- Combinatorics of Young tableaux
- Statistics on subsets of the symmetric group
- Pattern avoiding permutations
- Eulerian polynomials
My research interest was devoted to enumeration problems about
Young tableaux with bounded number of columns. We introduced an
infinite family of matrices, containing the numbers of Young
tableaux of given shapes. The recursive properties of such matrices
yield recurrences and explicit formulas for the numbers of Young
tableaux with 2 and 3 columns.
We then obtained recurrences and explicit formulas for the numbers
of integer partitions in 3 and 4 parts. By generating function
methods, we constructed bijections between sets of partitions in a
bounded number of parts.
The distribution of the descent statistic (also known as Eulerian
distribution) on the set of involutions of the symmetric group S_n
has been deeply investigated in recent years by several authors,
who studied the combinatorial properties of the generating
polynomial I_n(x) of the descent distribution. For example, Gessel
and Reutenauer proved that the coefficients of these polyomials are
symmetric (see Counting permutations with a given cycle structure
and descent set, J. Combin. Theory Ser. A, 13 (1972), 135-139),
while Guo and Zeng proved the unimodality of In(x) for every
integer n (see The Eulerian distribution on involutions is indeed
unimodal, J. Combin. Theory Ser. A 113 (2006), no. 6, 1061-1071).
Moreover, this distribution was conjectured to be log-concave by F.
Brenti (see the paper by Dukes Permutation statistics on
involutions, European J. Combin. 28 Issue 1 (2007), 186-198).
My research activity enters this vein from a new point of view. The
key tool is a formula that expresses the number i_{n,k} of
involutions on n objects with k descents in terms of the well known
sequence a_{n,s} counting semistandard tableaux with n cells on s
symbols. This approach gives a further and simpler proof of the
symmetry of the coefficients of the polynomial I_n(x) and allows to
confute the conjecture given by Brenti concerning the log-concavity
of the descent distribution on involutions.
The same approach can be applied to the solution of other
enumerative problems concerning some subsets of involutions, e.g.,
the set of involutions with no fixed points and the set J_2n of
centrosymmetric involutions on 2n objects, namely, involutions
corresponding to standard Young tableaux which are fixed under the
Schützenberger map. More precisely, the present tools yield an
explicit formula for the number j_{2n,k} of centrosymmetric
involutions on 2n objects with k descents.
Moreover, the same approach can be used also in the study of signed
eulerian numbers on involutions, and allows to exhibit an explicit
formula for such numbers.
In the last years I undertook the study of pattern avoiding
permutations. This theme was introduced by D.E. Knuth in his
fundamental volume "The Art of Computer Programming" and, in recent
years, it obtained the interest of many authors, so that there are
hundreds of papers on this subject in the current literature.
This study yielded four publications and two preprints.
