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arXiv 1801.08262math.CO
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连续模式的Wilf等价关系

Wilf equivalence relations for consecutive patterns

Tim Dwyer, Sergi Elizalde

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摘要：

如果对于每个$n$，避免$τ$作为连续模式的$S_n$中的置换的数目（即，在相邻位置）与避开$τ$的那些的数目相同。此外，如果对于每个$n$和$k$，包含作为连续模式的$π$的$k$出现的$S_n$中的排列数与$τ$相同，则$π$和$τ$是强c-Wilf等价的。在本文中，我们引入了第三个，更严格的等价关系，定义$π$和$τ$是超强c-Wilf等价的，如果上述条件成立的任何一组规定的位置的$k$发生。我们表明，当限制到非重叠排列，这三个等价关系相吻合。我们还给出了两个置换是强c-Wilf等价的一个必要条件。具体来说，我们表明，如果$S_m$中的$π,τ$是强c-Wilf等价的，那么$|π_m-π_1|=|τ_m-τ_1|$。在非重叠排列$π$和$τ$的特殊情况下，这证明了第二作者猜想的一个较弱版本，该猜想指出$π$和$τ$是c-Wilf等价的当且仅当$π_1=τ_1$和$π_m=τ_m$，直到平凡对称。最后，我们加强了Nakamura和Khoroshkin-Shapiro最近的一个结果，给出了强c-Wilf等价的充分条件。

英文摘要：

Two permutations $π$ and $τ$ are c-Wilf equivalent if, for each $n$, the number of permutations in $S_n$ avoiding $π$ as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding $τ$. In addition, $π$ and $τ$ are strongly c-Wilf equivalent if, for each $n$ and $k$, the number of permutations in $S_n$ containing $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In this paper we introduce a third, more restrictive equivalence relation, defining $π$ and $τ$ to be super-strongly c-Wilf equivalent if the above condition holds for any set of prescribed positions for the $k$ occurrences. We show that, when restricted to non-overlapping permutations, these three equivalence relations coincide. We also give a necessary condition for two permutations to be strongly c-Wilf equivalent. Specifically, we show that if $π,τ$ in $S_m$ are strongly c-Wilf equivalent, then $|π_m-π_1|=|τ_m-τ_1|$. In the special case of non-overlapping permutations $π$ and $τ$, this proves a weaker version of a conjecture of the second author stating that $π$ and $τ$ are c-Wilf equivalent if and only if $π_1=τ_1$ and $π_m=τ_m$, up to trivial symmetries. Finally, we strengthen a recent result of Nakamura and Khoroshkin-Shapiro giving sufficient conditions for strong c-Wilf equivalence.

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