Permutations Almost Avoiding Monotone Distant Patterns

In a previous work, B\'ona 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.