scholarly journals A bijection for nonorientable general maps

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jérémie Bettinelli
Keyword(s):  
Simple Form ◽  
Asymptotic Enumeration ◽  
International Audience ◽  
Enumeration Formula

International audience We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.

scholarly journals Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Christophe Aval ◽  
Philippe Duchon
Keyword(s):  
Alternating Sign Matrices ◽  
International Audience ◽  
Enumeration Formula ◽  
Half Turn

International audience The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestrited ASm's and the number of half-turn symmetric ASM's. L'objet de ce travail est d'énumérer les matrices à signes alternants (ASM) quasi-invariantes par rotation d'un quart-de-tour. La formule d'énumération, conjecturée par Duchon, fait apparaître trois facteurs, comprenant le nombre d'ASM quelconques et le nombre d'ASM invariantes par demi-tour.

scholarly journals On the asymptotic enumeration of accessible automata

2010 ◽  
Vol Vol. 12 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
Elcio Lebensztayn
Keyword(s):  
Asymptotic Estimate ◽  
Asymptotic Enumeration ◽  
Lagrange Inversion ◽  
Binomial Series ◽  
Letter Alphabet ◽  
International Audience ◽  
Generalized Binomial Series

Automata, Logic and Semantics International audience We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.

scholarly journals On trees, tanglegrams, and tangled chains

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
Frderick Matsen IV
Keyword(s):  
Asymptotic Formula ◽  
Computer Science ◽  
Explicit Formula ◽  
Binary Trees ◽  
Biological Research ◽  
Open Problems ◽  
International Audience ◽  
Enumeration Formula ◽  
New Formula ◽  
Simple Asymptotic Formula

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

scholarly journals The Selberg integral and Young books

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jang Soo Kim ◽  
Suho Oh
Keyword(s):  
Young Tableaux ◽  
Combinatorial Interpretation ◽  
Selberg Integral ◽  
Combinatorial Objects ◽  
Special Cases ◽  
International Audience ◽  
Enumeration Formula ◽  
Standard Young Tableaux

International audience The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained. L’intégrale de Selberg est une partie intégrante importante abord évaluée par Selberg en 1944. Stanley a trouvé une interprétation combinatoire de la Selberg aide en permutations. Dans ce papier, de nouveaux objets combinatoires “livres de Young” sont introduits et présentés à avoir un lien avec l’intégrale de Selberg. Cette connexion donne une formule d'énumération pour les livres de Young. Il est démontré que des cas spéciaux de livres de Young deviennent tableaux standards de Young de formes diverses: escaliers décalés, places, certaines formes gauches et certaines formes tronquées. En conséquence, l’énumération des formules pour tableaux standards de Young de ces formes sont obtenues.

scholarly journals Degree distribution in random planar graphs

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Omer Gimenez ◽  
Marc Noy
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Degree Distribution ◽  
Planar Graphs ◽  
Probability Generating Function ◽  
Asymptotic Estimates ◽  
Asymptotic Enumeration ◽  
Root Vertex ◽  
International Audience ◽  
Transfer Theorems

International audience We prove that for each $k \geq 0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimènez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.

scholarly journals Asymptotic enumeration on self-similar graphs with two boundary vertices

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
General Setting ◽  
Scaling Factor ◽  
Asymptotic Enumeration ◽  
Spanning Forests ◽  
International Audience ◽  
Self Similar ◽  
Connected Subgraphs ◽  
Graph Parameters ◽  
Number Of Spanning Trees

Combinatorics International audience We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.

scholarly journals Asymptotic enumeration of orientations

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Stefan Felsner ◽  
Eric Fusy ◽  
Marc Noy
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Linear Differential Equations ◽  
Asymptotic Enumeration ◽  
International Audience ◽  
Asymptotic Number ◽  
Complex Linear ◽  
Linear Differential ◽  
Prime 2

International audience We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations.

scholarly journals Matrix Ansatz, lattice paths and rook placements

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
S. Corteel ◽  
M. Josuat-Vergès ◽  
T. Prellberg ◽  
M. Rubey
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Laguerre Polynomials ◽  
Lattice Paths ◽  
Rook Placements ◽  
Combinatorial Interpretations ◽  
The Matrix ◽  
International Audience ◽  
Enumeration Formula

International audience We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern $13-2$, the generating function according to weak exceedances and crossings, and the $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials. Nous donnons deux interprétations combinatoires du Matrix Ansatz du PASEP en termes de chemins et de placements de tours. Cela donne deux preuves (presque) combinatoires d'une nouvelle formule pour la fonction de partition du PASEP. Cette formule donne aussi par exemple la fonction génératrice des permutations de taille donnée par rapport au nombre de montées et d'occurrences du motif $13-2$, la fonction génératrice par rapport au nombre d'excédences faibles et de croisements, et le $n^{\mathrm{ième}}$ moment de certains polynômes de $q$-Laguerre.

scholarly journals Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

10.37236/5512 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Vladimir Blinovsky ◽  
Catherine Greenhill
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Asymptotic Enumeration ◽  
Uniform Hypergraphs ◽  
Enumeration Formula ◽  
Linear Hypergraphs

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.

Background Fitting in the Low-Loss Region of Electron Energy Loss Spectra

1990 ◽  
Vol 48 (2) ◽  
pp. 56-57
Author(s):  
C P Scott ◽  
A J Craven ◽  
C J Gilmore ◽  
A W Bowen
Keyword(s):  
Energy Loss ◽  
Multiple Scattering ◽  
Simple Form ◽  
Single Scattering ◽  
Low Loss ◽  
Characteristic Energy ◽  
Normal Method ◽  
Fitting In ◽  
Electron Energy Loss ◽  
Electron Energy Loss Spectra

The normal method of background subtraction in quantitative EELS analysis involves fitting an expression of the form I=AE-r to an energy window preceding the edge of interest; E is energy loss, A and r are fitting parameters. The calculated fit is then extrapolated under the edge, allowing the required signal to be extracted. In the case where the characteristic energy loss is small (E < 100eV), the background does not approximate to this simple form. One cause of this is multiple scattering. Even if the effects of multiple scattering are removed by deconvolution, it is not clear that the background from the recovered single scattering distribution follows this simple form, and, in any case, deconvolution can introduce artefacts.The above difficulties are particularly severe in the case of Al-Li alloys, where the Li K edge at ~52eV overlaps the Al L2,3 edge at ~72eV, and sharp plasmon peaks occur at intervals of ~15eV in the low loss region. An alternative background fitting technique, based on the work of Zanchi et al, has been tested on spectra taken from pure Al films, with a view to extending the analysis to Al-Li alloys.

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