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Impulse parameter and a new equivalence between 123- and 132-avoiding permutations

Impulse parameter and a new equivalence between 123- and 132-avoiding permutations

In this paper, we enumerate permutations π=π_{1}π_{2}⋯π_{n}according to the number of indices *i* such that π_{i−}_{1}<*ℓ*≤π_{i}, where π_{0}=0 and *ℓ *is a fixed positive integer. We term such an index i an *ℓ*-impulse since it marks an occurrence where the bargraph representation of π rises above (or to the same level as) the horizontal line *y*=*ℓ*. We find an explicit formula for the distribution as well as a formula for the total number of *ℓ*-impulses in all permutations of [*n*]. Comparable distributions are also found for the τ-avoiding permutations of [*n*], where τ is a pattern of length three. Two markedly different distributions emerge, one for {213,312} and another for the remaining patterns in *S*_{3}. In particular, we obtain a new equidistribution result between 123- and 132-avoiding permutations. To prove our results, we make use of multiple arrays and systems of functional equations, employing the kernel method to solve the system in the case τ=123. We also provide a combinatorial proof of the aforementioned equidistribution result, which actually applies to a more general class of multi-set permutations.