论文标题
连续数字在排列中的聚类，避免了三个图案的图案，或避免了有限数量的简单模式
Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns
论文作者
论文摘要
对于$η\在s_3 $中，令$ s_n^{\ text {av}（η）} $表示避免$ s_n $中的一组排列，以避免模式$η$，然后让$ e_n^{\ text {\ text {av} {av}（η）} $表示预期的尊重，以指示预期$ s_n^{\ text {av}（η）} $。对于$ n \ ge k \ ge2 $和$τ\在s_k^{\ text {av}（η）} $中，令$ n_n^{（k）}（σ）$表示$ k $ copendece in Copectection in $ k $中的$ k $的出现数量出现在$ k $中的$ k $ in $ k $} $} $} av pextsect，in s_n^^\ in av { $ n_n^{（k;τ）}（σ）$表示$ k $编号的外观顺序是模式$τ$的此类事件的数量。 我们获得了$ e_n^{\ text {av}（η）} n_n^{（k;τ）} $和$ e_n^{\ text {av}（η）} n_n^{（k）n_n^{（k）} $的明确公式。 s_k^{\ text {av}（η）} $。然后，这些确切的公式将渐近公式作为$ n \ to \ infty $带有$ k $固定的$，而$ n \ to \ to \ infty $，$ k = k_n \ to \ to \ infty $。我们还获得了类似的结果 $ s_n^{\ text {av}（η_1，\ cdots，η_r）} $，由$ s_n $的子集组成的子集由避免模式避免$ \ {τ_i\} _ {i = 1} _ = 1}^r $，其中$τ_i\ in S__________i}的case {m____i}， $ \ {τ_i\} _ {i = 1}^n $都是简单的排列。其中的一个特殊情况是一组可分离排列，对应于$ r = 2 $，$τ_1= 2413，τ_2= 3142 $。
For $η\in S_3$, let $S_n^{\text{av}(η)}$ denote the set of permutations in $S_n$ that avoid the pattern $η$, and let $E_n^{\text{av}(η)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\text{av}(η)}$. For $n\ge k\ge2$ and $τ\in S_k^{\text{av}(η)}$, let $N_n^{(k)}(σ)$ denote the number of occurrences of $k$ consecutive numbers appearing in $k$ consecutive positions in $σ\in S_n^{\text{av}(η)}$, and let $N_n^{(k;τ)}(σ)$ denote the number of such occurrences for which the order of the appearance of the $k$ numbers is the pattern $τ$. We obtain explicit formulas for $E_n^{\text{av}(η)}N_n^{(k;τ)}$ and $E_n^{\text{av}(η)}N_n^{(k)}$, for all $2\le k\le n$, all $η\in S_3$ and all $τ\in S_k^{\text{av}(η)}$. These exact formulas then yield asymptotic formulas as $n\to\infty$ with $k$ fixed, and as $n\to\infty$ with $k=k_n\to\infty$. We also obtain analogous results for $S_n^{\text{av}(η_1,\cdots,η_r)}$, the subset of $S_n$ consisting of permutations avoiding the patterns $\{τ_i\}_{i=1}^r$, where $τ_i\in S_{m_i}$, in the case that $\{τ_i\}_{i=1}^n$ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to $r=2$, $τ_1=2413,τ_2=3142$.
