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A permutation $\pi$ avoids the simsun pattern $\tau$ if $\pi$ avoids the consecutive pattern $\tau$ and the same condition applies to the restriction of $\pi$ to any interval $[k].$ Permutations avoiding the simsun pattern $321$ are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length $3$ in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length $3$ together with any set of classical patterns of length $3.$ The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees.

We define a bijection between permutations and valued Dyck paths, namely, Dyck paths whose odd vertices are labelled with an integer that does not exceed their height. This map allows us to characterize the set of permutations avoiding the pattern 132 as the preimage of the set of Dyck paths with minimal labeling. Moreover, exploiting this bijection we associate to the set of ?$n$?-permutations a polynomial that generalizes at the same time Eulerian polynomials, Motzkin numbers, super-Catalan numbers, little Schröder numbers, and other combinatorial sequences. Lastly, we determine the Hankel transform of the sequence of such polynomials. Definiramo bijekcijo med permutacijami in označenimi Dyckovimi potmi, tj. Dyckovimi potmi, pri katerih je vsako liho vozlišče označeno s celim številom, ki ne presega njegove višine. S pomočjo te preslikave lahko karakteriziramo množico permutacij, ki ne vsebujejo vzorca 132, kot prasliko množice Dyckovih poti z minimalno označitvijo. To bijekcijo izkoristimo tudi za to, da množici ?$n$?-permutacij priredimo polinom, ki hkrati posplošuje Eulerjeve polinome, Motzkinova števila, super-Catalanova števila, mala Schröderjeva števila in druga kombinatorična zaporedja. Določimo tudi Hankelovo transformiranko zaporedja takih polinomov.

We examine three permutations on Dyck words. The first one, α, is related to the Baker and Norine theorem on graphs, the second one, β, is the symmetry, and the third one is the composition of these two. The first two permutations are involutions and it is not difficult to compute the number of their fixed points, while the third one has cycles of different lengths. We show that the lengths of these cycles are odd numbers. This result allows us to give some information about the interplay between α and β, and a characterization of the fixed points of α ź β .