On Packing Densities of Set Partitions

We study packing densities for set partitions, which is a generalization of packing words. We use results from the literature about packing densities for permutations and words to provide packing densities for set partitions. These results give us most of the packing densities for partitions of the set $\{1,2,3\}$. In the final section we determine the packing density of the set partition $\{\{1,3\},\{2\}\}$.Comment: 12 pages, to appear in the Permutation Patterns edition of the Australasian Journal of Combinatoric

The largest singletons of set partitions

Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let $A_{n,k}$ denote the number of partitions of $\{1,2,\dots, n+1\}$ with the largest singleton $\{k+1\}$ for $0\leq k\leq n$. In this paper, several explicit formulas for $A_{n,k}$, involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods, many combinatorial identities involving $A_{n,k}$ and Bell numbers are presented by operator methods, and congruence properties of $A_{n,k}$ are also investigated. It will been showed that the sequences $(A_{n+k,k})_{n\geq 0}$ and $(A_{n+k,k})_{k\geq 0}$ (mod $p$) are periodic for any prime $p$, and contain a string of $p-1$ consecutive zeroes. Moreover their minimum periods are conjectured to be $N_p=\frac{p^p-1}{p-1}$ for any prime $p$.Comment: 14page

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 {\sigma,\tau}, where \sigma\ is a pattern of size three with at least two distinct symbols and \tau\ is an arbitrary pattern of size k that avoids \sigma. 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,\tau}, where \tau\ 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 \tau\ 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 {\sigma,\tau}, where \sigma\ and \tau\ have size four.Comment: 37 pages. Corrected a typ

Crossings and nestings in colored set partitions

Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call \emph{$r$-colored set partitions}). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that $r$-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in \NN^r, generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further revised, additional section adde