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Christian Krattenthaler

Publications and source records attributed to Christian Krattenthaler.

At least 37 records · Page 2Linked to original sources
arXiv2023

Boundary dents, the arctic circle and the arctic ellipse

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 the 2D Coulomb energy of the corresponding system of electric charges, we consider Aztec diamond regions with unit square defects along two adjacent sides. We show that for large regions, if these defects are at fixed distances from a corner, the ratio between the number of domino tilings of the Aztec diamond with defects and the number of tilings of the entire Aztec diamond approaches a Delannoy number. When the locations of the defects are not fixed but instead approach given points on the boundary of the scaling limit $S$ (a square) of the Aztec diamonds, we prove that, provided the line segment connecting these points is outside the circle inscribed in $S$, this ratio has the same asymptotics as the Delannoy number corresponding to the locations of the defects; if the segment crosses the circle, the asymptotics is radically different. We use this to deduce (under the assumption that an arctic curve exists) that the arctic curve for domino tilings of Aztec diamonds is the circle inscribed in $S$. We also discuss counterparts of this phenomenon for lozenge tilings of hexagons.

math.CO↗
arXiv2023

Domino tilings of generalized Aztec triangles

Di Francesco introduced Aztec triangles as combinatorial objects for which their domino tilings are equinumerous with certain sets of configurations of the twenty-vertex model that are the main focus of his article. We generalize Di Francesco's construction of Aztec triangles. While we do not know whether there is again a correspondence with configurations in the twenty-vertex model, we prove closed-form product formulas for the number of domino tilings of our generalized Aztec triangles. As a special case, we obtain a proof of Di Francesco's conjectured formula for the number of domino tilings of his Aztec triangles, and thus for the number of the corresponding configurations in the twenty-vertex model.

math.CO↗
arXiv2023

Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity

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 $n$ produces numbers of alternating sequences of integers that are bounded from below and from above. We show that this reciprocity extends to families of non-intersecting bounded up-down paths and certain arrays of alternating sequences which we call alternating tableaux. We provide as well weighted versions of these results. Our proofs are based on Viennot's theory of heaps of pieces and on the combinatorics of non-intersecting lattice paths. An unexpected application leads to a refinement of a result of Bousquet-Mélou and Viennot on the width-height-area generating function of parallelogram polyominoes. Finally, we exhibit the relation of the arising alternating tableaux to plane partitions of strip shapes.

math.CO↗
arXiv2022

An asymptotic approach to Borwein-type sign pattern theorems

The celebrated (First) Borwein Conjecture predicts that for all positive integers~$n$ the sign pattern of the coefficients of the ``Borwein polynomial'' $$(1-q)(1-q^2)(1-q^4)(1-q^5) \cdots(1-q^{3n-2})(1-q^{3n-1})$$ is $+--+--\cdots$. It was proved by the first author in [Adv. Math. 394 (2022), Paper No. 108028]. In the present paper, we extract the essentials from the former paper and enhance them to a conceptual approach for the proof of ``Borwein-like'' sign pattern statements. In particular, we provide a new proof of the original (First) Borwein Conjecture, a proof of the Second Borwein Conjecture (predicting that the sign pattern of the square of the ``Borwein polynomial'' is also $+--+--\cdots$), and a partial proof of a ``cubic'' Borwein Conjecture due to the first author (predicting the same sign pattern for the cube of the ``Borwein polynomial''). Many further applications are discussed.

math.CO↗
arXiv2022

The Rank Enumeration of Certain Parabolic Non-Crossing Partitions

We consider $m$-divisible non-crossing partitions of $\{1,2,\ldots,mn\}$ with the property that for some $t\leq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapoton's $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.

math.CO↗
arXiv2021

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

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 correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi [J. Math. Anal. Appl. 431} (2015), 1253-1274] (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami [Comm. Math. Phys. 214 (2000), 111-135], and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

math.CA↗
arXiv2021

Infinite product formulae for generating functions for sequences of squares

We state and prove product formulae for several generating functions for sequences $(a_n)_{n\ge0}$ that are defined by the property that $Pa_n+b^2$ is a square, where $P$ and $b$ are given integers. In particular, we prove corresponding conjectures of the second author. We show that, by means of the Jacobi triple product identity, all these generating functions can be reduced to a linear combination of theta function products. The proof of our formulae then consists in simplifying these linear combinations of theta products into single products. We do this in two ways: (1) by using modular function theory, and (2) by applying the Weierstraßaddition formula for theta products.

math.NT↗
arXiv2021

Lucas congruences for the Apéry numbers modulo $p^2$

The sequence $A(n)_{n \geq 0}$ of Apéry numbers can be interpolated to $\mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $\mathbb{Z}$-linear combination of multiple zeta values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.

math.NT↗
arXiv2020

Evaluation of binomial double sums involving absolute values

We show that double sums of the form $$ \sum_{i,j=-n} ^{n} |i^sj^t(i^k-j^k)^β| \binom {2n} {n+i} \binom {2n} {n+j} $$ can always be expressed in terms of a linear combination of just four functions, namely $\binom {4n}{2n}$, ${\binom {2n}n}^2$, $4^n\binom {2n}n$, and $16^n$, with coefficients that are rational in $n$. We provide two different proofs: one is algorithmic and uses the second author's computer algebra package Sigma; the second is based on complex contour integrals. In many instances, these results are extended to double sums of the above form where $\binom {2n}{n+j}$ is replaced by $\binom {2m}{m+j}$ with independent parameter $m$.

math.CO↗
arXiv2020

Moments of q-Jacobi Polynomials and q-Zeta Values

We explore some connections between moments of rescaled little q-Jacobi polynomials, q-analogues of values at negative integers for some Dirichlet series, and the q-Eulerian polynomials of wreath products of symmetric groups.

math.CO↗
arXiv2019

Spectral decomposition of discrepancy kernels on the Euclidean ball, the special orthogonal group, and the Grassmannian manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of $\mathbb{R}^d$. For restrictions to the Euclidean ball in odd dimensions, to the rotation group $SO(3)$, and to the Grassmannian manifold $\mathcal{G}_{2,4}$, we compute the kernels' Fourier coefficients and determine their asymptotics. The $L_2$-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For $SO(3)$, the nonequispaced fast Fourier transform is publicly available, and, for $\mathcal{G}_{2,4}$, the transform is derived here. We also provide numerical experiments for $SO(3)$ and $\mathcal{G}_{2,4}$.

math.NA↗
arXiv2019

On $q$-analogues of some series for $π$ and $π^2$

We obtain a new $q$-analogue of the classical Leibniz series $\sum_{k=0}^\infty(-1)^k/(2k+1)=π/4$, namely \begin{equation*} \sum_{k=0}^\infty\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\frac{(q^2;q^2)_{\infty}(q^8;q^8)_{\infty}}{(q;q^2)_{\infty}(q^4;q^8)_{\infty}}, \end{equation*} where $q$ is a complex number with $|q|<1$. We also show that the Zeilberger-type series $\sum_{k=1}^\infty(3k-1)16^k/(k\binom{2k}k)^3=π^2/2$ has two $q$-analogues with $|q|<1$, one of which is $$\sum_{n=0}^\infty q^{n(n+1)/2} \frac {1-q^{3n+2}} {1-q} \cdot\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2 \frac{(q^2;q^2)^4_\infty}{(q;q^2)^4_\infty}.$$

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arXiv2018

Regular sequences of symmetric polynomials

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 necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.

math.AC↗
arXiv2018

Some determinants of path generating functions, II

We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant evaluations have numerous corollaries. In particular, one consequence is that the Hankel determinant of Motzkin prefix numbers equals 1, regardless of the size of the Hankel matrix.

math.CO↗
arXiv2018

A Joint Central Limit Theorem for the Sum-of-Digits Function, and Asymptotic Divisibility of Catalan-like Sequences

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within the framework of quasi-additive functions. As application, we show that most elements of "Catalan-like" sequences - by which we mean integer sequences defined by products/quotients of factorials - are divisible by any given positive integer.

math.NT↗
arXiv2018

Hypergeometry inspired by irrationality questions

We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $π^2$, 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.

math.NT↗
arXiv2018

Motzkin numbers and related sequences modulo powers of $2$

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 $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Théorie Nombres Bordeaux 27 (2015), 245-288]. Analogous results are also shown to hold for related combinatorial sequences, namely for the Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and for the sequence of hex tree numbers.

math.CO↗