Avoiding patterns with three distinct letters in Canon permutations

Source abstract

We study classical pattern avoidance in canon permutations from two complementary points of view. First, for arbitrary alphabet size, we resolve three conjectures of Laudone by giving canonical structural decompositions. Second, we fix the alphabet size at three and determine $c_3^k(Λ)$ for every $Λ\subseteq\mathfrak{S}_3$. A relabeling reduction and a six-class lattice-word theorem show that each fixed-underlying-permutation component has cardinality $$0,\qquad 1,\qquad C_k,\qquad B_k,\qquad Q_k,\qquad\text{or}\qquad T_k$$ where $C_k$ denote the $k$-th Catalan number, $T_k=f^{(k,k,k)}$ is the unrestricted rectangular-tableau number, $B_k$ counts three-row lattice words in which every $1$ precedes every $3$, and $Q_k$ counts $321$-avoiding three-row lattice words. The sixty-four forbidden sets yield twelve enumerative formulas. We classify the forbidden sets whose descent polynomials are uniformly palindromic or $γ$-positive and give Boolean toggle actions of an elementary abelian $2$-group for all restricted $γ$-positive classes. Finally, we derive a closed formula and a cubic algebraic equation for the generating function of $B_k$, an exact finite multisum for $Q_k$, and pose several questions and conjectures.