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We enumerate involutions avoiding any single pattern of length three according to the statistic 'number of visible pairs', namely, the number of non-adjacent columns in the bargraph representation of a given permutation that is mutually visible to each other. The proofs are combinatorial and reside on well-known bijections between pattern avoiding involutions and lattice paths. [ABSTRACT FROM AUTHOR]

We enumerate involutions according to the joint distribution of left-to-right and right-to-left maxima. From this computation we deduce the distribution of the static "number of visible pairs", namely, the number of non-adjacent columns in the bargraph representation of a given permutation that are mutually visible to each other. The corresponding generating functions are expressible in terms of formal continued fractions. We obtain the same distribution over the set of involutions avoiding either the pattern 3412 or 4321. The proofs reside on a well-known bijection between involutions and labeled Motzkin paths. [ABSTRACT FROM AUTHOR]

We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in Sn satisfying the property σ(i) + σ(n + 1 - i) = n + 1 for every i = 1 ... n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points. [ABSTRACT FROM AUTHOR]