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A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with a precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.

We study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when not.

The results of this paper have been subsumed by those of our new paper arXiv:0910.1858

In literature, many important combinatorial properties of subsets of N have been studied both with nonstandard techniques and from the point of view of N. In this thesis we mix these two different approaches in a technique that, at the same time, incorporates nonstandard tools and ultrafilters.

We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived.

We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived.

An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.

An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.

We show, in great detail, how the perturbative tools of quantum field theory allow one to rigorously obtain: a ``categorified'' Faa di Bruno type formula for multiple composition, an explicit formula for reversion and a proof of Lagrange-Good inversion, all in the setting of multivariable power series. We took great pains to offer a self-contained presentation that, we hope, will provide any mathematician who wishes, an easy access to the wonderland of quantum field theory.

Buch and Fulton conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono proved this conjecture as an immediate consequence of their ``component formula''. We present an alternative proof of the component formula by substituting combinatorics for Grobner degeneration. We relate the component formula to the work of Buch, Kresch, Tamvakis and the author where a ``splitting'' formula for Schubert polynomials in terms of quiver coefficients was obtained. We prove analogues of this latter result for the type BCD-Schubert polynomials of Billey and Haiman.

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an interval of the form [x, \infty) or just a single point. The inequality is exact and the optimal values of c and k are given explicitly. It improves Kwapie\'n's inequality in the case of the Rademacher series. We also provide a new and very short proof of the Littlewood-Offord problem without using Sperner's Theorem. Finally, an extension to odd Lipschitz functions is given.

We present a study of three families of Kronecker coefficients, which we describe in terms of reduced Kronecker coefficients. This study is grounded on the generating function of the coefficients, proved by a bijection between two combinatorial objects. This study includes the connection between plane partitions and these three families of reduced Kronecker coefficients, providing us their combinatorial interpretation. As an application, we verify that the saturation hypothesis holds for our three families of reduced Kronecker coefficients. The study also includes other interpretation in terms of the quasipolynomials that govern these families. We specify the degree and the period of these quasipolynomials. Finally, the direct relation between Kronecker coefficients and reduced Kronecker coefficients allows us to give some observations about the rate of growth of the Kronecker coefficients associated to the reduced Kronecker coefficients of the study.

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity $k$. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.

We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, $\widetilde{\mathbf G}_n(R)$ and $\exp_n(X)$ which were introduced and studied by V.~Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets $\exp_n(S^m)$ which leads to a negative answer to a question of Vassilev raised at the workshop ``Geometric Combinatorics'' (MSRI, February 1997).

We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, $\widetilde{\mathbf G}_n(R)$ and $\exp_n(X)$ which were introduced and studied by V.~Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets $\exp_n(S^m)$ which leads to a negative answer to a question of Vassilev raised at the workshop ``Geometric Combinatorics'' (MSRI, February 1997).

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth's decomposition theorem, Mirsky's theorem, Hall's marriage theorem and the Erd\H{o}s-Szekeres theorem. Dilworth's decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky's theorem is a dual of Dilworth's decomposition theorem, which states that in any finite poset, the size of a smallest antichain cover and a largest chain are the same. We use Dilworth's theorem in the proofs of Hall's Marriage theorem and the Erd\H{o}s-Szekeres theorem. The combinatorial objects involved in these theorems are sets and sequences. All the proofs are formalized in the Coq proof assistant. We develop a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets.

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.

An Horizontal Visibility Graph (for short, HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos [B. Luque, et al., Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics [P. Flajolet and M. Noy, Discrete Math., 204 (1999) 203-229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique we determine asymptotically the average number of edges of HVGs.

We establish a relation between the two-party Bell inequalities for two-valued measurements and a high-dimensional convex polytope called the cut polytope in polyhedral combinatorics. Using this relation, we propose a method, triangular elimination, to derive tight Bell inequalities from facets of the cut polytope. This method gives two hundred million inequivalent tight Bell inequalities from currently known results on the cut polytope. In addition, this method gives general formulas which represent families of infinitely many Bell inequalities. These results can be used to examine general properties of Bell inequalities.

We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's "special" bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.

We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's "special" bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.

We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's "special" bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.

We study the determinant of the pxp circulant matrix whose first row is (1,-x,0,...,0,-y,0,...,0), the -y being in position q+1. The coefficients of this polynomial are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomial x^ry^s have the same sign, and we determine that sign. We prove that a monomial x^ry^s appears if and only if p divides r+sq. Finally, we show that the size of the largest coefficient of the monomials that appear grows exponentially with p. We do this by proving that the permanent of the circulant whose first row is (1,1,0,...,0,1,0,...,0) is the sum of the absolute values of the coefficients of the monomials in the original determinant.

Papers on Connectivity of Smarandachely Line Splitting Graphs, Equitable Coloring of Helm Graph and Gear Graph, Some Results on Pair Sum Labeling of Graphs, Entire Semitotal-Point Domination in Graphs, and other topics.

Speaker: Francis Su

Speaker: Francis Su

Lecture notes given at the summer school ``Applications of random matrices to physics", Les Houches, June 2004.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published   quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-Euclidean geometry and topology and their applications to other sciences.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published   quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-Euclidean geometry and topology and their applications to other sciences.

From blurb: "Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. In order to make it self-contained, the authors give full coverage of the underlying mathematics and give a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes throughout the book to aid understanding. The book can be used as a reference for researchers, as a textbook for an advanced undergraduate or a graduate course on the subject, or for self-study."

We use the combinatorial and number-theoretical methods developed in previous work by the authors to study black hole entropy in the new proposal put forward by Engle, Noui and Perez. Specifically we give the generating functions relevant for the computation of the entropy and use them to derive its asymptotic behavior including the value of the Immirzi parameter and the coefficient of the logarithmic correction.

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not reflection arrangements.

We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n (the labels) are bijectively attached to the $n$ north steps of the above-bounding path, with the condition that they appear in increasing values along consecutive north steps. We calculate the Frobenius characteristic of the action of the symmetric group S_n on these labels. All these enumeration results are refined to take into account the area of these polyominoes. We make a connection between our enumeration results and the theory of operators for which the intergral Macdonald polynomials are joint eigenfunctions. We also explain how these same polyominoes can be used to explicitly construct a linear basis of a ring of SL_2-invariants.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $\P$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published   quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-Euclidean geometry and topology and their applications to other sciences.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published   quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-Euclidean geometry and topology and their applications to other sciences.

Papers on Ruled Surfaces in Minkowski 3-Space, Enumeration of k-Fibonacci Paths Using Infinite Weighted Automata, The Natural Lift Curves and Geodesic Curvatures of the Spherical Indicatrices of The Spacelike-Timelike Bertrand Curve Pair, Magic Properties of Special Class of Trees, and other topics.

In this paper we describe a method for significantly reducing the computational complexity required for observations-track gating in multiple target tracking. We define the gating process as follows: given a set of N(R) observations and N(T) tracks, identify all observation-track pairs whose scores fall above a chosen threshold. The score for observation-track pair (i,j) is defined as the function: In our analysis and numerical simulations we employ algorithms which find near neighbors of points in l-dimensional position space, where nearness is defined by the Euclidean metric. They are used, for example, to find the tracks with mean positions near a given observation position. Essential is that: (1) these algorithms find all the neighbors in some expected optimal or near optimal time, and (2),that their performance be relatively insensitive to spatial distributions. After examining several search algorithms, we selected a BLD-enhanced k-d search tree for our tests.