Path systems in acyclic directed graphs

Linear Algebra and its Applications

2005

234

185

This is a complement to my previous article "Advanced Determinant Calculus" (Séminaire 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 and the author, Experiment. Math. 12 (2003), 441-456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems.2000 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A18 05A30 05E10 05E15 11B68 11B73 11C20 11Y60 15A15 33C45 33D45 33E05.The prime factorisation of the second-to-last number is (we are using Mathematica here) In[1]:= FactorInteger[477638700]3 The writing N ICE(n) is borrowed from Doron Zeilberger [193, Recitation III]. The technical term for a formula of the type (2.5) is "hypergeometric term", see [144, Sec. 3.2], whereas, most often, the colloquial terms "closed form" or "nice formula" are used for it, see [193, Recitation II]. More recently, some authors call sequences given by formulae of that type sequences of "round" numbers, see [117, Sec. 6].4 Rate is available from http://igd.univ-lyon1.fr/~kratt. It is based on a rather simple algorithm which involves rational interpolation. In contrast to what I read, with great surprise, in [46], the explanations of how Rate works in Appendix A of [109] can be read and understood without any knowledge about determinants and, in particular, without any knowledge of the fifty or so pages that precede Appendix A in [109]. 5 Rate will always be able to guess a formula of the type (2.5) if there are enough initial terms of the sequence available. However, there is a larger class of sequences which have the property that the size of the primes in the prime factorisation of the terms of the sequence grows only slowly with n. These are sequences given by formulae containing "Abelian" factors, such as n n . Unfortunately, Rate does not know how to handle such factors. Recently, Rubey [158] proposed an algorithm for covering Abelian factors as well. His implementation Guess is written in Axiom and is available at

Section: Figure 2 a Hexagon With Triangular Holementioning

confidence: 99%

Advanced determinant calculus: A complement

Linear Algebra and its Applications

2005

234

185

“…5.3] and in [28,26]; for a more detailed historical account see Footnote 6 in [32]). Since, as is the case in most applications, we do not need the theorem in its most general form, we state the special case that serves our purposes.…”

Section: Preliminariesmentioning

confidence: 99%

Determinants of (generalised) Catalan numbers

Journal of Statistical Planning and Inference

2010

“…By a curious coincidence, Lindström's result (the motivation of which was matroid theory!) was rediscovered in the 1980s at about the same time in three different communities, not knowing from each other at that time: in statistical physics by Fisher [17, Sec.5.3] in order to apply it to the analysis of vicious walkers as a model of wetting and melting, in combinatorial chemistry by John and Sachs[30] and Gronau, Just, Schade, Scheffler and Wojciechowski[28] in order to compute Pauling's bond order in benzenoid hydrocarbon molecules, and in enumerative combinatorics by Gessel and Viennot[24,25] in order to count tableaux and plane partitions. Since only Gessel and…”

mentioning

confidence: 99%

Watermelon configurations with wall interaction: exact and asymptotic results

2006

J. Phys.: Conf. Ser.

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 the wall. We improve and extend the earlier (partially nonrigorous) results by Brak, Essam and Owczarek, providing new exact results, and more precise and more general asymptotic results, in particular full asymptotic expansions for the partition function and the mean number of contacts. Furthermore, we relate this circle of problems to earlier results in the combinatorial and statistical literature.2000 Mathematics Subject Classification. Primary 82B20; Secondary 05A15 05A16 60G50 82B23 82B26 82B41 82B43.