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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 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 ∘ β . [ABSTRACT FROM AUTHOR]

We describe a map Γ from the set of Dyck paths of given semilength to itself that is the analog of the Schützenberger involution on standard Young tableaux. Afterwards, we examine the behavior of Γ with respect to Knuth’s correspondence between pairs of standard Young tableaux of the same shape with at most two rows and Dyck paths. Finally, we exploit the previous results to describe a bijection between the set of 321 -avoiding centrosymmetric permutations of even length and the set of 321 -avoiding involutions of the same length. [ABSTRACT FROM AUTHOR]

This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space ( V, ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups ℍ, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin's complex of differential forms in ℍ. [ABSTRACT FROM AUTHOR]

We show that the distribution of the major index over the set of involutions in S n that avoid the pattern 321 is given by the q -analogue of the n -th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson–Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside a ⌊ n 2 ⌋ × ⌈ n 2 ⌉ box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions. [ABSTRACT FROM AUTHOR]

Abstract: We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions. [Copyright &y& Elsevier]

Abstract: We exploit Krattenthaler’s bijection between the set of permutations in avoiding the classical pattern and Dyck -paths to study the joint distribution over the set of a given consecutive pattern of length 3 and of descents. We utilize a involution on Dyck paths due to E. Deutsch to show that these consecutive patterns split into 3 equidistribution classes. In addition, we state equidistribution theorems concerning quadruplets of statistics relative to occurrences of consecutive patterns of length 3 and of descents in a permutation. [Copyright &y& Elsevier]

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]

Abstract: Let denote the set of partitions of the integer into exactly parts, and the subset of containing all partitions whose two largest parts coincide. We present a bijection between and for a suitable in the cases . Such bijections yield recurrence formulas for the numbers and of partitions of n into 3 and 4 parts. Furthermore, we show that the present approach can be extended to the case . [Copyright &y& Elsevier]

Abstract: We establish a combinatorial connection between the sequence counting the involutions on letters with descents and the sequence enumerating the semistandard Young tableaux on cells with symbols. This allows us to show that the sequences are not log-concave for some values of , hence answering a conjecture due to F. Brenti. [Copyright &y& Elsevier]