Mark Shattuck

A general two-term recurrence and its solution

We find a general explicit formula for all sequences satisfying a two-term recurrence of a certain kind. This generalizes familiar formulas for the Stirling and Lah numbers.

On multiple pattern avoiding set partitions

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T. Our goal is to classify the equivalence classes among two-element pattern sets of several general types. First, we focus on pairs of patterns {@s,@t}, where @s is a pattern of size three with at least two distinct symbols and @t is an arbitrary pattern of size k that avoids @s. We show that pattern-pairs of this type determine a small number of equivalence classes; in particular, the classes have on average exponential size in k. We provide a (sub-exponential) upper bound for the number of equivalence classes, and provide an explicit formula for the generating function of all such avoidance classes, showing that in all cases this generating function is rational. Next, we study partitions avoiding a pair of patterns of the form (1212,@t), where @t is an arbitrary pattern. Note that partitions avoiding 1212 are exactly the non-crossing partitions. We provide several general equivalence criteria for pattern-pairs of this type, and show that these criteria account for all the equivalences observed when @t has size at most six. In the last part of the paper, we perform a full classification of the equivalence classes of all the pairs {@s,@t}, where @s and @t have size four.

Pattern avoidance in inversion sequences

Abstract A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question.

Restricted partitions and q-Pell numbers

In this paper, we provide new combinatorial interpretations for the Pell numbers pn in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by pn. By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of pn. Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.

Counting humps and peaks in generalized Motzkin paths

Let us call a lattice path in ZxZ from (0,0) to (n,0) using U=(1,k), D=(1,-1), and H=(a,0) steps and never going below the x-axis, a (k,a)-path (of order n). A super (k,a)-path is a (k,a)-path which is permitted to go below the x-axis. We relate the total number of humps in all of the (k,a)-paths of order n to the number of super (k,a)-paths, where a hump is defined to be a sequence of steps of the form UH^iD, i>=0. This generalizes recent results concerning the cases when k=1 and a=1 or a=~. A similar relation may be given involving peaks (consecutive steps of the form UD).

A Recurrence Related to the Bell Numbers

Abstract. In this paper, we solve a general, four-parameter recurrence by both algebraic and combinatorial methods. The Bell numbers and some closely related sequences are solutions to the recurrence corresponding to particular choices of the parameters.

Nine classes of permutations enumerated by binomial transform of Fine’s sequence

Abstract The problem of avoidance of a single permutation pattern or of a pair of patterns of length four has been well studied. Less is known concerning the avoidance of three 4 -letter patterns. In this paper, we determine up to symmetry all triples of 4 -letter patterns such that the number of members of S n avoiding any one of them is given by the binomial transform of Fine’s sequence (see A033321 in OEIS). We make use of both algebraic and combinatorial proofs in order to establish our results. In a couple of cases, we introduce certain auxiliary statistics on S n which give rise to a system of functional equations that can be solved using the kernel method. In another case, a direct bijection is defined between members of the avoidance class in question and the set of skew Dyck paths.

Counting permutations by the number of successions within cycles

In this note, we consider the problem of counting (cycle) successions, i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on S n for the statistics recording the number of cycles and adjacencies of the form j , j + d where d 0 which extends earlier results.

Combinatorial proofs of some Stirling number formulas

Abstract In this note, we provide bijective proofs of some recent identities involving Stirling numbers of the second kind, as previously requested. Our arguments also yield generalizations in terms of a well known q-Stirling number.

Counting subwords in flattened partitions of sets

In this paper, we consider the problem of avoidance of subword patterns in flattened partitions, which extends recent work of Callan. We determine in all cases explicit formulas and/or generating functions for the number of set partitions of size n which avoid a single subword pattern of length three. The asymptotic behavior of the resulting counting sequences turns out to depend quite heavily on the specific pattern. For the cases of 312 and 213, we make use of the kernel method to determine the generating function which counts the members of the avoidance class. Furthermore, in the cases of 132, 231, and 123, we also find formulas concerning the distribution on the set of partitions for the statistics recording the number of occurrences of the pattern in question and some related bijective proofs are given. Finally, in each of these cases, it is shown that the number of occurrences of the pattern asymptotically follows a normal distribution.