This is a summary of my current results for the generalization of the
factorial ratio problem to hooks of partitions. In particular it
includes a proof of Landau's theorem in that case, as well as a full
classification for height 1, based on Bober's work for factorial
ratios.
Most classical constructions of low-discrepancy point sets are based
on generalizations of the one-dimensional binary van der Corput sequence
whose im- plementation requires non-trivial bit-operations. As an
alternative we introduce the quasi-regular golden ratio sequences which
are based on the fractional part of successive integer multiples of the
golden ratio. By leveraging results from number theory we show that point
sets which evenly cover the unit square or disc can be computed by a
simple incremental permutation of a generator golden ratio sequence. We
compare ambient occlusion images generated with a Monte Carlo ray-tracer
based on random, Hammersley, blue noise and golden ratio point sets.
The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other
Weyl groups. Finally, we compute some asymptotics results when n tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.
Discrete Mathematics (311) 23-24, pp. 2690--2702 (2011).
A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of
them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir, Raza
and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset
hook-content formula is also proved.
Canadian Mathematics Bulletin 54 (2011), no.2, 255--269.
We explain an identity due to Bump, Diaconis, Tracy and Widom. They obtained this identity by computing asymptotics for the determinants of finite rank reductions of minors of Toeplitz matrices in two different ways, spanning different branches of mathematics: "Toeplitz minors", J.Combinatorial Theory Ser. A, 97 (2002), pp.252--271 uses Laguerre polynomials and representation theory of symmetric groups, while "On the limit of some Toeplitz-like determinants", SIAM J.Matrix Anal. Appl., 23 (2002), pp. 1194--1196 uses the Wiener-Hopf factorization.
We show that this identity is essentially a differentiated version of the Jacobi-Trudi identity, under the action of a differential operator on symmetric functions reminiscent of vertex operators. We deduce new results on the asymptotics of minors.
Through an extension of the Heine identity, this is also relevant for Random Matrix Theory considerations.
DMTCS Proc. Formal Power Series and Algebraic Combinatorics 2010, vol. 12.
We present a simple technique to compute moments of derivatives of unitary characteristic polynomials.
The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over
unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those
polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski.
This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted
Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain
alternative expressions involving hypergeometric functions of matrix arguments.
See also the poster, presented at FPSAC 2010. The content of the poster is
largely different from the conference paper.
We investigate the joint moments of the 2k-th power of the
characteristic polynomial of random unitary matrices with the 2h-th
power of the derivative of this same polynomial. We prove that for a
fixed h, the moments are given by rational functions of k, up to a
well-known factor that already arises when h=0.
We fully describe the denominator in those rational functions (this had
already been done by Hughes experimentally), and define the numerators
through various formulas, mostly sums over partitions.
We also use this to formulate conjectures on joint moments of the zeta
function and its derivatives, or even the same questions for the Hardy
function, if we use a ``real'' version of characteristic polynomials.
Our methods should easily be applicable to other similar problems, for
instance with higher derivatives of characteristic polynomials.
J. Combin. Theory Ser. A 114 (2007), no. 7, 1278--1292.
We compute $E_G (\prod_i \tr(g^{\lambda_i}))$, where $G=Sp(2n)$ or $SO(m) (m=2n, 2n+1)$ with Haar measure. This was first obtained by Persi Diaconis and Mehrdad Shahshahani, but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions $E_G f_n$ are affected when we introduce a Weyl character $\chi^G_\lambda$ into the integrand.
We show that the value of $E_G \chi^G_\lambda f_n / E_G f_n$ approaches a constant for large $n$. More surprisingly, the ratio we obtain only changes with $f_n$ and $\lambda$ and is independent of the Cartan type of $G$. Even in the unitary case, Daniel Bump and Persi Diaconis have obtained the same ratio. Finally, those ratios can be combined with asymptotics for $E_G f_n$ due to Kurt Johansson and provide asymptotics for $E_G \chi^G_\lambda f_n$.
A classic riddle: One hundred prisoners, a lightbulb and a sadistic warden. We analyzed various solutions, their complexity and robustness to changes in the initial problem.
The classification of finite flag-transitive linear spaces is almost complete. For the thick case, this result was announced by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl, and in the thin case (where the lines have 2 points), it amounts to the classification of $2$-transitive groups, which is generally considered to follow from the classification of finite simple groups. These two classifications actually leave an open case, which is the so-called 1-dimensional case.
In this paper, we work with two additional assumptions. These two conditions, namely (2T)_1 and RWPRI, are taken from another field of study in Incidence Geometry and allow us to obtain a complete classification, which we present at the end of this paper. In particular, for the $1$-dimensional case, we show that the only (2T)_1 flag-transitive linear spaces are AG(2,2) and AG(2,4), with $A\Gamma L(1,4)$ and $A\Gamma L (1,16)$ as respective automorphism groups.
We analyze the structure of the group of non-zero multiplicative arithmetical functions, with group law given by the usual Dirichlet product. In particular, we prove that this group is isomorphic to a complete direct product of certain subgroups of the multiplicative group of infinite upper-triangular matrices. We also show that this group is divisible.
