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enumerative combinatorics of Simplicial and Cell Complexes: Kirchhoff and Trent Type Theorems

DISCRETE & COMPUTATIONAL GEOMETRY 2019年第1期61卷 1-41页

作者： Cappell, Sylvain E. Miller, Edward Y. NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA

This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of sub-objects. The matrices are: the mesh matrix for integral d-cycles, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral (d-1)-chains. Trent's theorem states that the determinant of the mesh matrix on 1-cycles of a connected graph is equal to the number of spanning trees (Trent in Proc Nat Acad Sci USA 40:1004-1007, 1954;J Acoust Soc Am 27(3):500-527, 1955). Here this theorem is extended to the mesh matrix on d-cycles in an arbitrary cell complex and, new even for graphs, to enumerative combinatorial interpretation of all of the coefficients of its characteristic polynomial. This last is well defined once a basis for the integral d-cycles is chosen. Additionally, a parallel result for the mesh matrix for integral d-boundaries is proved. Kirchhoff's classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497-508, 1847) states that the product of the non-zero eigenvalues of the Kirchhoff matrix, i.e., combinatorial Laplacian, for connected graphs equals n times the number of spanning trees, with n the number of vertices. Kalai (Isr J Math 45(4): 337-351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. Lyons has generalized Kirchhoff's result on the product of the non-zero eigenvalues of the Kirchhoff or combinatorial Laplacian on (d-1)-chains to cell complexes for d>1 (Lyons in J Topol Anal 1(2), 153-175, 2009;Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete

关键词： Combinatorial Laplacian Simplicial complexes Complexes Kirchhhoff's theorem Trent's theorem Reidemeister-Franz torsion enumerative combinatorics Geometrical combinatorics Spanning trees High dimensional generalizations of graph theory Primary 05E45 05C30 05C05 05A99 Secondary 55U10 05C50 57M15 57Q10

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Duality of codes supported on regular lattices, with an application to enumerative combinatorics

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DESIGNS CODES AND CRYPTOGRAPHY 2018年第9期86卷 2035-2063页

作者： Ravagnani, Alberto Univ Neuchatel Inst Math Emile Argand 11 CH-2000 Neuchatel Switzerland

We introduce a general class of regular weight functions on finite abelian groups, and study the combinatorics, the duality theory, and the metric properties of codes endowed with such functions. The weights are obtained by composing a suitable support map with the rank function of a graded lattice satisfying certain regularity properties. A regular weight on a group canonically induces a regular weight on the character group, and invertible MacWilliams identities always hold for such a pair of weights. Moreover, the Krawtchouk coefficients of the corresponding MacWilliams transformation have a precise combinatorial significance, and can be expressed in terms of the invariants of the underlying lattice. In particular, they are easy to compute in many examples. Several weight functions traditionally studied in Coding Theory belong to the class of weights introduced in this paper. Our lattice-theory approach also offers a control on metric structures that a regular weight induces on the underlying group. In particular, it allows to show that every finite abelian group admits weight functions that, simultaneously, give rise to MacWilliams identities, and endow the underlying group with a metric space structure. We propose a general notion of extremality for (not necessarily additive) codes in groups endowed with semi-regular supports, and establish a Singleton-type bound. We then investigate the combinatorics and duality theory of extremal codes, extending classical results on the weight and distance distribution of error-correcting codes. Finally, we apply the theory of MacWilliams identities to enumerative combinatorics problems, obtaining closed formulae for the number of rectangular matrices over a finite having prescribed rank and satisfying some linear conditions.

关键词： Additive code (finite abelian group) Poset lattice Support map Weight function MacWilliams identity Extremal code Weight distribution Distance distribution enumerative combinatorics

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Robust Network Covert Communications Based on TCP and enumerative combinatorics

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IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING 2012年第6期9卷 890-902页

作者： Luo, Xiapu Chan, Edmond W. W. Zhou, Peng Chang, Rocky K. C. Hong Kong Polytech Univ Dept Comp Hong Kong Hong Kong Peoples R China

The problem of communicating covertly over the Internet has recently received considerable attention from both industry and academic communities. However, the previously proposed network covert channels are plagued by their unreliability and very low data rate. In this paper, we show through a new class of timing channels coined as Cloak that it is possible to devise a 100 percent reliable covert channel and yet offer a much higher data rate (up to an order of magnitude) than the existing timing channels. Cloak is novel in several aspects. First, Cloak uses the different combinations of N packets sent over X flows in each round to represent a message. The combinatorial nature of the encoding methods increases the channel capacity largely with (N, X). Second, based on the well-known 12-fold Way, Cloak offers 10 different encoding and decoding methods, each of which has a unique tradeoff among several important considerations, such as channel capacity and camouflage capability. Third, the packet transmissions modulated by Cloak can be carefully crafted to mimic normal TCP flows for evading detection. We have implemented Cloak and evaluated it in the PlanetLab and a controlled testbed. The results show that it is not uncommon for Cloak to have an order of channel goodput improvement over the IP Timing channel and JitterBug. Moreover, Cloak does not suffer from any message loss under various loss and reordering scenarios.

关键词： Network covert channel timing channel enumerative combinatorics TCP covert channel detection

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The Interval Posets of Permutations Seen from the Decomposition Tree Perspective

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ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS 2025年 1-21页

作者： Bouvel, Mathilde Cioni, Lapo Izart, Benjamin Univ Lorraine CNRS INRIA LORIA F-54000 Nancy France Univ Firenze Dipartimento Matemat & Informat U Dini Florence Italy

The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in Tenner (Order 39(3), 523-536 2022). We study this poset from the perspective of the decomposition trees of permutations, describing a procedure to obtain the former from the latter. We then give alternative proofs of some of the results in Tenner (Order 39(3), 523-536 2022), and we solve the open problems that it posed (and some other enumerative problems) using techniques from symbolic and analytic combinatorics. Finally, we compute the M & ouml;bius function on such posets.

关键词： Permutation Interval Poset Decomposition tree Substitution decomposition enumerative combinatorics

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Lifted inference beyond first-order logic

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ARTIFICIAL INTELLIGENCE 2025年 342卷

作者： Malhotra, Sagar Bizzaro, Davide Serafini, Luciano TU Wien Vienna Austria Fdn Bruno Kessler Trento Italy Univ Padua Padua Italy

Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general (#P- complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers (C2) is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in C2, or first order logic in general. In this work, we expand the domain liftability of C2 with multiple such properties. We show that any C2 sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of counting by splitting. Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.

关键词： Lifted inference enumerative combinatorics Weighted model counting First order logic Directed acyclic graphs Connected graphs Counting quantifiers Statistical relational learning

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Directed animals, forests and permutations

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DISCRETE MATHEMATICS 1999年第1-3期204卷 41-71页

作者： Barcucci, E Del Lungo, A Pergola, E Pinzani, R Univ Florence Dipartimento Sistemi & Informat I-50134 Florence Italy

In this paper, we illustrate a method to enumerate two-dimensional directed animals, with compact sources, on both the square and the triangular lattice. We give a recursive description of these structures from which we deduce their generating function, according to various parameters: the area, the right half-width and the number of compact sources. We determine the average number of compact sources in both lattices. Referring to single-source animals on the square lattice, we show the bijections connecting them to the forests of 1-2 frees and to permutations with the forbidden subsequence: 321, 4 (1) over bar 523. In a similar way we describe the bijections between single-source animals on the triangular lattice and forests of binary flees. Each of these two objects presents a bijection with two permutations with the forbidden subsequences: 4132, 4231, 4312, 4321. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： enumerative combinatorics directed animals trees permutations

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Exhaustive generation of atomic combinatorial differential operators

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THEORETICAL COMPUTER SCIENCE 2014年 536卷 62-69页

作者： Tremblay, Hugo Labelle, Gilbert Brlek, Srecko Masse, Alexandre Blondin Univ Quebec LaCIM Montreal PQ H2X 3Y7 Canada Univ Quebec Chicoutimi Chicoutimi PQ G7H 2B1 Canada

Labelle and Lamathe introduced in 2009 a generalization of the standard combinatorial differential species operator D, by giving a combinatorial interpretation to Omega (X,D) F(X), where Omega (X,T) and F (X) are two-sort and one-sort species respectively. One can show that such operators can be decomposed as sums of products of simpler operators called atomic combinatorial differential operators. In their paper, Labelle and Lamathe presented a list of the first atomic differential operators. In this paper, we describe an algorithm that allows to generate (and enumerate) all of them, subject to available computer resources. We also give a detailed analysis of how to compute the molecular components of Omega (X,D) F (X). (C) 2014 Elsevier B.V. All rights reserved.

关键词： enumerative combinatorics Species theory Molecular species Differential operator Algorithm

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On compact representations for the solutions of linear difference equations with variable coefficients

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 318卷 411-421页

作者： Abderraman Marrero, J. Tomeo, V. UPM Tech Univ Madrid Sch Telecommun Engn Dept Math Appl Informat Technol Avda Complutense S-NCiudad Univ Madrid 28040 Spain Univ Complutense Fac Stat Studies Dept Algebra Avda Puerta Hierro S-NCiudad Univ E-28040 Madrid Spain

A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both nth and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested sums of their variable coefficients. It makes explicit all the sum of products involved in the previous representations of such solutions. Some basic applications are also illustrated. (C) 2016 Elsevier B.V. All rights reserved.

关键词： enumerative combinatorics Hessenbergian Linear difference equation Nested sum

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Statistics on non-crossing trees

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DISCRETE MATHEMATICS 2002年第1-3期254卷 75-87页

作者： Deutsch, E Noy, M Polytech Univ Dept Math Brooklyn NY 11201 USA Univ Politecn Catalunya Dept Matemat Aplicada 2 E-08028 Barcelona Spain

A non-crossing tree is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments and do not cross. Continuing previous research on non-crossing trees, we study several new statistics: number of end-points and boundary edges, number of vertices of a given degree, maximum degree, height, and path-length. In some cases, we obtain closed formulae while in others we deduce asymptotic estimates. Our approach is based on generating functions and on several bijections between NC-trees and various other combinatorial objects. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： non-crossing tree enumerative combinatorics asymptotic enumeration

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Enumeration and limit laws of dissections on a cylinder

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DISCRETE MATHEMATICS 2010年第19期310卷 2519-2541页

作者： Rue, Juanjo Ecole Polytech Lab Informat F-91128 Palaiseau France

We compute the generating function for triangulations on a cylinder, with the restriction that all vertices belong to its boundary and that the intersection of a pair of different faces is either empty, a vertex or an edge. We generalize these results to maps with either constant ({k}-dissections) or unrestricted (unrestricted dissections) face degree. We apply singularity analysis to the resulting generating functions to obtain asymptotic estimates for their coefficients, as well as limit distributions for natural parameters. (C) 2010 Elsevier B.V. All rights reserved.

关键词： enumerative combinatorics Analytic combinatorics Symbolic method Map enumeration Singularity analysis

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