Pattern avoidance for set partitions \`a la Klazar

In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.Comment: 21 page

Two Vignettes On Full Rook Placements

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ru\v{s}kuc.Comment: 9 pages, 4 figure

Egge triples and unbalanced Wilf-equivalence

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 $\tau=4132$.Comment: 20 pages, 6 figures (published version