Permutations Almost Avoiding Monotone Distant Patterns
Nicholas Van Nimwegen
TL;DR
This work studies almost distant monotone patterns in permutation avoidance by formalizing the $M_{k,j,i}$ family and connecting the avoidance of a single-square distant pattern to a set of $k+1$ classical patterns. It establishes a large Wilf-Equivalence class among the patterns $M_{k,i+1,i+1}$ through a constructive injection and reverse-complement symmetry, yielding $L(M_{k,i+1,i+1})=(k-1)^2+1$ for those families and providing bounds $L(D_i)\in[(k-1)^2,(k-1)^2+1]$ for $D_i=1\dots i\square (i+1)\dots k$, improving previous bounds. The paper also derives unbalanced Wilf-Equivalences such as an explicit bijection between Av$_n(M_{k,2,2})$ and Av$_n(M_{k,2,1})$, and discusses limitations where $M_{k,j,j}$ is not always equivalent to $M_{k,j,j\pm1}$, introducing finite obstruction sets as a tool. It closes with open questions on exact growth rates across broader index configurations, including non-monotone underlying patterns, and highlights several conjectural relations that could guide future work on pattern-avoidance growth.
Abstract
In a previous work, Bóna and Pantone studied permutations that avoided all but one pattern of length $k$ that began with a length $k-1$ increasing subsequence. We draw the connection between that idea and distant patterns, first discussed heavily in a work by Dimitrov, and study similar permutation classes, where the index not part of the increasing subsequence can vary. We find a large class of Wilf-Equivalences between $k+1$ classes of $k$ patterns of length $k+1$, and outline several classes of unbalanced Wilf-Equivalences related to the first class. Using this, we are also find new bounds on the exponential growth rate on all monotone distant patterns with a single gap constraint.