Combinatorial properties of the numbers of tableaux of bounded height

We introduce an infinite family of lower triangular matrices ¡(s), where °s n;i counts the standard Young tableaux on n cells and with at most s columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number ¿s(n) of tableaux on n cells and with at most s columns

Two permutation classes enumerated by the central binomial coefficients

We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a bijection that allows us to determine some notable features of these permutations, such as the distribution of the statistics "number of ascents", "number of left-to-right maxima", "first element", and "position of the maximum element"Comment: 26 pages, 3 figure

The Eulerian numbers on restricted centrosymmetric permutations

We study the descent distribution over the set of centrosymmetric permutations that avoid the pattern of length 3. Our main tool in the most puzzling case, namely, $\tau=123$ and $n$ even, is a bijection that associates a Dyck prefix of length $2n$ to every centrosymmetric permutation in $S_{2n}$ that avoids 123.Comment: 17 pages, 6 figure

Descent sets on 321-avoiding involutions and hook decompositions of partitions

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 an n/2 by 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

Restricted involutions and Motzkin paths

AbstractWe 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

Post-Newtonian evolution of massive black hole triplets in galactic nuclei.

Massive black-hole binaries (MBHBs) are thought to be the main source of gravitational waves (GWs) in the low-frequency domain surveyed by Pulsar Timing Array (PTA) campaigns and space-borne missions (LISA). However, many MBHBs in realistic astrophysical environments may not reach separations small enough to allow significant GW emission. This final-parsec problem can be eased by the appearance of a third massive black hole (MBH) whose action can force, under certain conditions, the former MBHB to merge. A detailed assessment of the process requires a general relativistic treatment and the inclusion of environmental effects. In this thesis, I developed a three-body Post-Newtonian (PN) code framed in a galactic potential, including PN terms up to 2.5PN order, orbital hardening and dynamical friction. With the code I performed a vast exploration of the parameter space represented by MBH triplets. I found that a non-negligible fraction (_ 30%) of the otherwise stalled binaries can actually merge because of the perturbation of the third body. By combining these results with a cosmological semi-analytical code of galaxy formation, I drew robust predictions about the nHz stochastic background of GWs, which in the most pessimistic scenario is suppressed by only a factor of two with respect to the standard models in which MBHBs can efficiently merge. This result has important consequences for PTA and implies that a detection of the GW background could be claimed in the near future

and

Abstract. We study the descent distribution over the set of centrosymmetric permutations that avoid a pattern of length 3. In the most puzzling case, namely, τ = 123 and n even, our main tool is a bijection that associates a Dyck pre x of length 2n to every centrosymmetric permutation in S2n that avoids 123. Mathematics Subject Classi cation(2000). 05A05, 05A15, 05A19