Abstract:
We construct a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley limit for the class of 4231-avoiding permutations is bounded below by 9.35. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Wilf-Stanley limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2.

Abstract:
The extension of pattern avoidance from ordinary permutations to those on multisets gave birth to several interesting enumerative results. We study permutations on regular multisets, i.e., multisets in which each element occurs the same number of times. For this case, we close a gap in the work of Heubach and Mansour (2006) and complete the study of permutations avoiding a pair of patterns of length three. In all studied cases, closed enumeration formulae are given and well-known sequences appear. We conclude this paper by some remarks on a generalization of the Stanley-Wilf conjecture to permutations on multisets and words.

Abstract:
We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.

Abstract:
We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also conjecture that, for any $k\ge 0$, the set of 1324-avoiding permutations with $k$ inversions contains at least as many permutations of length $n+1$ as those of length $n$. We show that if this is true then the Stanley-Wilf limit for 1324 is at most $e^{\pi\sqrt{2/3}} \simeq 13.001954$.

Abstract:
For a permutation $\pi$, let $S_{n}(\pi)$ be the number of permutations on $n$ letters avoiding $\pi$. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that $L(\pi)= \lim_{n \to \infty} S_n(\pi)^{1/n}$ exists and is finite. Backed by numerical evidence, it has been conjectured by many researchers over the years that $L(\pi)=\Theta(k^2)$ for every permutation $\pi$ on $k$ letters. We disprove this conjecture, showing that $L(\pi)=2^{k^{\Theta(1)}}$ for almost all permutations $\pi$ on $k$ letters.

Abstract:
Let S(n,k) be the Stirling number of the second kind. Wilf conjectured that the alternating sum of S(n,k) for k from 0 to n is not zero for all n>2. In this paper, we prove that Wilf conjecture is true except at most one number with the properties of weighted Motzkin number.

Abstract:
The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits sqrt[n](|S_n(tau)|) and further works suggest asymptotic ordering of layered versus monotone patterns. Yet, Bona has provided the only known up to now result of its type on ordering of permutations: |S_n(1342)|<|S_n(1234)|<|S_n(1324)| for n>6. We give a different proof of this result by ordering S_3 up to the stronger shape-Wilf-order: |S_Y(213)|<=|S_Y(123)|<=|S_Y(312)| for any Young diagram Y, derive as a consequence that |S_Y(k+2,k+1,k+3,tau)|<=|S_Y(k+1,k+2,k+3,tau)|<= |S_Y(k+3,k+1,k+2,tau)| for any tau in S_k, and find out when equalities are obtained. (In particular, for specific Y's we find out that |S_Y(123)|=|S_Y(312)| coincide with every other Fibonacci term.) This strengthens and generalizes Bona's result to arbitrary length permutations. While all length-3 permutations have been shown in numerous ways to be Wilf-equivalent, the current paper distinguishes between and orders these permutations by employing all Young diagrams. This opens up the question of whether shape-Wilf-ordering of permutations, or some generalization of it, is not the ``true'' way of approaching pattern-avoidance ordering.

Abstract:
Recently, Dokos et al. conjectured that for all $k, m\geq 1$, the patterns $ 12\ldots k(k+m+1)\ldots (k+2)(k+1) $ and $(m+1)(m+2)\ldots (k+m+1)m\ldots 21 $ are $maj$-Wilf-equivalent. In this paper, we confirm this conjecture for all $k\geq 1$ and $m=1$. In fact, we construct a descent set preserving bijection between $ 12\ldots k (k-1) $-avoiding permutations and $23\ldots k1$-avoiding permutations for all $k\geq 3$. As a corollary, our bijection enables us to settle a conjecture of Gowravaram and Jagadeesan concerning the Wilf-equivalence for permutations with given descent sets.

Abstract:
Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.