We propose a new definition of effective formulas for problems in enumerative combinatorics. We outline the proof of the fact that every linear recurrence sequence of integers has such a formula. It follows from a lower bound that can be deduced from the Skolem-Mahler-Lech theorem and the Subspace Theorem. We will give details of this deduction that is due to P. Corvaja in the full version of this extended abstract.

Martin Klazar

We develop a theory of real numbers as rational Cauchy sequences, in which any two of them, $(a_n)$ and $(b_n)$, are equal iff $\lim\,(a_n-b_n)=0$. We need such reals in the Countable Mathematical Analysis ([4]) which allows to use only hereditarily at most countable (HMC) sets.

Martin Klazar

We show how to prove by means of the Lagrange inversion the limit of Arnol'd that $$ \lim_{x\to0}\frac{\sin(\tan x)-\tan(\sin x)}{\arcsin(\arctan x)-\arctan(\arcsin x)}=1\,. $$ In fact, we obtain a more general result in terms of formal power series.

Martin Klazar

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}).

Martin Klazar Adam Marcus

We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n.log n upper bound on the ordered graph extremal function of F=({1,3}, {1,5}, {2,3}, {2,4}) due to Z. Furedi to the n.(log n)^2.(loglog n)^3 upper bound in the hypergraph case. We use Davenport-Schinzel sequences to derive almost linear upper bounds in terms of the inverse Ackermann function. We obtain such upper bounds for the extremal functions of forests consisting of stars whose all centers precede all leaves.

Martin Klazar

We investigate extremal functions ex_e(F,n) and ex_i(F,n) counting maximum numbers of edges and maximum numbers of vertex-edge incidences in simple hypergraphs H which have n vertices and do not contain a fixed hypergraph F; the containment respects linear orderings of vertices. We determine both functions exactly if F has only distinct singleton edges or if F is one of the 55 hypergraphs with at most four incidences (we give proofs only for six cases). We prove some exact formulae and recurrences for the numbers of hypergraphs, simple and all, with n incidences and derive rough logarithmic asymptotics of these numbers. Identities analogous to Dobinski's formula for Bell numbers are given.

Martin Klazar

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].

Martin Klazar

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.

Martin Klazar

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 Szemer\'edi'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.

Martin Klazar

We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval. We review in its framework in detail two derivations of the Stirling formula. The first approximates log(1)+log(2)+...+log(n) with an integral and the second uses the classical gamma function and a Fubini-type result. We mention two more integral representations of n!.

Martin Klazar