Permutation Patterns, Continued Fractions, and a Group Determined by an Ordered Set
We present six articles:
In the first and second article we give the first few results on generalized pattern avoidance, focusing on patterns of type (1,2) or (2,1). There are twelve such patterns, and they fall into three classes with respect to being equidistributed. We use 1-23, 1-32, and 2-13 as representatives for these classes. We prove that
|Sn(1-23)| = |Sn(1-32)|= Bn and |Sn(2-13)| = Cn,
where Bn is the nth Bell number and Cn is the nth Catalan number. A complete solution for the number of permutations avoiding any pair of patterns of type (1,2) or (2,1) is also given.
In the third article we present an ordinary generating function for the number of permutations containing one occurrence of 1-23 (or 1-32). We also give the distribution of 2-13 in the form of a continued fraction, and explicit formulas for the number of permutations containing r occurrences of 2-13 when r=1, 2, or 3.
In the fourth article the notion of a σ-segmented permutation is introduced: A permutation π is σ-segmented if every occurrence of σ in π is a contiguous subword in π. A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We show that 132-segmented permutations of length n with k occurrences of 132 are in one-to-one correspondence with bicoloured Dyck paths of length 2n-4k with k red up-steps. Similarly, we show that 123-segmented permutations of length n with k occurrences of 123 are in one-to-one correspondence with bicoloured Dyck paths of length 2n-4k with k red up-steps, each of height less than 2. We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths.
Continued fractions and patterns are the two main topics of the fifth article. Let ek(π) be the number of increasing subsequences of length k+1 in π. We prove that any Stieltjes continued fractions with monic monomial numerators is the generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly infinite) linear combination of the eks. Moreover, there is an invertible linear transformation that translates between linear combinations of eks and the corresponding continued fractions.
In the sixth article we study a permutation group determined by an ordered set. Let P be a finite ordered set, and let J(P) be the distributive lattice of order ideals of P. The covering relations of J(P) are naturally associated with elements of P; in this way, each element of P defines an involution on the set J(P). Let Γ(P) be the permutation group generated by these involutions. We show that if P is connected then Γ(P) is either the alternating or the symmetric group.