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Martin Klazar

Publications and source records attributed to Martin Klazar.

At least 37 records · Page 2Linked to original sources
arXiv2017

Complex derivatives are continuous - three self-contained proofs. Part 1

We prove in three ways the basic fact of analysis that complex derivatives are continuous. The first, classical, proof of Cauchy and Goursat uses integration. The second proof of Whyburn and Connell is topological and is based on the winding number. The third proof of Adel'son-Vel'skii and Kronrod employs graphs of real bivariate functions. We give short and concise presentation of the first proof (in textbooks it often spreads over dozens of pages). In the second and third proof we on the contrary fill in and expand omitted steps and auxiliary results. This part 1 presents the first two proofs. The third proof, treated in the future part 2, takes one on a tour through theorems and results due to Jordan, Brouwer, Urysohn, Tietze, Vitali, Sard, Fubini, Komarevsky, Young, Cauchy, Riemann and others.

math.CV↗
arXiv2013

The Potts model and chromatic functions of graphs

The $U$-polynomial of Noble and Welsh is known to have intimate connections with the Potts model as well as with several important graph polynomials. For each graph $G$, $U(G)$ is equivalent to Stanley's symmetric bad colouring polynomial $XB(G)$. Moreover Sarmiento established the equivalence between $U$ and the polychromate of Brylawski. Loebl defined the $q$-dichromate $B_q(G,x,y)$ as a function of a graph $G$ and three independent variables $q,x,y$, proved that it is equal to the partition function of the Potts model with variable number of states and with a certain external field contribution, and conjectured that the $q$-dichromate is equivalent to the $U$-polynomial. He also proposed a stronger conjecture on integer partitions. The aim of this paper is two-fold. We present a construction disproving Loebl's integer partitions conjecture, and we introduce a new function $B_{r,q}(G;x,k)$ which is also equal to the partition function of the Potts model with variable number of states and with a (different) external field contribution, and we show that $B_{r,q}(G;x,k)$ is equivalent to the $U$-polynomial and to Stanley's symmetric bad colouring polynomial.

math.CO↗
arXiv2012

Polymath's combinatorial proof of the density Hales-Jewett theorem

This is an exposition of the combinatorial proof of the density Hales--Jewett theorem, due to D.\,H.\,J. Polymath in 2012. The theorem says that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,k\}^n$ with $|A|\ge\de k^n$ contains a combinatorial line. It implies Szemerédi's theorem, which claims that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,n\}$ with $|A|\ge\de n$ contains a $k$-term arithmetic progression.

math.CO↗
arXiv2011

On the theorem of Duminil-Copin and Smirnov about the number of self-avoiding walks in the hexagonal lattice

This is an exposition of the theorem from the title, which says that the number of self-avoiding walks with n steps in the hexagonal lattice has asymptotics (2cos(pi/8))^{n+o(n)}. We lift the key identity to formal level and simplify the part of the proof bounding the growth constant from below. In our calculation the lower bound comes from an identity asserting that a linear combination of 288 generating functions counting self-avoiding walks in a certain domain by length, final edge direction and winding number modulo 48 equals the geometric series 2cos(pi/8)x + (2cos(pi/8))^2x^2 + ... .

math.CO↗
arXiv2008

Overview of some general results in combinatorial enumeration

This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicity of numbers of MSOL-definable structures.

math.CO↗
arXiv2007

Generalizations of Khovanskii's theorems on growth of sumsets in abelian semigroups

We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $χ$ is a coloring of the lattice points in the orthant such that the color $χ(a+b)$ depends only on the colors $χ(a)$ and $χ(b)$, then the number of colors of the lattice points in the dilation $nP$ of $P$ is for large $n$ given by a polynomial (or, for rational $P$, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanski\uı on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanski\uı's theorem. Another result of Khovanski\uı states that the size of the image of a finite set after $n$ applications of mappings from a finite family of mutually commuting mappings is for large $n$ a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem.

math.CO↗
arXiv2007

On growth rates of permutations, set partitions, ordered graphs and other objects

For classes O of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order cont (containment of permutations, subgraph relation etc.), we investigate restrictions on the function f(n) counting objects with size n in a lower ideal in (O, cont). We present a framework of edge P-colored complete graphs (C(P), cont) which includes many of these situations, and we prove for it two such restrictions (jumps in growth): f(n) is eventually constant or f(n) >= n for all n>0; f(n) 0 for a constant c>0 or f(n) >= F_n for all n>0, F_n being the Fibonacci numbers. This generalizes a fragment of a more detailed theorem of Balogh, Bollobas and Morris on hereditary properties of ordered graphs.

math.CO↗
arXiv2006

Note on the number of proper colorings of a graph

We present a simpler proof of a bound on the number of proper colorings of a graph that was obtained recently by Liu and Murty using Tur'an sieve (in fact, we prove a stronger inequality). We also point out that these results are subsumed in a stronger result due to Lazebnik in 1990.

math.CO↗
arXiv2006

Counting Keith numbers

A Keith number is a positive integer N with the decimal representation a_1a_2...a_n such that n>=2 and N appears in the sequence (K_m) given by the recurrence K_1=a_1,...,K_n=a_n and K_m=K_{m-1}+K_{m-2}+...+K_{m-n} for m>n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a_1=a_2=...=a_n), and that the set of Keith numbers is of asymptotic density zero.

math.NT↗
arXiv2006

On the maximal order of numbers in the "factorisatio numerorum" problem

Let m(n) be the number of ordered factorizations of n in factors larger than 1. We prove that for every eps>0 n^{rho} m(n) < exp[(log n)^{1/rho}/(loglog n)^{1+eps}] holds for all integers n>n_0, while, for a constant c>0, n^{rho} m(n) > exp[c(log n)^{1/ρ}/(loglog n)^{1/rho}] holds for infinitely many positive integers n, where rho=1.72864... is the real solution to zeta(rho)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).

math.NT↗
arXiv2006

Non-Holonomicity of Sequences Defined via Elementary Functions

We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results; e.g., non-holonomicity of the logarithmic sequence is extended to rational functions involving $\log n$. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at odd integers is not holonomic.

math.CO↗
arXiv2005

Extensions of the linear bound in the Furedi-Hajnal conjecture

We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1's in an n by n 0-1 matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Branden and Mansour.We then extend the original Furedi-Hajnal problem from ordinary matrices to d-dimensional matrices and show that the number of 1's in a d-dimensional 0-1 matrix with side length n which avoids a d-dimensional permutation matrix is O(n^{d-1}).

math.CO↗
arXiv2005

On identities concerning the numbers of crossings and nestings of two edges in matchings

Let M,N be two matchings on [2n]={1, 2, ..., 2n} (possibly M=N) and for a nonnegative integer l let T(M,l) be the set of those matchings on [2n+2l] which can be obtained from M by successively adding l times in all ways the first edge, and similarly for T(N,l). Let s,t in {cr,ne} where cr is the statistic of the number of crossings (in a matching) and ne is the statistic of the number of nestings (possibly s=t). We prove that if the statistics s and t coincide on the sets of matchings T(M,l) and T(N,l) for l=0,1, they must coincide on these sets for every l >= 0; similar identities hold for the joint statistic of cr and ne. These results are instances of a general identity in which crossings and nestings are weighted by elements from an abelian group.

math.CO↗
arXiv2004

Counting set systems by weight

Applying the enumeration of sparse set partitions, we show that the number of set systems H such that the emptyset is not in H, the total cardinality of edges in H is n, and the vertex set of H is {1, 2, ..., m}, equals (1/log(2)+o(1))^nb_n where b_n is the n-th Bell number. The same asymptotics holds if H may be a multiset. If vertex degrees in H are restricted to be at most k, the asymptotics is (1/alpha_k+o(1))^nb_n where alpha_k is the unique root of x^k/k!+...+x^1/1!-1 in (0,1].

math.CO↗