arXiv subjects
Publications and source records attributed to Christian Krattenthaler.
Stankova and West showed that for any non-negative integer $s$ and any permutation $γ$ of $\{4,5,\dots,s+3\}$ there are as many permutations that avoid $231γ$ as there are that avoid $312γ$. We extend this result to the setting of words.
We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaev's $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.
Let $E_n(f)_{α,β,γ}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_{α,β,γ})$ on the triangle $\{(x,y): x, y \ge 0, x+y \le 1\}$, where $\varpi_{α,β,γ}(x,y) := x^αy ^β(1-x-y)^γ$ for $α,β,γ> -1$. Our main result gives a sharp estimate of $E_n(f)_{α,β,γ}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. The result also leads to a characterization of $E_n(f)_{α,β,γ}$ by a weighted $K$-functional.
We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves right, up, left, or down along the spiral.
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 $-h, \dots, -1, +1, \dots, +h$. The case $h=1$ is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case $h=2$ corresponds to "basketball" walks, which we treat in full detail. Then we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called "kernel method", leads to explicit formulas for the number of walks of length $n$, for any $h$, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.
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 identity in terms of symmetry classes of lozenge tilings of a hexagon on the triangular lattice. Our generalization consists of allowing the hexagon have certain symmetrically placed holes along its horizontal symmetry axis. The special case when there are no holes can be viewed as a new, simpler proof of the enumeration of symmetric plane partitions.
We discuss a partial normalisation of a finite graph of finite groups $(Γ(-), X)$ 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 inequality for the number of edges in a Stallings decomposition $Γ\cong π_1(Γ(-), X)$ of a finitely generated virtually free group, (ii) the proof of equivalence of a number of conditions for such a group to be `large', as well as (iii) the classification up to isomorphism of virtually free groups of (free) rank $2$. We also discuss some number-theoretic consequences of the last result.
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 establishes. We prove as well a "Knuth-type" extension stating a similar equi-enumeration result between generalised oscillating tableaux and semistandard tableaux.
We compute the congruence class modulo 16 of the number of unique path partitions of $n$ (as defined by Olsson), thus generalising previous results by Bessenrodt, Olsson and Sellers [Ann. Combin. 13 (2013), 591-602].
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations - a well-known open problem from computational geometry.
In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $Γ$ 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 number of free subgroups of index~$λ$ in $Γ$ is - essentially - congruent to a binomial coefficient times a rational function in $λ$ modulo a power of a prime that divides a certain invariant of the group $Γ$, respectively to a binomial sum involving such numbers. These results, apart from their intrinsic interest, in particular allow for a much more efficient computation of congruences for free subgroup numbers in these cases compared to the direct recursive computation of these numbers implied by the generating function results in [J. London Math. Soc. (2) 44 (1991), 75-94].
We consider discretisations of the Macdonald--Mehta integrals from the theory of finite reflection groups. For the classical groups, $\mathrm{A}_{r-1}$, $\mathrm{B}_r$ and $\mathrm{D}_r$, we provide closed-form evaluations in those cases for which the Weyl denominators featuring in the summands have exponents $1$ and $2$. Our proofs for the exponent-$1$ cases rely on identities for classical group characters, while most of the formulas for the exponent-$2$ cases are derived from a transformation formula for elliptic hypergeometric series for the root system $\mathrm{BC}_r$. As a byproduct of our results, we obtain closed-form product formulas for the (ordinary and signed) enumeration of orthogonal and symplectic tableaux contained in a box.
(Dieudonné 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 Dwork's lemma. As applications, we provide several results concerning lower bounds on the $p$-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian $p$-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers modulo $p$ for free products of finite Abelian $p$-groups, results on $p$-divisibility of permutation numbers with restrictions on their cycle structure, and a curious "supercongruence" for a certain binomial sum.
For the Möbius 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.
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 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 Poincaré-Einstein metric, and relies on combinatorial recurrence identities related to the dual Hahn polynomials.
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.
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".