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Markus Fulmek

Publications and source records attributed to Markus Fulmek.

At least 19 recordsLinked to original sources
arXiv2024

Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture

In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting determinants as generating functions of nonintersecting lattice paths: this proof employs the reflection principle, the Lindstr\"om-Gessel-Viennot-method and a certain construction involving reflections and overlays of nonintersecting lattice paths. Shortly after this bijective proof was presented here, Cigler provided a shorter proof based on earlier results.

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arXiv2023

Tilings of damaged hexagons

In a recent paper, Byun presented nice formulas for the enumeration of lozenge tilings of certain hexagonal regions with intrusions. This paper attempts to generalise some of Byun's investigations.

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arXiv2021

The generating function of lozenge tilings for a "quarter" of a hexagon, obtained with non--intersecting lattice paths

In a recent preprint, Lai and Rohatgi compute the generating functions of lozenge tilings of "quartered hexagons with dents" by applying the method of "graphical condensation". The purpose of this note is to exhibit how (a generalization of) Theorems 2.1 and 2.2 in Lai and Rohatgi's preprint can be achieved by the Lindström--Gessel--Viennot method of non--intersecting lattice paths and a certain determinant evaluation.

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arXiv2020

The quotient of generating functions of lozenge tilings for certain regions derived from hexagons, obtained with non--intersecting lattice paths

In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents", which differ only in width, factors nicely, and the same is true for the quotient of generating functions of weighted lozenge tilings of two "quarter hexagons with lateral dents". Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice paths. For the case of "half hexagons", basically the same observation, but restricted to mere enumeration (i.e., all weights of lozenge tilings are equal to $1$), is contained in a recent preprint of Condon.

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arXiv2020

A certain ratio of generating functions of lozenge tilings, obtained with non--intersecting lattice paths

In a recent preprint, Lai worked out the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents" which differ only in width. Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice paths in a quite simple way. Basically the same observation, but restricted to mere enumeration (i.e., all weights of lozenge tilings are equal to $1$), is contained in a recent preprint of Condon.

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arXiv2019

Enumeration of symmetric Gelfand--Tsetlin patterns by linear algebra

We shall present a ``linear algebraic'' proof (involving some calculations in the algebra of linear operators on a vector space of polynomials and some manipulations of determinants) of the formula for the enumeration of symmetric Gelfand--Tsetlin patterns with fixed bottom row, which was proved by Tri Lai in the context of enumerating symmetric lozenge tilings of a ``halved'' hexagon with ``dents''.

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arXiv2018

A bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruplet of statistics considered by Behrend, Di Francesco and Zinn--Justin

We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the inversion words of permutations and the "usual" representation of descending plane partitions as families of non--intersec\-ting lattice paths.

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arXiv2017

A simple bijection between permutation matrices and descending plane partitions without special parts

We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of permutations and the (well-known) representation of descending plane partitions as families of non--intersecting lattice paths. (Taking a short detour, we will also exhibit how the (well--known) enumeration of descending plane partitions follows easily from the evaluation of Andrew's determinant.)

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arXiv2013

A combinatorial proof for Cayley's identity

In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof involving only combinatorial arguments together with a generalization of Laplace's Theorem, for which a "purely combinatorial" proof is already known.

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arXiv2010

Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot interpretation of semistandard Young tableaux and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides very simple "graphical proofs" for classical determinantal identities like the Cauchy--Binet formula, Dodgson's condensation formula, the Plücker relations and Laplace's expansion. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which might be new.

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arXiv2009

Graphical condensation, overlapping Pfaffians and superpositions of matchings

The purpose of this note is to exhibit clearly how the "graphical condensation" identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn-Percus method for the enumeration of matchings. Knuth termed the relevant identities "overlapping Pfaffian" identities and the key concept of proof "superpositions of matchings". In our uniform presentation of the material, we also give an apparently unpublished general "overlapping Pfaffian" identity of Krattenthaler. A previous version of this paper contained an erroneous application of the Kasteleyn-Percus method, which is now corrected.

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arXiv2009

Bijective proofs for Schur function identities

Gurevich, Pyatov and Saponov recently stated an expansion for the product of two Schur functions and gave a proof based on the Pluecker relations. Here we show that this identity is in fact a special case of a quite general Schur function identity, which was stated and proved in a paper by Fulmek and Kleber, where it was used to prove bijectively Dodgsons condensation formula and the Pluecker relations, but was not paid further attention: So we take the opportunity to make obvious the range of applicability of this identity by giving concrete examples, accompanied by many graphical illustrations.

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arXiv2006

Asymptotics of the average height of 2--watermelons with a wall

We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length $n$) to the case of $p$--watermelons with a wall (i.e., to a certain family of $p$ nonintersecting Dyck paths; simple Dyck paths being the special case $p=1$.) We work out this asymptotics for the case $p=2$ only, since the computations involved are already quite complicated (but might be of some interest in their own right).

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arXiv2004

Nonintersecting lattice paths on the cylinder

We show how a formula concerning ``vicious walkers'' (which basically are nonintersecting lattice paths) on the cylinder given by P.J. Forrester can be proved and generalized by using the Lindström--Gessel--Viennot method, after having things set up in the right way. We apply the corresponding results to the (thermodynamic limit of the) free energy of the ``lock step model of vicious walkers'', thus completing (and in one instance correcting) the work of Forrester . Moreover, we also show how a related formula given by I. Gessel and C. Krattenthaler can be obtained from the same ``point of view''.

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arXiv2002

Enumeration of permutations containing a prescribed number of occurrences of a pattern of length 3

We consider the problem of enumerating the permutations containing exactly $k$ occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with ``jumps'', which were first considered by Krattenthaler.

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arXiv2000

Bijective proofs for Schur function identities which imply Dodgson's condensation formula and Plücker relations

We present a ``method'' for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson's condensation formula, Plücker relations and a recent identity of the second author.

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