Generalizing the results in our previous paper, we consider pseudo-involutions in the Riordan group where the generating function g for the first column of a Riordan array satisfies a functional equation of certain types involving a polynomial. For those types of equations, we find the pseudo-involutory companion of g. We also develop a general method for finding B-functions of Riordan pseudo-involutions in the cases we consider, and show that these B-functions involve Chebyshev polynomials. We apply our method for several families of Riordan arrays, obtaining new results and deriving known results more efficiently.

We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of patterns {12345, 12354} and {45123, 45213} that extends the list of 10 related conjectures proved in the literature in a series of papers. To achieve our goals, we prove generalized versions of shape-Wilf-equivalence results of Backelin, West, and Xin and use a particular result on shape-Wilf-equivalence of monotone patterns. We also derive general results on shape-Wilf-equivalence of certain classes of partially ordered patterns and use their specialization (also appearing in a paper by Bloom and Elizalde) as an essential piece in proving the conjecture. Our results allow us to show (shape)Wilf-equivalence of large classes of sets of patterns, including 11 out of 12 classes found by Bean et al. in relation to the conjecture.

In this note, we prove some and conjecture other results regarding the distribution of descent top and descent bottom sets on some pattern-avoiding permutations. In particular, for 3-letter patterns, we show bijectively that the set of descent tops and the set of descent bottoms are jointly equidistributed on the avoiders of 231 and 312. We also conjecture similar equidistributions for 4-letter patterns, in particular, that the set of descent tops and the set of descent bottoms are jointly equidistributed on the avoiders of 3142, 3241, 4132.

We show that the distribution of the number of peaks at height $i$ modulo $k$ in $k$-Dyck paths of a given length is independent of $i\in[0,k-1]$ and is the reversal of the distribution of the total number of peaks. Moreover, these statistics, together with the number of double descents, are jointly equidistributed with any of their permutations. We also generalize this result to generalized Motzkin paths and generalized ballot paths.

We construct an injection from the set of permutations of length n that contain exactly one copy of the decreasing pattern of length k to the set of permutations of length n+2 that avoid that pattern. We then prove that the generating function counting the former is not rational, and in the case when k is even and k ≥ 4, it is not even algebraic. We extend our injection and our nonrationality result to a larger class of patterns.

We consider pseudo-involutions in the Riordan group where the generating function $g$ for the first column of a Riordan array satisfies a palindromic or near-palindromic functional equation. For those types of equations, we find, for very little work, the pseudo-involutory companion of $g$ and have a pseudo-involution in a $k$-Bell subgroup. There are only slight differences in the ordinary and exponential cases. In many cases, we also develop a general method for finding B-functions of Riordan pseudo-involutions in $k$-Bell subgroups, and show that these B-functions involve Chebyshev polynomials. We apply our method for many families of Riordan arrays, both new and already known. We also have some duality and reciprocity results. Since many of the examples we discuss have combinatorial significance, we conclude with a few remarks on the general framework for a combinatorial interpretation of some of the generating function results we obtain.

In this paper, we enumerate Dumont permutations of the fourth kind avoiding or containing certain permutations of length 4. We also conjecture a Wilf-equivalence of two 4-letter patterns on Dumont permutations of the first kind.

We study the distribution and the popularity of some patterns in k-ary faro words, i.e. words over the alphabet {1,2,…,k} obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the x-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Finally, we consider some interesting subclasses of faro words that are permutations, involutions, derangements, or subexcedent words.

Four of these classes were separately shown previously to be enumerated by this sequence, while the five others are new. In chronological order of publication, see [1, Lemma 10] for T2, [4, Example 6.4] for T9, [9, Theorems 3.4 and 3.5] for T4, and [3, Section 3.3] for T8. The same sequence also enumerates unimodal inversion sequences [8]. An inversion sequence (subexcedant sequence, a reversal of a Lehmer code) is a sequence e = e1e2 . . . en (n > 0) such that ei ∈ [0, i− 1] for all i. We generalize these findings in two different ways.

Egge conjectured that permutations avoiding the set of patterns $\{2143,3142,\tau\}$, where $\tau\in\{246135,254613,263514,524361,546132\}$, are enumerated by the large Schr\"oder numbers. Consequently, $\{2143,3142,\tau\}$ with $\tau$ as above is Wilf-equivalent to the set of patterns $\{2413,3142\}$. Burstein and Pantone proved the case of $\tau=246135$. We prove the remaining four cases. As a byproduct of our proof, we also enumerate the case $\tau=4132$.

We prove that the set of patterns {1324, 3416725} is Wilf-equivalent to the pattern 1234 and that the set of patterns {2143, 3142, 246135} is Wilf-equivalent to the set of patterns {2413, 3142}. These are the first known unbalanced Wilf-equivalences for classical patterns between finite sets of patterns.

We give a direct combinatorial proof of the equidistribution of two pairs of permutation statistics, (des, aid) and (lec, inv), which have been previously shown to have the same joint distribution as (exc, maj), the major index and the number of excedances of a permutation. Moreover, the triple (pix, lec, inv) was shown to have the same distribution as (fix, exc, maj), where fix is the number of fixed points of a permutation. We define a new statistic aix so that our bijection maps (pix, lec, inv) to (aix, des, aid). We also find an Eulerian partner das for a Mahonian statistic mix defined using mesh patterns, so that (das, mix) is equidistributed with (des, inv).

One of the central problems in botanical epidemiology is whether disease spreads within crops in a regular pattern or follows a random process. In this study, we consider a row of n plants in which m are infected. We then develop a rigorous mathematical approach to investigate the total number of ways to obtain k isolated individuals among m infected plants. We give a recurrence relation in three parameters that describes the problem, then we find a closed-form solution, and give two different approaches to tackle the proof. Finally, we find interesting formulae for the expectation and variance of the random variable that represents the number of infected and isolated plants.

We study the total number of occurrences of several vincular (also called generalized) patterns and other statistics, such as the major index and the Denert statistic, on permutations avoiding a pattern of length 3, extending results of Bona (2010, 2012) and Homberger (2012). In particular, for 2-3-1-avoiding permutations, we find the total number of occurrences of any vincular pattern of length 3. In some cases the answer is given by simple expressions involving binomial coefficients. The tools we use are bijections with Dyck paths, generating functions, and block decompositions of permutations.

We give a recursive formula for the Mobius function of an interval [sigma, pi] in the poset of permutations ordered by pattern containment in the case where pi is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2, ... , k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Mobius function in the case where sigma and pi are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. We also show that the Mobius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Mobius function of an interval [sigma, pi] of separable permutations is bounded by the number of occurrences of sigma as a pattern in pi. Another consequence is that for any separable permutation pi the Mobius function of (1, pi) is either 0, 1 or -1. (C) 2011 Elsevier Inc. All rights reserved.

We give a short proof for J. Noonan's result on the number of permutations containing pattern 321 exactly once.

Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this question has been tangentially considered previously, this is the first article focusing exclusively on it. We find the packing density for various special sets of permutations and study permutation and pattern co-occurrence.

Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors have shown that Patience Sorting provides an algorithmic description for permutations avoiding the barred (generalized) permutation pattern $3-\bar{1}-42$. Motivated by this and a recently formulated geometric form for Patience Sorting in terms of certain intersecting lattice paths, we study the related themes of restricted input and avoidance of similar barred permutation patterns. One such result is to characterize those permutations for which Patience Sorting is an invertible algorithm as the set of permutations simultaneously avoiding the barred patterns $3-\bar{1}-42$ and $3-\bar{1}-24$. We then enumerate this avoidance set, which involves convolved Fibonacci numbers.