Let us start with two results of technical character.

#### Proof

From Lemma 2.5 and the Stirling formula [14] we get that there exists an increasing sequence of natural numbers {*n*(*u*) : *u* ∈ ℕ} such that *S*_{n(u)} ∖ *S*_{n(u)}(*u*, *u*) ≠ ∅ for every *u* ∈ ℕ. Let
${\stackrel{~}{p}}_{u}\in ({S}_{n(u)}\mathrm{\setminus}{S}_{n(u)}(u,u))$ for every *u* ∈ ℕ.

Let us fix *k* ∈ ℕ, *k* > 1, and let {*I*_{u}} be the increasing sequence of intervals (which means that *i* < *j* for any *i* ∈ *I*_{u}, *j* ∈ *I*_{u + 1}, *u* ∈ ℕ) forming a partition of ℕ and satisfying condition card *I*_{u} = *k* ⋅ *n*(*u*), for every *u* ∈ ℕ. Let us put
$${J}_{u}=[(k-2)n(u)+min{I}_{u},(k-1)n(u)+min{I}_{u}]$$
for *u* ∈ ℕ.

The expected permutation *p* is defined in the following way
$$\begin{array}{}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}p(i\phantom{\rule{thinmathspace}{0ex}}n(u)+j+min{I}_{u})=(i+1)n(u)+j+min{I}_{u},\\ p((k-2)n(u)+j+min{I}_{u})=(k-1)n(u)+{\stackrel{~}{p}}_{u}(j+1)-1+min{I}_{u},\\ p((k-1)n(u)+j+min{I}_{u})={\stackrel{~}{p}}_{u}^{-1}(j+1)-1+min{I}_{u},\end{array}$$
for *i* = 0, 1, …, *k* – 3 and *j* = 0, 1,…, *n*(*u*) − 1.

The verification of condition *p*^{k} = id_{ℕ} is trivial, whereas the fact that *p*^{i} ∈ (𝔓 ∖ *G*) for every *i* = 1, 2, …,*k* − 1 is a consequence of the fact that composition of the following three mappings: the increasing mapping of interval [1, *n*(*u*)] onto interval *J*_{u}, the restriction of *p* to interval *J*_{u} and the increasing mapping of interval [(*k* − 1)*n*(*u*) + min *I*_{u}, max *I*_{u}] onto interval [1, *n*(*u*)], is equal to
${\stackrel{~}{p}}_{u}$
for every *u* ∈ ℕ.

Now, let us suppose that there exist permutations *p*_{i} ∈ 𝔓, *i* = 1, 2, …, *n*, such that
$$p={p}_{n}\phantom{\rule{thinmathspace}{0ex}}{p}_{n-1}\dots {p}_{1}$$
and
$$\text{either}\phantom{\rule{1em}{0ex}}{p}_{i}\in \mathfrak{C}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{p}_{i}^{-1}\in \mathfrak{C},$$
for every *i* = 1, 2, …,*n*.

Let *J* be an interval of natural numbers. We denote by *p*_{J} and *p*_{i,J} the restrictions of permutations *p* and *p*_{i}, respectively, to sets *J* and
$${p}_{i-1}{p}_{i-2}\dots {p}_{1}\phantom{\rule{thinmathspace}{0ex}}{p}_{0}(J),$$
respectively, for each *i* = 1, 2, …,*n*, where *p*_{0} := id_{ℕ}. Then the following decomposition holds
$$\begin{array}{}{q}_{n,J}{p}_{j}{q}_{1,J}^{-1}=({q}_{n},Jpn,J{q}_{n-1,J}^{-1})({q}_{n}-1,Jpn-1,J{q}_{n-2,J}^{-1})\dots ({q}_{2,J}{p}_{2,J}{q}_{1,J}^{-1})({q}_{1,J}{p}_{1,J}{q}_{0,J}^{-1}),\end{array}$$(3)
where *q*_{i,J}, 1 ≤ *i* ≤ *n*, is the increasing mapping of set *p*_{i}*p*_{i − 1} … *p*_{1}(*J*) onto interval [1, card (*J*)] and *q*_{0,J} := id_{ℕ}.

Let us notice that for each *i* = 1, 2, …,*n* we have *q*_{i,J} *p*_{i,J}
$\begin{array}{}{q}_{i-1,J}^{-1}\end{array}$ ∈ *S*_{card(J)} and this is the *b*–connected permutation for *b* = *c*(*p*_{i}) or *b* = *c*(
$\begin{array}{}{p}_{i}^{-1}\end{array}$), depending on the fact whether *p*_{i} ∈ ℭ or
$\begin{array}{}{p}_{i}^{-1}\end{array}$ ∈ ℭ. Justification is needed only for the last property.

So let us suppose that there exists 1 ≤ *i* ≤ *n* such that *p*_{i} ∈ ℭ and simultaneously ξ = *q*_{i,J}*p*_{i,J}
$\begin{array}{}{q}_{i-1,J}^{-1}\end{array}$ is not the *b*–connected permutation. Then there exists subinterval Λ of interval [1, card (*J*)] such that the set ξ(Λ) is a union of at least (*b* + 1) **MSI**. Let 𝔍 be the family of **m**utually separated intervals of natural numbers forming the decomposition of set ξ(Λ) . Auxiliarly we take
$$\begin{array}{}{\displaystyle \alpha =min{q}_{i-1,J}^{-1}(\mathrm{\Lambda})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\beta =max{q}_{i-1,J}^{-1}(\mathrm{\Lambda}).}\end{array}$$

Then from the fact that
$\begin{array}{}{q}_{i-1,J}^{-1}\end{array}$
is the increasing mapping we get the following relations
$$\begin{array}{}{\displaystyle \alpha ={q}_{i-1,J}^{-1}(min\mathrm{\Lambda})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\beta ={q}_{i-1,J}^{-1}(max\mathrm{\Lambda})}\end{array}$$(4)
and
$$\begin{array}{}[\alpha ,\beta ]\cap {q}_{i-1,J}^{-1}([1,\text{card}(J)])={q}_{i-1,J}^{-1}(\mathrm{\Lambda}).\end{array}$$(5)

Let us notice one more property. For any Θ, Ξ ∈ 𝔍, Θ ≠ Ξ,
$$\begin{array}{}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{q}_{i,J}^{-1}(\mathrm{\Theta})<{q}_{i,J}^{-1}(\mathrm{\Xi})\text{\hspace{0.17em}then the following relation holds}\\ (max{q}_{i,J}^{-1}(\mathrm{\Theta}),min{q}_{i,J}^{-1}(\mathrm{\Xi}))\cap {p}_{i}{q}_{i,J}^{-1}([1,\text{card}(J)])\ne \mathrm{\varnothing}.\end{array}$$(6)

From the relations (4), (5) and (6) we are able to deduce that the set *p*_{i}([*α*, *β*]), as well as the set ξ(Λ), is the union of at least (*b* + 1)**MSI**, since the numbers from the set *p*_{i}([*α*, *β*] ∖
$\begin{array}{}{q}_{i-1,J}^{-1}\end{array}$(Λ)) “do not fill” completely all the“holes” between sets
$\begin{array}{}{q}_{i,J}^{-1}\end{array}$(Θ) and
$\begin{array}{}{q}_{i,J}^{-1}\end{array}$(Ξ) for any Θ, Ξ ∈ 𝔍, Θ ≠ Ξ. Thereby we get the contradiction with definition of number *b* and, in consequence, with the assumption that ξ is not the *b*–connected permutation.

Thus, if we assume that
$$\begin{array}{}v>n+{\displaystyle \underset{1\le i\le n}{max}\{c({q}_{i}):{q}_{i}={p}_{i}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}if\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}{p}_{i}\in \mathfrak{C},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{q}_{i}={p}_{i}^{-1}\phantom{\rule{thinmathspace}{0ex}}\text{otherwise}\}}\end{array}$$
then *p*_{u} = *q*_{n,J} *pJ*
$\begin{array}{}{q}_{1,J}^{-1}\end{array}$ ∉ *S*_{n(u)}(*u*, *u*) for *J* = *J*_{u} and for every *u* ∈ ℕ, *u* > *v*, which contradicts the assumption.□

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.