Anick-type resolutions and consecutive pattern avoidance

For permutations avoiding consecutive patterns from a given set, we present a combinatorial formula for the multiplicative inverse of the corresponding exponential generating function. The formula comes from homological algebra considerations in the same sense as the corresponding inversion formula for avoiding word patterns comes from the well known Anick’s resolution.

💡 Research Summary

The paper investigates the enumeration of permutations that avoid a prescribed set of consecutive patterns and provides a combinatorial formula for the multiplicative inverse of the associated exponential generating function (EGF). Drawing inspiration from Anick’s resolution—a homological construction originally devised for word‑pattern avoidance—the authors develop an analogous “Anick‑type” resolution tailored to the non‑commutative algebra of permutations.

The authors begin by encoding a set P of forbidden consecutive patterns as relations in a free associative algebra generated by elementary transpositions. These relations generate a two‑sided ideal, and the quotient algebra A(P) captures precisely the combinatorial constraints of pattern avoidance. By constructing a minimal free resolution of the trivial A(P)‑module, they identify a basis of chains (called Anick‑chains) that correspond to minimal overlapping configurations of the forbidden patterns. Each chain records the exact way in which patterns can overlap in a permutation, thereby reflecting the non‑commutative nature of the problem.

Using the filtered structure of the resolution, the authors derive an explicit expression for the inverse EGF: \