Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342-avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.

We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method completely in terms of the patterns `abc',`cab' and `abcd'. 0.

. Consider, for a permutation oe 2 S k , the number F (n; oe) of permutations in Sn which avoid oe as a subpattern. The conjecture of Stanley and Wilf is that for every oe there is a constant c(oe) ! 1 such that for all n, F (n; oe) c(oe) n . All the recent work on this problem also mentions the "stronger conjecture" that for every oe, the limit of F (n; oe) 1=n exists and is finite. In this short note we prove that the two versions of the conjecture are equivalent, with a simple argument involving subadditivity. We also discuss n-permutations, containing all oe 2 S k as subpatterns. We prove that this can be achieved with n = k 2 , we conjecture that asymptotically n (k=e) 2 is the best achievable, and we present Noga Alon's conjecture that n (k=2) 2 is the threshold for random permutations. Mathematics Subject Classification: 05A05,05A16. 1. Introduction Consider, for a permutation oe 2 S k , the set A(n; oe) of permutations 2 S n which avoid oe as a subpattern, and it...

Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123-avoiding permutations. The rewriting rule automatically gives a functional equation satis ed by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series.

A permutation   is said to ¡ be –avoiding if it does not contain any subsequence having all the same pairwise comparisons ¡ as. This paper concerns the characterization and enumeration of permutations which avoid a set ¢¤ £ of subsequences increasing both in number and in length at the same time. Let ¢ £ be the set of subsequences of the “¥§¦©¨�������¦©¨����� � form ¥ ”, being any permutation ��������������¨� � on. ¨��� � For the only subsequence in ¢�� ���� � is and ���� � the –avoiding permutations are enumerated by the Catalan numbers; ¨��� � for the subsequences in ¢� � are, ������ � and the (������������������ � –avoiding permutations are enumerated by the Schröder numbers; for each other value ¨ of greater � than the subsequences in ¢ £ ¨� � are and their length ¦©¨����� � is; the permutations avoiding ¨�� these subsequences are enumerated by a number ������ � �� � � sequence such �������������� � that �� � , being � the –th Catalan number. For ¨ each we determine the generating function of permutations avoiding the subsequences in ¢� £ , according to the length, to the number of left minima and of non-inversions.

We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as Young tableau, planar graph theory, and simplicial complexes.

We consider the problem of enumerating the permutations containing exactly k occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with “jumps”, which were first considered by Krattenthaler.

. Write p1p2 : : : pm for the permutation matrix ffi p i ;j . Let Sn (M) be the set of n \Theta n permutation matrices which do not contain the m \Theta m permutation matrix M as a submatrix. In [2] Simion and Schmidt show bijectively that jSn(123)j = jSn(213)j. In the present work this is generalised to a bijection between Sn(12 : : : tp t+1 : : : pm) and Sn (t : : : 21p t+1 : : : pm ). This result was established for t = 2 in [5] and for t = 3 in [8]. 1. Introduction and main theorem A permutation matrix of order n is a traversal of the n by n square diagram, in other words a placement of n non-attacking rooks on an n by n board. We can easily generalise this definition of a traversal to boards of general shape, and will do so below. Given a permutation matrix M of order m, a permutation matrix N of order n ? m will be said to contain the smaller matrix if there exist two subsets of the index set [n], R = fr 1 ! r 2 ! \Delta \Delta \Delta ! r m g and C = fc 1 ! c 2 ! \Delta ...

Let Bn be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let Bn(T) be the set of all signed permutations in Bn which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets Bn(T) where T ⊆ B2. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers. 1.

Define Sk n (α) to be the set of permutations of {1, 2,...,n} with exactly k fixed points which avoid the pattern α ∈ Sm. Letsk n(α) be the size of Sk n(α). We investigate S0 n(α) for all α ∈ S3 as well as show that sk n (132)=skn (213)=skn (321) and skn (231)=skn (312) for all 0 ≤ k ≤ n.