Avoidance of Partitions of a Three-element Set

We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let $\op_{n,k}(p)$ be the number of ordered set partitions of $\{1,2,\ldots,n\}$ into $k$ blocks that avoid a permutation pattern $p$. We establish an explicit identity between the number $\op_{n,k}(p)$ and the numbers of words avoiding the inverse of $p$. This identity allows us to easily translate results on pattern-avoiding words obtaine...

A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of interest to this work. We first characterize the restricted growth functions of several avoidance classes based on partitions of size four, enumerate these avoidance classes, and consider the distribution of the Wachs and White statistics acro...

An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(\sigma)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $\sigma$ and obtained the formula for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $2$. Later, Chen, Dai and...

Sergey Kitaev has shown that the exponential generating function for permutations avoiding the generalized pattern $\sigma$-$k$, where $\sigma$ is a pattern without dashes and $k$ is one greater than the biggest element in $\sigma$, is determined by the exponential generating function for permutations avoiding $\sigma$. We show that this also holds for permutations avoiding all the generalized patterns $\sigma_1$-$k_1$, $...$, $\sigma_n$-$k_n$, where $\sigma_1$, $...$, $\si...

Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".

The notion of containment and avoidance provides a natural partial ordering on set partitions. Work of Sagan and of Goyt has led to enumerative results in avoidance classes of set partitions, which were refined by Dahlberg et al. through the use of combinatorial statistics. We continue this work by computing the distribution of the dimension index (a statistic arising from the supercharacter theory of finite groups) across certain avoidance classes of partitions. In doing so ...

The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well as a pair of two patterns. Several counting sequences, namely Catalan numbers, powers of two, Fibonacci numbers and Motzkin numbers arise. We also consider other combinatorial statistics, namely runs and inversions, and establish some bijec...

Enumeration of pattern-avoiding objects is an active area of study with connections to such disparate regions of mathematics as Schubert varieties and stack-sortable sequences. Recent research in this area has brought attention to colored permutations and colored set partitions. A colored partition of a set $S$ is a partition of $S$ with each element receiving a color from the set $[k]=\{1,2,...,k\}$. Let $\Pi_n\wr C_k$ be the set of partitions of $[n]$ with colors from $[k]$...

An ordered partition of $[n]=\{1, 2, \ldots, n\}$ is a partition whose blocks are endowed with a linear order. Let $\mathcal{OP}_{n,k}$ be set of ordered partitions of $[n]$ with $k$ blocks and $\mathcal{OP}_{n,k}(\sigma)$ be set of ordered partitions in $\mathcal{OP}_{n,k}$ that avoid a pattern $\sigma$. Recently, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of $[n]$ with 3 blocks and the number of ordered partitions of $[n]$ with ...

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-...