, p_{n, d}$ put*

$$f(i,j)=\left\{\max \
f\leqslant i: f\ \mbox{ is painted into color }j\right\}.$$

If there is no such $f$ then the word from $B^p(i)$ at position $j$ is assumed to be equal to $\theta$, otherwise to the $p$-tail that begins from the $f(i,j)$-th letter.

Informally speaking, we observe the speed of "evolution” of tails in their chains when the sequence of positions in $W$ is considered as the time axis.

The sets $B^p(i)$ and the process at positions
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\[c:lem\] Given a sequence $S$ of length $|S|$ consisting of words of 