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Christian Krattenthaler

Publications and source records attributed to Christian Krattenthaler.

At least 109 records · Page 6Linked to original sources
arXiv2006

Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes

We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and nestings of matchings and set partitions in the larger context of the enumeration of fillings of Ferrers shape on which one imposes restrictions on their increasing and decreasing chains. While Chen et al. work with Robinson-Schensted-like insertion/deletion algorithms, we use the growth diagram construction of Fomin to obtain our results. We extend the results by Chen et al., which, in the language of fillings, are results about $0$-$1$-fillings, to arbitrary fillings. Finally, we point out that, very likely, these results are part of a bigger picture which also includes recent results of Jonsson on $0$-$1$-fillings of stack polyominoes, and of results of Backelin, West and Xin and of Bousquet-Mélou and Steingr\'ımsson on the enumeration of permutations and involutions with restricted patterns. In particular, we show that our growth diagram bijections do in fact provide alternative proofs of the results by Backelin, West and Xin and by Bousquet-Mélou and Steingr\'ımsson.

math.CO↗
arXiv2005

2-adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman--Friedland--Loewy on the parity of the degrees of projective varieties of $n\times n$ complex symmetric matrices of rank at most $k$. We also characterize the parity of the degrees of projective varieties of $n\times n$ complex skew symmetric matrices of rank at most $2p$. We give recursive relations which determine the parity of the degrees of projective varieties of $m\times n$ complex matrices of rank at most $k$. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of $n\times n$ skew symmetric real matrices and of $m\times n$ real matrices containing a nonzero matrix of rank at most $k$. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.

math.NT↗
arXiv2005

Advanced Determinant Calculus: A Complement

This is a complement to my previous article "Advanced Determinant Calculus" (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441-456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems.

math.CO↗
arXiv2005

Watermelon configurations with wall interaction: exact and asymptotic results

We perform an exact and asymptotic analysis of the model of $n$ vicious walkers interacting with a wall via contact potentials, a model introduced by Brak, Essam and Owczarek. More specifically, we study the partition function of watermelon configurations which start on the wall, but may end at arbitrary height, and their mean number of contacts with the wall. We improve and extend the earlier (partially non-rigorous) results by Brak, Essam and Owczarek, providing new exact results, and more precise and more general asymptotic results, in particular full asymptotic expansions for the partition function and the mean number of contacts. Furthermore, we relate this circle of problems to earlier results in the combinatorial and statistical literature.

math.CO↗
arXiv2005

How can we escape Thomae's relations?

In 1879, Thomae discussed the relations between two generic hypergeometric $_3F_2$-series with argument 1. It is well-known since then that there are 120 such relations (including the trivial ones which come from permutations of the parameters of the hypergeometric series). More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over $\mathbf Q$ between two convergent $_3F_2$-series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae's relations. We show that Sato's counter-examples can be naturally embedded into two families of infinitely many $_3F_2$-relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain $_3F_2$-transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and $_3F_2$-series.

math.CA↗
arXiv2004

Trace identities from identities for determinants

We present new identities for determinants of matrices $(A_{i,j})$ with entries $A_{i,j}$ equal to $a_{i,j}$ or $a_{i,0}a_{0,j}-a_{i,j}$, where the $a_{i,j}$'s are indeterminates. We show that these identities are behind trace identities for $SL(2,\Bbb C)$ matrices found earlier by Magnus in his study of trace algebras.

math.AC↗
arXiv2004

Approximants de Padé des $q$-polylogarithmes

We solve a Padé-type problem of approximating three specific functions simultaneously by $q$-analogues of polylogarithms, respectively by powers of the logarithm. This problem is intimately related to recent results of the authors and Wadim Zudilin ["Séries hypergéométriques basiques, fonction $q$-zêta et séries d'Eisenstein", J. Inst. Math. Jussieu (to appear)] on the dimension of the vector space generated by $q$-analogues of values of the Riemann zeta function at integers. We also show that our result can be considered as a $q$-analogue of a result of Stéphane Fischler and the second author [J. Math. Pures Appl. {\bf 82} (2003), 1369-1394].

math.NT↗
arXiv2004

Recent progress in the study of representations of integers as sums of squares

In this article, we collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. We start with a sketch of Milne's proof of two of these conjectures. We also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of $4s^2$, respectively $4s(s+1)$, triangular numbers. This approach is inspired by Zagier's proof of the Kac--Wakimoto formulas via modular forms. We end the survey with recent conjectures of the first author and Chua.

math.NT↗
arXiv2004

Non-commutative Hopf algebra of formal diffeomorphisms

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the multiplication, while the second group is the set of formal diffeomorphisms with the composition. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization in quantum field theory, and to the renormalization functor given by the double tensor algebra on a bi-algebra.

math.QA↗
arXiv2003

An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series

We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev which were proposed as tools in the study of the arithmetic behaviour of values of the Riemann zeta function at integers. Our proof is based on limiting cases of a basic hypergeometric identity of Andrews.

math.CA↗
arXiv2003

Descending plane partitions and rhombus tilings of a hexagon with triangular hole

It is shown that the descending plane partitions of Andrews can be geometrically realized as cyclically symmetric rhombus tilings of a certain hexagon where an equilateral triangle of side length 2 has been removed from its centre. Thus, the lattice structure for descending plane partitions, as introduced by Mills, Robbins and Rumsey, allows for an elegant visualization.

math.CO↗
arXiv2003

On a hypergeometric identity of Gelfand, Graev and Retakh

A hypergeometric identity equating a triple sum to a single sum, originally found by Gelfand, Graev and Retakh [Russian Math. Surveys 47 (1992), 1-88] by using systems of differential equations, is given hypergeometric proofs. As a bonus, several $q$-analogues can be derived.

math.CA↗
arXiv2003

Equations in finite semigroups: Explicit enumeration and asymptotics of solution numbers

We study the number of solutions of the general semigroup equation in one variable, $X^\al=X^\be$, as well as of the system of equations $X^2=X, Y^2=Y, XY=YX$ in $H\wr T_n$, the wreath product of an arbitrary finite group $H$ with the full transformation semigroup $T_n$ on $n$ letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concerning the first mentioned problem generalize earlier results by Harris and Schoenfeld (J. Combin. Theory Ser. A 3 (1967), 122-135) on the number of idempotents in $T_n$, and a partial result of Dress and the second author (Adv. in Math. 129 (1997), 188-221). Among the asymptotic tools employed are Hayman's method for the estimation of coefficients of analytic functions and the Poisson summation formula.

math.CO↗
arXiv2002

Some new formulas for $π$

We show how to find series expansions for $π$ of the form $π=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$, $a=(-4)^k$, for any $k$, and give explicit examples for such expansions for small values of $m$, $p$ and $a$.

math.NT↗
arXiv2002

Proof of a summation formula for an $\tilde A_n$ basic hypergeometric series conjectured by Warnaar

A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1ψ_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.

math.CA↗
arXiv2002

Vicious walkers, friendly walkers and Young tableaux III: Between two walls

We derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls, as well as for the analogous problem for $\infty$-friendly walkers. Our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series.

cond-mat.stat-mech↗