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Considering the bijections of an arbitrary group onto itself which commute with all inner automorphisms, we establish the general properties. In particular, the automorphisms constitute the group that includes the group of central automorphisms. Also, we fully describe for the dihedral groups , with . [ABSTRACT FROM AUTHOR]

For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes with their triangulations), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into independent sets. More generally, we give bijections between facet partitions whose parts have minimal distance ≥ s and vertex partitions whose parts have minimal distance ≥ s + 1 . [ABSTRACT FROM AUTHOR]

Recently, Merca and Schmidt proved a number of identities relating partitions of an integer with two classic number-theoretic functions, namely the Möbius function and Euler's totient function. Their demonstrations were mainly algebraic. We give bijective proofs of some of these results. Our main tools are the concept of a rooted partition and an operation which we call the direct sum, which combines a partition and a rooted partition. [ABSTRACT FROM AUTHOR]

Graphs are a powerful tool for solving various mathematical problems. One such task is the representation of discrete structures. Combinatorial generation methods make it possible to obtain algorithms that can create discrete structures with specified properties. This article is devoted to issues related to the construction of such combinatorial generation algorithms for a wide class of directed lattice paths. The main method used is based on the representation of a given combinatorial set in the form of an AND/OR tree structure. To apply this method, it is necessary to have an expression for the cardinality function of a combinatorial set that satisfies certain requirements. As the main result, we have found recurrence relations for enumerating simple directed lattice paths that satisfy the requirements of the applied method and have constructed the corresponding AND/OR tree structure. Applying the constructed AND/OR tree structure, we have developed new algorithms for ranking and unranking simple directed lattice paths. Additionally, the obtained results were generalized to the case of directed lattice paths. [ABSTRACT FROM AUTHOR]

A hybrid binary tree is a complete binary tree where each internal node is labeled with 1 or 2, but with no left (1, 1)-edges. In this paper, we consider enumeration of the set of hybrid binary trees according to the number of internal nodes and some other combinatorial parameters. We present enumerative results by giving Riordan arrays, bivariate generating functions, as well as closed formulas. As a consequence, we obtain some new combinatorial matrices, one of which is analogous to the Borel triangle. We also present a bijection between the set of all hybrid binary trees with n internal nodes and the set of generalized Schröder paths from (0, 0) to (2n, 0) which are consist of up steps u = (1 , 1) , horizontal steps h = (2 , 0) , down steps d = (1 , - 1) , and double up steps U = (2 , 2) , and never travel below the x-axis. [ABSTRACT FROM AUTHOR]

We solve two problems regarding the enumeration of lattice paths in $\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings. The first problem considers crossings of a single path with a fixed horizontal line. The second one counts pairs of paths with respect to the number of times they cross each other. Our proofs introduce lattice path bijections with convenient visual descriptions, and the answers are given by remarkably simple formulas involving $q$-binomial coefficients.Mathematics Subject Classifications: 05A19, 05A15, 05A30Keywords: Lattice path, major index, crossing, valley, bijection

In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the 2-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks. [ABSTRACT FROM AUTHOR]

Graphs are a powerful tool for solving various mathematical problems. One such task is the representation of discrete structures. Combinatorial generation methods make it possible to obtain algorithms that can create discrete structures with specified properties. This article is devoted to issues related to the construction of such combinatorial generation algorithms for a wide class of directed lattice paths. The main method used is based on the representation of a given combinatorial set in the form of an AND/OR tree structure. To apply this method, it is necessary to have an expression for the cardinality function of a combinatorial set that satisfies certain requirements. As the main result, we have found recurrence relations for enumerating simple directed lattice paths that satisfy the requirements of the applied method and have constructed the corresponding AND/OR tree structure. Applying the constructed AND/OR tree structure, we have developed new algorithms for ranking and unranking simple directed lattice paths. Additionally, the obtained results were generalized to the case of directed lattice paths.

Volumetric mapping is a ubiquitous and difficult problem in Geometry Processing and has been the subject of research in numerous and various directions. While several methods show encouraging results, the field still lacks a general approach with guarantees regarding map bijectivity. Through this work, we aim at opening the door to a new family of methods by providing a novel framework based on the concept of progressive expansion. Starting from an initial map of a tetrahedral mesh whose image may contain degeneracies but no inversions, we incrementally adjust vertex images to expand degenerate elements. By restricting movement to so-called expansion cones, it is done in such a way that the number of degenerate elements decreases in a strictly monotonic manner, without ever introducing any inversion. Adaptive local refinement of the mesh is performed to facilitate this process. We describe a prototype algorithm in the realm of this framework for the computation of maps from ball-topology tetrahedral meshes to convex or star-shaped domains. This algorithm is evaluated and compared to state-of-the-art methods, demonstrating its benefits in terms of bijectivity. We also discuss the associated cost in terms of sometimes significant mesh refinement to obtain the necessary degrees of freedom required for establishing a valid mapping. Our conclusions include that while this algorithm is only of limited immediate practical utility due to efficiency concerns, the general framework has the potential to inspire a range of novel methods improving on the efficiency aspect. [ABSTRACT FROM AUTHOR]

Andrews, Hirschhorn, and Sellers studied the partition function ped(n) which enumerates the number of partitions of n with even parts distinct, and obtained a number of interesting congruences. This paper aims to introduce a partition statistic to investigate the partition function ped(n). We give combinatorial interpretations for some properties of ped(n) including the infinite families of congruences given by Andrews et al. [ABSTRACT FROM AUTHOR]

In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schröder–Bernstein theorem. Some additional results address statements in higher-order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher-order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice. [ABSTRACT FROM AUTHOR]

In this paper, we consider various theorems of P.A. MacMahon and M.V. Subbarao. For a non-negative integer n, MacMahon proved that the number of partitions of n wherein parts have multiplicity greater than 1 is equal to the number of partitions of n in which odd parts are congruent to 3 modulo 6. We give a new bijective proof for this theorem and its generalization, which consequently provides a new proof of Andrews' extension of the theorem. Considering Subbarao's finitization of Andrews' extension, we generalize this result of Subbarao. Our generalization is based on Glaisher's extension of Euler's mapping for odd-distinct partitions and as a result, a bijection given by Sellers and Fu is also extended. Unlike in the case of Sellers and Fu where two residue classes are fixed, ours takes into consideration all possible residues. Furthermore, some arithmetic properties of related partition functions are derived. [ABSTRACT FROM AUTHOR]

We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes. Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition. Hence, given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints, we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints. To successfully apply this idea, we develop two key techniques. Firstly, an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips. Secondly, edge-based remeshing is used to optimize the regularity of the scaffold mesh containing flips, thereby improving practical robustness. Compared to state-of-the-art methods, our method is much more robust. We demonstrate the capability and feasibility of our method on a large number of complex meshes. [ABSTRACT FROM AUTHOR]

Abstract We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes. Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition. Hence, given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints, we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints. To successfully apply this idea, we develop two key techniques. Firstly, an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips. Secondly, edge-based remeshing is used to optimize the regularity of the scaffold mesh containing flips, thereby improving practical robustness. Compared to state-of-the-art methods, our method is much more robust. We demonstrate the capability and feasibility of our method on a large number of complex meshes.

In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.

In this paper, we analyze the homology of the simplicial complex induced by a given pair of RNA secondary structures, R = (S , T) . Such a pair induces a bi-secondary structure, whose associated loop nerve X is the simplicial complex obtained by loop intersections. We will provide an algebraic proof of the fact that H 1 (X) = 0 . We will provide a combinatorial interpretation for the generators of H 2 (X) in terms of crossing components of the bi-structure and establish that the rank of H 2 (X) equals the total number of such crossing components. Finally, we shall prove that each crossing component naturally encodes a triangulation of a 2-sphere and provide an analysis of the geometric realization of X. [ABSTRACT FROM AUTHOR]

We give a different presentation of a recent bijection due to Chapuy and Dolega 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 nonorientable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and this allows us to recover a famous asymptotic enumeration formula found by Gao. [ABSTRACT FROM AUTHOR]

We answer some questions on the asymptotics of ballot walks raised in [S. B. Ekhad and D. Zeilberger, April 2021] and prove that these models are not D-finite. This short note demonstrates how the powerful tools developed in the last decades on lattice paths in convex cones help us to answer some challenging problems that were out of reach for a long time. On the way we generalize tandem walks to the family of large tandem walks whose steps are of arbitrary length and map them bijectively to a generalization of ballot walks in three dimensions. [ABSTRACT FROM AUTHOR]

We propose iFlowGAN that learns an invertible flow (a sequence of invertible mappings) via adversarial learning and exploit it to transform a source distribution into a target distribution for unsupervised image-to-image translation. Existing GAN-based generative model such as CycleGAN [1], StarGAN [2], AGGAN [3] and CyCADA [4] needs to learn a highly under-constraint forward mapping $\mathcal {F}: X \rightarrow Y$ F : X → Y from a source domain $X$ X to a target domain $Y$ Y . Researchers do this by assuming there is a backward mapping $\mathcal {B}: Y \rightarrow X$ B : Y → X such that $\boldsymbol{x}$ x and $\boldsymbol{y}$ y are fixed points of the composite functions $\mathcal {B} \circ \mathcal {F}$ B ∘ F and $\mathcal {F} \circ \mathcal {B}$ F ∘ B . Inspired by zero-order reverse filtering [5], we (1) understand $\mathcal {F}$ F via contraction mappings on a metric space; (2) provide a simple yet effective algorithm to present $\mathcal {B}$ B via the parameters of $\mathcal {F}$ F in light of Banach fixed point theorem; (3) provide a Lipschitz-regularized network which indicates a general approach to compose the inverse for arbitrary Lipschitz-regularized networks via Banach fixed point theorem. This network is useful for image-to-image translation tasks because it could save the memory for the weights of $\mathcal {B}$ B . Although memory can also be saved by directly coupling the weights of the forward and backward mappings, the performance of the image-to-image translation network degrades significantly. This explains why current GAN-based generative models including CycleGAN must take different parameters to compose the forward and backward mappings instead of employing the same weights to build both mappings. Taking advantage of the Lipschitz-regularized network, we not only build iFlowGAN to solve the redundancy shortcoming of CycleGAN but also assemble the corresponding iFlowGAN versions of StarGAN, AGGAN and CyCADA without breaking their network architectures. Extensive experiments show that the iFlowGAN version could produce comparable results of the original implementation while saving half parameters. [ABSTRACT FROM AUTHOR]

Finding distortion-minimizing homeomorphisms between surfaces of arbitrary genus is a fundamental task in computer graphics and geometry processing. We propose a simple method utilizing intrinsic triangulations, operating directly on the original surfaces without going through any intermediate domains such as a plane or a sphere. Given two models A and B as triangle meshes, our algorithm constructs a Compatible Intrinsic Triangulation (CIT), a pair of intrinsic triangulations over A and B with full correspondences in their vertices, edges and faces. Such a tessellation allows us to establish consistent images of edges and faces of A's input mesh over B (and vice versa) by tracing piecewise-geodesic paths over A and B. Our algorithm for constructing CITs, primarily consisting of carefully designed edge flipping schemes, is empirical in nature without any guarantee of success, but turns out to be robust enough to be used within a similar second-order optimization framework as was used previously in the literature. The utility of our method is demonstrated through comparisons and evaluation on a standard benchmark dataset. [ABSTRACT FROM AUTHOR]

In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established. [ABSTRACT FROM AUTHOR]

A permutation π of a multiset is said to be a quasi-Stirling permutation if there do not exist four indices i < j < k < ℓ such that π i = π k , π j = π ℓ and π i ≠ π j . Define Q ¯ M (t , u , v) = ∑ π ∈ Q ¯ M t des (π) u asc (π) v plat (π) , where Q ¯ M denotes the set of quasi-Stirling permutations of the multiset M , and asc (π) (resp. des (π) , plat (π) ) denotes the number of ascents (resp. descents, plateaux) of π . Denote by M σ the multiset { 1 σ 1 , 2 σ 2 , ... , n σ n } , where σ = (σ 1 , σ 2 , ... , σ n) is an n-composition of K for positive integers K and n. In this paper, we show that Q ¯ M σ (t , u , v) = Q ¯ M τ (t , u , v) for any two n-compositions σ and τ of K. This is accomplished by establishing an (asc , des , plat) -preserving bijection between Q ¯ M σ and Q ¯ M τ . As applications, we obtain generalizations of several results for quasi-Stirling permutations on M = { 1 k , 2 k , ... , n k } obtained by Elizalde and solve an open problem posed by Elizalde. [ABSTRACT FROM AUTHOR]

Combinatorial enumeration of various RNA secondary structures and protein contact maps is of great interest for both combinatorists and computational biologists. Counting protein contact maps is much more difficult than that of RNA secondary structures due to the significant higher vertex degree. The state of art upper bound for vertex degree in previous works is two. This paper proposes a solution for counting general stacks with arbitrary vertex degree upper bound. By establishing a bijection between such general stacks and m -regular Λ-avoiding DLU -paths, and counting these pattern avoiding lattice paths, we obtain a unified system of equations for the generating functions of the number of general stacks. We further show that previous enumeration results for RNA secondary structures and linear stacks of protein contact maps can be derived from the equations for general stacks as special cases. [ABSTRACT FROM AUTHOR]

Inspired by Andrews and Merca's recent work on the number of even parts over all partitions into distinct parts, we introduce a new kind of Beck type identities, which can be viewed as the dual form of Beck's original conjectures. Based on a refinement of a special case of Franklin's theorem, we obtain two interesting identities in great generality, corresponding to Yang's generalization of Beck's conjectures. Both analytic and combinatorial proofs of these results are provided. [ABSTRACT FROM AUTHOR]

We give two simple combinatorial proofs directly on derangements to the well-known identity d n = n d n - 1 + (- 1) n of derangement numbers. Inversion properties on derangements are given. Partial i n v q -analogues are established. [ABSTRACT FROM AUTHOR]

A noncrossing forest is a forest drawn on n points numbered in counterclockwise order on a circle in such a way that its edges are rectilinear and do not cross. Utilizing analytic combinatorics, Flajolet and Noy obtained the number of noncrossing forests with n vertices and k components. In this paper, we will give a new representation for noncrossing forests. Based on such respresentation, we establish a decomposition algorithm and a merging algorithm, which leads to a bijection between labeled noncrossing forests and sets of rooted matches. As an application, we derive a new formula, which is a refinement of the formula given by Flajolet and Noy. [ABSTRACT FROM AUTHOR]

Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. They proved that P D (3 n + 2) is divisible by 3. Later, Chen, Ji, Jin, and the author introduced the pd-rank for partitions with designated summands and gave a combinatorial interpretation of the congruence. In this paper, we define a crank named the d-crank of partitions with designated summands to provide a new combinatorial interpretation for the congruence of Andrews, Lewis, and Lovejoy for P D (3 n + 2) . In addition, we derive the generating functions for the d-crank differences of partitions with designated summands modulo 2, 3, 4, and 6. Based on these generating functions, we obtain an equality and some inequalities for d-cranks of partitions with designated summands. We also introduce the d-crank moments weighted by the parity of d-cranks μ 2 k , d (- 1 , n) and show the positivity of (- 1) n μ 2 k , d (- 1 , n) . [ABSTRACT FROM AUTHOR]

The number of down-steps between pairs of up-steps in kt-Dyck paths, a generalization of Dyck paths consisting of steps f(1; k); (1;-1)g such that the path stays (weakly) above the line y = -t, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between kt-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers. [ABSTRACT FROM AUTHOR]

We define a dual of a graph, generalizing the definition of Goulden et al. (2002), which only applies to trees; then we reprove their main result using our new definition. We in fact give two characterizations of the dual: a graph-theoretic one and an algebraic one, showing that they are in fact the same, and are also the same (on trees) as the topological dual developed by Goulden and Yong. Goulden and Yong use their dual to define a bijection between the vertex labeled trees and the factorizations of the permutation into transpositions, showing that their bijection has a particular structural property. We give a combinatorial, non-topological, proof of their result, using our dual.

Recently, Andrews and Merca obtained two identities concerning the number of even parts in all partitions of n into distinct parts. In this paper, we provide bijective proofs of these two results. [ABSTRACT FROM AUTHOR]

We consider a sequence of four variable polynomials by refining Stieltjes's continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections, we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a byproduct, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several companion permutation statistics for two bistatistics in a conjecture of Baril and Kirgizov. [ABSTRACT FROM AUTHOR]

In the spirit of Göllnitz's "big" partition theorem of 1967, we present a new mod-6 partition identity. Alladi et al. provided a four-parameter refinement of Göllnitz's big theorem in 1995 via a key identity of generating functions and the method of weighted words. By means of this technique, two similar mod-6 identities of this type were discovered—one by Alladi in 1999 and one by Alladi and Andrews in 2015. We finish the picture by presenting and proving the fourth and final possible mod-6 identity in this spirit. Furthermore, we provide a complete generalization of mod-n identities of this type. Finally, we apply a similar argument to generalize an identity of Alladi et al. from 2003. [ABSTRACT FROM AUTHOR]

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Andrews ['Binary and semi-Fibonacci partitions', J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi- $m$ -Fibonacci partitions of  $n$ and the set of partitions of  $n$ into powers of  $m$ in which all parts appear with multiplicity not divisible by  $m$. We also give a new characterisation of semi- $m$ -Fibonacci partitions and some congruences satisfied by the associated number sequence. [ABSTRACT FROM AUTHOR]

The study of numerical solutions for nonlinear equations is one of the trending areas of research in pure and applied mathematics. As it is applied to several areas such as optimization, mathematical modelling, operation research and so on. This research work compared three methods (Bijection, Secant and Newton-Raphson) for solving nonlinear equations. It was observed that secant may converge faster than Bisection method, while Newton-Raphson method converge faster than secant method. Therefore, we concluded that Newton-Raphson method is the most efficient and fastest method for solving nonlinear equation than secant method and Bijection method because it converges at the three alterations. [ABSTRACT FROM AUTHOR]