Using homological duality in consecutive pattern avoidance

Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.

💡 Research Summary

The paper addresses the problem of consecutive pattern avoidance in permutations by introducing a homological‑duality framework that yields a concrete sufficient condition for two collections of patterns to share the same exponential generating function (EGF) for the number of avoiding permutations. The condition consists of a length‑preserving bijection between the two collections that is identical on the unordered sets of overlaps of any pair of patterns. In other words, if every pair of patterns in the first collection overlaps in exactly the same way (as a set of positions and lengths) as the corresponding pair in the second collection under the bijection, then the two collections generate identical EGFs.

To prove this, the author develops a direct algorithm for computing the inverse generating function (F(x)=1/E(x)). The algorithm translates the overlap structure into a graph, builds a chain complex whose boundary maps encode the ways patterns can be concatenated, and then computes homology to obtain the coefficients of (F(x)) recursively. Because the overlap sets are preserved by the bijection, the chain complexes for the two collections are isomorphic, which forces their homology—and consequently their inverse generating functions—to coincide. This homological argument replaces more ad‑hoc combinatorial manipulations and provides a systematic, algebraic route to the result.

The paper further identifies a broad class of pattern families for which the algorithm runs in polynomial time. In these families the overlap graph has bounded degree, so the associated chain complex has low dimension. As a consequence the inverse generating function satisfies a linear ordinary differential equation with polynomial coefficients: \