A reciprocity approach to computing generating functions for permutations with no pattern matches

Jones, Miles Eli ; Remmel, Jeffrey

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011).

Résumé

In this paper, we develop a new method to compute generating functions of the form $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ where $τ$ is a permutation that starts with $1, \mathcal{NM_n}(τ )$ is the set of permutations in the symmetric group $S_n$ with no $τ$ -matches, and for any permutation $σ ∈S_n$, $LRMin(σ )$ is the number of left-to-right minima of $σ$ and $des(σ )$ is the number of descents of $σ$ . Our method does not compute $NM_τ (t,x,y)$ directly, but assumes that $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ where $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ so that $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. We then use the so-called homomorphism method and the combinatorial interpretation of $NM_τ (t,1,y)$ to develop recursions for the coefficient of $U_τ (t,y)$.

Détail

DOI : 10.46298/dmtcs.2933