Combinatorial proof
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Pros and cons of consensus algorithm proof of stake. Difference in the network safety in proof of work and proof of stake Open
The consensus algorithm is a mechanism that allows you to protect the network against attacks.The work of the algorithm is to provide rules that act on the network members.Proof of Work is one of the consensus algorithms based on the calcu…
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Combinatorial Calabi flows on surfaces Open
For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of the smooth Calabi flow. We prove that the solution to the combinatorial Calabi flow exists for all time and converges if and only if the Thurston…
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Maximum independent sets on random regular graphs Open
We determine the asymptotics of the independence number of the random d-regular graph for all ${d\\geq d_0}$ . It is highly concentrated, with constant-order fluctuations around ${n\\alpha_*-c_*\\log n}$ for explicit constants ${\\alpha_*(…
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Hook Formulas for Skew Shapes II. Combinatorial Proofs and Enumerative Applications Open
The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of …
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The Ramanujan–Dyson identities and George Beck’s congruence conjectures Open
Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions tha…
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Using Grassmann calculus in combinatorics: Lindström-Gessel-Viennot lemma and Schur functions Open
Grassmann (or anti-commuting) variables are extensively used in theoretical physics. In this paper we use Grassmann variable calculus to give new proofs of celebrated combinatorial identities such as the Lindström-Gessel-Viennot formula fo…
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Existence of unimodular triangulations — positive results Open
Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course, combinatorics. In this article, we review several classes of polytopes that do have unimodular triangulati…
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An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings Open
In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfe…
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Combinatorial Perspectives on the Crank and Mex Partition Statistics Open
Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistic, the mex, and Fro…
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On 2-Level Polytopes Arising in Combinatorial Settings Open
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-leve…
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Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions Open
This paper proves a combinatorial rule giving all maximal and minimal\npartitions $\\lambda$ such that the Schur function $s_\\lambda$ appears in a\nplethysm of two arbitrary Schur functions. Determining the decomposition of\nthese plethys…
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Eulerian Polynomials, Stirling Permutations of the Second Kind and Perfect Matchings Open
In this paper, we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles. Moreover, we get an expansion of the ordinary d…
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On the sum of parts with multiplicity at least 2 in all the partitions of n Open
In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is…
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Moments of moments of characteristic polynomials of random unitary\n matrices and lattice point counts Open
In this note we give a combinatorial and non-computational proof of the\nasymptotics of the integer moments of the moments of the characteristic\npolynomials of Haar distributed unitary matrices as the size of the matrix goes\nto infinity.…
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Combinatorial Flows and Their Normalisation Open
This paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic p…
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Combinatorial proofs of identities for the generalized Leonardo numbers Open
In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by \mathcal{L}_{k,n}, as well as derive some new formulas. To do so, we interpret \mathca…
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q-Stirling Identities Revisited Open
We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel q-Stir…
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A Topological Characterization of Modulo-<i>p</i> Arguments and Implications for Necklace Splitting Open
The classes PPA-p have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p. However, these classes were not known to have complete problems of a to…
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Efficiency of Learning from Proof Blocks Versus Writing Proofs Open
Proof Blocks is a software tool that provides students with a scaffolded proof-writing experience, allowing them to drag and drop prewritten proof lines into the correct order instead of starting from scratch. In this paper we describe a r…
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Untangling mechanized proofs Open
Proof assistants like Coq, Lean, or HOL4 rely heavily on stateful meta-programs called scripts to assemble proofs. Unlike pen-and-paper proofs, proof scripts only describe the steps to take (induct on x, apply a theorem, …), not the states…
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Combinatorial models for Schubert polynomials Open
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other kno…
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The problem of proof identity, and why computer scientists should care about Hilbert's 24th problem Open
In this short overview article, I will discuss the problem of proof identity and explain how it is related to Hilbert's 24th problem. I will also argue that not knowing when two proofs are ‘the same’ has embarrassing consequences not only …
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Notes on the combinatorial fundamentals of algebra Open
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is ent…
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A combinatorial proof of the extension property for partial isometries Open
We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.
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A new family of combinatorial numbers and polynomials associated with peters numbers and polynomials Open
The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties…
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Probability Models of Distributed Proof Generation for zk-SNARK-Based Blockchains Open
The paper is devoted to the investigation of the distributed proof generation process, which makes use of recursive zk-SNARKs. Such distributed proof generation, where recursive zk-SNARK-proofs are organized in perfect Mercle trees, was fo…
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How to encode a tree Open
We construct bijections giving three "codes" for trees. These codes follow naturally from the Matrix Tree Theorem of Tutte and have many advantages over the one produced by Prufer in 1918. One algorithm gives explicitly a bijection that is…
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Determinants of Toeplitz–Hessenberg matrices with generalized Fibonacci entries Open
In this paper, we evaluate several families of Toeplitz-Hessenberg matrices whose entries are generalized Fibonacci numbers.In particular, we find simple formulas for several determinants whose entries are translates of the Chebyshev polyn…
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Partitions with bounded differences between largest and smallest parts. Open
This is a freely-available open access publication.
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Combinatorial Auslander-Reiten quivers and reduced expressions Open
In this paper, we introduce the notion of combinatorial Auslander-Reiten(AR) quiver for commutation classes $[\widetilde{w}]$ of $w$ in finite Weyl group. This combinatorial object visualizes the convex partial order $\prec_{[\widetilde{w}…