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Johann Cigler

Publications and source records attributed to Johann Cigler.

At least 19 recordsLinked to original sources
arXiv2026

Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1

As is well known the Catalan numbers can be interpreted as numbers of Dyck paths. The Narayana polynomials and the q-Narayana polynomials for q=-1 can analogously be interpreted as certain weights of these paths. In the present note we try to get some information about the corresponding weights of bounded Dyck paths. This also leads to some number sequences and analogues of Pascals triangle which previously have occurred in other contexts. We consider some examples and obtain some results and conjectures.

math.CO↗
arXiv2026

Continued fractions related to Narayana polynomials

The generating functions of some sequences of Catalan numbers and Narayana polynomials have simple expansions as continued fractions of Jacobi type. We give an overview of these facts and prove analogous results for q-Narayana polynomials at q=-1.

math.CO↗
arXiv2024

Factorization of spread polynomials

We present a proof of a conjecture of Goh and Wildberger on the factorization of the spread polynomials. We indicate how the factors can be effectively calculated and exhibit a connection to the factorization of Fibonacci numbers into primitive parts.

math.NT↗
arXiv2023

Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers

Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other columns which are suggested by computer experiments. By prepending zero rows to admissible matrices we also consider Hankel determinants of backward shifts.

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↗