Christian Krattenthaler

• University of Vienna

Let $f=\sum_{n=0}^\infty f_n x^n \in \overline{\mathbb Q}[[x]$ be a solution of an algebraic differential equation $Q(x,y(x), \ldots, y^{(k)}(x))=0$, where $Q$ is a multivariate polynomial with coefficients in $\overline{\mathbb Q}$. The sequence $(f_n)_{n\ge 0}$ satisfies a non-linear recurrence, whose expression involves a polynomial $M$ of degre...

The original motivation for this paper goes back to the mid-1990's, when James Propp was interested in natural situations when the number of domino tilings of a region increases if some of its unit squares are deleted. Guided in part by the intuition one gets from earlier work on parallels between the number of tilings of a region with holes and th...

We present a formula that expresses the Hankel determinants of a linear combination of length~$d+1$ of moments of orthogonal polynomials in terms of a $d\times d$ determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented cor...

The theme of this article is a "reciprocity" between bounded up-down paths and bounded alternating sequences. Roughly speaking, this ``reciprocity" manifests itself by the fact that the extension of the sequence of numbers of paths of length $n$, consisting of diagonal up- and down-steps and being confined to a strip of bounded width, to negative $...

To derive an eigenvalue problem for the associated Askey–Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey–Wilson polynomials introduced by Ismail and Rahman. The Askey–Wilson operator, applied in each variable separately, maps this function to the ordinary Askey–Wilson polynomials with...

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalising known results for Catalan numbers.

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps \(-h, \dots , -1, +1, \dots ,...

Recent methods used in lattice path combinatorics and various related branches of enumerative combinatorics are grouped together and presented in this volume, together with relevant applications. Contributions are mainly comprised of research articles though an attractive bonus is the inclusion of several captivating, expository articles on the lif...

We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying `permutation group' structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.

We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.

We discuss a partial normalisation of a finite graph of finite groups $(\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include (i) an important inequa...

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never attain altitude $0$ in-between. We first discuss the case of walks on the integers with steps $-h, \dots, -1,...

We show that the generating function $\sum_{n\ge0}M_n\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\sum _{e\ge0} ^{} {z^{4^e}}/( {1-z^{2\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$...

We show that the generating function $\sum_{n\ge0}M_n\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\sum _{e\ge0} ^{} {z^{4^e}}/( {1-z^{2\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$...

We discuss a partial normalisation of a finite graph of finite groups $(\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include (i) an important inequa...

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining cases, in which the sequence of free subgroup numbers is not ultimately periodic modulo a given prime power, the...

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining cases, in which the sequence of free subgroup numbers is not ultimately periodic modulo a given prime power, the...

This is a survey of results in the enumeration of lattice paths.

These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.

We solve the problem of effectively computing the a-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic solution.

We characterise the behaviour of (generalised) Apéry numbers modulo 9, thereby in particular establishing two conjectures in Krattenthaler and Müller [10].

(Dieudonn\'e and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the coefficients of the exponential of $S(z)$, assuming weaker conditions on the coefficients of $S(z)$ than in Dwo...

We prove that the number of oscillating tableaux of length $n$ with at most $k$ columns, starting at $\emptyset$ and ending at the one-column shape $1^m$, is equal to the number of standard Young tableaux of size~$n$ with $m$ columns of odd length, all columns of length at most $2k$. This refines a conjecture of Burrill, which it thereby establishe...

We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the associated Poincar\'e-Einstein m...

We completely characterise when the sequence of free subgroup numbers of a finitely generated virtually free group is ultimately periodic modulo a given prime power.

In this paper we present a combinatorial generalization of the fact that the number of plane partitions that fit in a $2a\times b\times b$ box is equal to the number of such plane partitions that are symmetric, times the number of such plane partitions for which the transpose is the same as the complement. We use the equivalent phrasing of this ide...

A specialisation of a transformation formula for multi-dimensional elliptic hypergeometric series is used to provide compact, non-determinantal formulae for the generating function with respect to the major index of standard Young tableaux of skew shapes of the form "staircase minus rectangle".

We characterise the modular behaviour of (generalised) Ap\'ery number modulo $9$, thereby in particular establishing two conjectures in "A method for determining the mod-$3^k$ behaviour of recursive sequences" [arXiv:1308.2856].

It is shown that the number fλfλ of free subgroups of index 6λ in the modular group PSL2(Z)PSL2(Z), when considered modulo a prime power pαpα with p⩾5p⩾5, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing PSL2(Z)PSL2(Z) as a special case), and for a one-...

We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finit...

A rhombus tiling of a hexagon is said to be centered if it contains the central lozenge. We compute the number of vertically symmetric rhombus tilings of a hexagon with side lengths $a, b, a, a, b, a$ which are centered. When $a$ is odd and $b$ is even, this shows that the probability that a random vertically symmetric rhombus tiling of a $a, b, a,...

In this paper we introduce a counterpart structure to the shamrocks studied in the paper "A dual of Macmahon's theorem on plane partitions" by M. Ciucu and C. Krattenthaler (Proc. Natl. Acad. Sci. USA, vol. 110 (2013), 4518-4523), which, just like the latter, can be included at the center of a lattice hexagon on the triangular lattice so that the r...

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\binom {12n} {3n}$...

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg's equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are brie...

It is shown that the number $f_\lambda$ of free subgroups of index $6\lambda$ in the modular group $\PSL_2(\Z)$, when considered modulo a prime power $p^\al$ with $p\ge5$, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing $\PSL_2(\Z)$ as a special case),...

To derive an eigenvalue problem for the associated Askey-Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey-Wilson polynomials introduced by Ismail and Rahman. The Askey-Wilson operator, applied in each variable separately, maps this function to the ordinary Askey-Wilson polynomials with...

We evaluate a determinant of generalized Fibonacci numbers, thus providing a common generalization of several determinant evaluation results that have previously appeared in the literature, all of them extending Cassini's identity for Fibonacci numbers.

We introduce Fuß-Catalan complexes as d-dimensional generalisations of triangulations of a convex polygon. These complexes are used to refine Catalan numbers and Fuß-Catalan numbers, by introducing colour statistics for triangulations and Fuß-Catalan complexes. Our refinements consist in showing that the number of triangulations, respectively of Fu...

We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fu\ss-Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions o...

We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2.

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-spe...

We consider a multi-parameter family of canonical coordinates and mirror maps o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and\ Parkes [Nucl. Phys. B...

We refine Catalan numbers and Fu{\ss}-Catalan numbers by introducing colour statistics for triangulations of polygons and $d$-dimensional generalisations there-of which we call Fu{\ss}-Catalan complexes. Our refinements consist in showing that the number of triangulations, respectively Fu{\ss}-Catalan complexes, with a given colour distribution of...

We show how to determine the asymptotics of a certain Selberg-type integral by means of tools available in the theory of (generalised) hypergeometric series. This provides an alternative derivation of a result of Carr\'e, Deneufch\^atel, Luque and Vivo [arXiv:1003.5996]. Comment: AmS-TeX, 6 pages

The Hilbert depth of a module M is the maximum depth that occurs among all modules with the same Hilbert function as M. In this note we compute the Hilbert depths of the powers of the irrelevant maximal ideal in a standard graded polynomial ring.

Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients

We prove that the generalised non-crossing partitions associated with well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by D. Armstrong [Mem. Am. Math. Soc. 202, No. 949, 1–159 (2009; Zbl 1191.05095)], and by D. Bessis and V. Reiner [Ann. Comb. 15, No. 2, 197–222 (2011; Zbl 1268...

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove tha...

We develop a new setting for the exponential principle in the context of multisort species, where indecomposable objects are generated intrinsically instead of being given in advance. Our approach uses the language of functors and natural transformations (composition operators), and we show that, somewhat surprisingly, a single axiom for the compos...

We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers—most no...

For the M\"obius spheres $S^{q,p}$, we give alternative elementary proofs of the recursive formulas for GJMS-operators and $Q$-curvatures due to the first author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These proofs make essential use of the theory of hypergeometric series.

Stanley decompositions of multigraded modules $M$ over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module $M$ is always at least the (classical) depth of $M$. In this paper we in...

We continue our study begun in "On the integrality of the Taylor coefficients of mirror maps" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential eq...

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that...

We continue our study begun in "On the integrality of the Taylor coefficients of mirror maps" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential eq...

We prove Obnservation 2 in arXiv:math/0402386 by Gert Almkvist and Wadim Zudilin. Comment: 2 pages, AmS-LaTeX

Bipartitional relations were introduced by Foata and Zeilberger in their characterization of relations which give rise to equidistribution of the associated inversion statistic and major index. We consider the natural partial order on bipartitional relations given by inclusion. We show that, with respect to this partial order, the bipartitional rel...

We prove that a Schur function of rectangular shape (M n ) whose variables are specialized to \(x_{1},x_{1}^{-1},\dots,x_{n},x_{n}^{-1}\) factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at −x 1,…,−x n , if M is even, while it factorizes into a product of a symplectic character and an even o...

The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler char...

We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;\lambda)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-\lambda)}, \qquad |q|>1, \quad \lambda\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($\lambda=0$) and the $q$-exponential function ($\lambda=1$). In particular, we prove t...

We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogo...

In this paper we give an analytic proof of the identity A 5,3,3(n)=B 5,3,30(n), where A 5,3,3(n) counts the number of partitions of n subject to certain restrictions on their parts, and B 5,3,30(n) counts the number of partitions of n subject to certain other restrictions on their parts, both too long to be stated in the abstract. Our proof estab...

It is shown how Andrews’ multidimensional extension of Watson’s transformation between a very-well-poised 8 ϕ 7 -series and a balanced 4 ϕ 3 -series can be used to give a straightforward proof of a conjecture of W. Zudilin and the second author [Math. Ann. 326, No. 4, 705–721 (2003; Zbl 1028.11046)] on the arithmetic behaviour of the coefficients o...

Considérons la série $$ \zeta q\left( s \right) = \sum\limits_{k = 1}^\infty {k^{s - 1} \tfrac{{q^k }} {{1 - q^k }},}

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the...

We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\mathbf...

Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A nece...

Just as the authors anxiously waited for the identities of the Top Seven Songs of the week years ago, readers of this article must now be brimming with unbridled excitement to learn the identities of the Top Ten Most Fascinating Formulas from Ra-manujan's Lost Notebook. The choices for the Top Ten Formulas were made by the authors. However, motivat...

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindström–Gessel–Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from Krattenthaler (1990). This approach leads also naturally to extensions and generalisations.

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This approach leads also naturally to extensions and ge...

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that...

A regular ring of Josephson junctions, connected in parallel, is studied analytically and numerically. We show that, depending on the strength of the r-well cosine potential the energy landscape of the Hamiltonian can have of the order of rN/N locally stable minima separated by large barriers specified by unstable saddle points. The counting proble...

A regular ring of Josephson junctions, connected in parallel, is studied analytically and numerically. We show that, depending on the strength of the r-well cosine potential the energy landscape of the Hamiltonian can have of the order of rN/N locally stable minima separated by large barriers specified by unstable saddle points. The counting proble...

Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\si_1\si_2 cdots\si_d$ of a Coxeter element $c$ of $W$, such that $\si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$,...

We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over Q spanned by 1, zeta(m), zeta(m + 2),..., zeta(m + 2h), where m and h are integers such that m >...

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 c...

Introduction et plan de l'article Arriere plan Les resultats principaux Consequences diophantiennes du Theoreme $1$ Le principe des demonstrations des Theoremes $1$ a $6$ Deux identites entre une somme simple et une somme multiple Quelques explications Des identites hypergeometrico-harmoniques Corollaires au Theoreme $8$ Corollaires au Theoreme $9$...

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...

This paper deals with 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 group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to i...

The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum_{u,w\in P} ^{}\mu(u,w) x^{\rk u}y^{\rk w}$, where $\mu(.,.)$ is the M\"obius function of $P$. We compute the $M$-triangle of Armstrong's poset of $m$-divisible non-crossing partitions for the root systems of type $E_7$ and $E_8$. For the other types except $D_n$...

Nous étudions la nature arithmétique de $q$-analogues des valeurs $\zeta(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)=\sum_{k=1}^{\infty}q^k\sum_{d\mid k}d^{s-1}$, $s=1,2,\dots$, où $q$ est un nombre complexe, $|q|<1$ (ces fonctions sont intimement liées au monde automorphe). Le théorème principal de cet artic...

We study the arithmetic properties of q-analogues of values zeta(s) of the Riemann zeta function, in particular of the values of the functions zeta(q)(s) = Sigma(k=1 q)(infinity)(k) Sigma(d vertical bar k) d(s-1), s=1, 2,..., where q is a number with vertical bar q vertical bar < 1 (these functions are also connected with the automorphic world). Th...

In this article we present a review of the work on Group theory of ordinary and basic Hypergeometric transformations and the application of the transformations of ordinary hypergeometric series in Quantum Theory of Angular Momentum.

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 us...

We prove a conjecture by Kreiman and Lakshmibai on a combinatorial description of multiplicities of points on Schubert varieties in Graßmannian in terms of certain sets of reflections in the corresponding Weyl group. The proof is accomplished by relating these sets of reflections to the author's previous combinatorial interpretation of these multip...

We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results generalize previous determinant evaluations due to the first and third author [SIAM J. Matrix Anal. Appl. 23 (2...

The $F$-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the $F$-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the "$M=F$ Conjecture" of Armstrong which predicts a surprising relation between the $F$-triangle and the M\"obius fun...

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 determ...

This is a complement to my previous article "Advanced Determinant Calculus" (S\'eminaire 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 a...

We show that the number of fully packed loop configurations corresponding to a matching with $m$ nested arches is polynomial in $m$ if $m$ is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11 (2004), Article #R13].

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,...

We consider a ∞agged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing difie...

The M-triangle of a ranked locally finite poset P is the generating function ∑ u,w∈P μ(u,w)x rku y rkw , where μ(·,·) is the Möbius function of P. We compute the M-triangle of D. D. Armstrong’s poset of m-divisible non-crossing partitions for the root systems of types E 7 and E 8 . For the other types except D n this had been accomplished in [C. F....

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.