Martin Klazar

On abab -free and abba -free set partitions

These are partitions of[l]={1, 2, . . . , l}intonblocks such that no four-term subsequence of [l]induces the mentioned pattern and eachkconsecutive numbers of[l]fall into different blocks. These structures are motivated by Davenport–Schinzel sequences. We summarize and extend known enumeriative results for the patternp=ababand give an explicit formula for the numberp(abab, n, l, k)of such partitions. Our main tools are generating functions. We determine the corresponding generating function forp=abbaandk=1, 2, 3.Fork=2there is a connection with the number of directed animals. We solve exactly two related extremal problems.

Bell numbers, their relatives, and algebraic differential equations

We prove that the ordinary generating function of Bell numbers satisfies no algebraic differential equation over C(x) (in fact, over a larger field). We investigate related numbers counting various set partitions (the Uppuluri-Carpenter numbers, the numbers of partitions with j mod i blocks, the Bessel numbers, the numbers of connected partitions, and the numbers of crossing partitions) and prove for their ogfs analogous results. Recurrences, functional equations, and continued fraction expansions are derived.

Generalized Davenport-Schinzel sequences

The extremal functionEx(u, n) (introduced in the theory of Davenport-Schinzel sequences in other notation) denotes for a fixed finite alternating sequenceu=ababa... the maximum length of a finite sequencev overn symbols with no immediate repetition which does not containu. Here (following the idea of J. Nešetřil) we generalize this concept for arbitrary sequenceu. We summarize the already known properties ofEx(u, n) and we present also two new theorems which give good upper bounds onEx(ui,n). We use these theorems to describe a wide class of sequencesu (“linear sequences”) for whichEx(u, n)=O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems aboutEx(u, n).

Counting Pattern-free Set Partitions I

A partition u of k= {1, 2,? , k } is contained in another partition v of l if l has a k -subset on which v inducesu . We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of the Stanley?Wilf conjecture is proposed. We prove that the generating function (GF) counting v is rational if (i)R is finite and the number of parts of v is fixed or if (ii)u has only singleton parts and at most one doubleton part. In fact, (ii) is an application of (i). As another application of (i) we prove that for each k the GF counting partitions with k pairs of crossing parts belongs toZ(1 ? 4 x).

Twelve Countings with Rooted Plane Trees

The average numbers of (1) antichains, (2) maximal antichains, (3) chains, (4) infima closed sets, (5) connected sets, (6) independent sets, (7) maximal independent sets, (8) brooms, (9) matchings, (10) maximal matchings, (11) linear extensions and (12) drawings in (of) a rooted plane tree onnvertices are investigated. Using generating functions we determine the asymptotics and give some explicit formulae and identities. In conclusion, we discuss the extremal values of the above quantitites and pose some problems.

Exact Algorithms for L (2,1)-Labeling of Graphs

The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O*((k+1)n) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k=4, where the running time of our algorithm is O(1.3006n). Furthermore we show that dynamic programming can be used to establish an O(3.8730n) algorithm to compute an optimal L(2,1)-labeling.

The Füredi-Hajnal Conjecture Implies the Stanley-Wilf Conjecture

We show that the Stanley-Wilf enumerative conjecture on permutations follows easily from the Furedi-Hajnal extremal conjecture on 0–1 matrices. We apply the method, discovered by Alon and Friedgut, that derives an (almost) exponential bound on the number of some objects from a (almost) linear bound on their sizes. They proved by it a weaker form of the Stanley-Wilf conjecture. Using bipartite graphs, we give a simpler proof of their result.

Generalized Davenport-Schinzel sequences with linear upper bound

In 1965 Davenport and Schinzel posed the problem of determining the maximum length of a sequence on II letters with no immediate repetition of the same letter, not containing any subsequence of type ababa (i.e., the occurrences of two letters can give no configuration of type a . . . b . . . a . . . b * . . a). Originally this problem arose as a combinatorial problem connected with differential equations [ 1,2]. Later Davenport-Schinzel sequences were studied by Hart and Sharir and connections with path compression algorithms in combinatorics were discovered [3], as well as further applications in combinatorial geometry, [4,5]. See [7,8] for a related work. In 1986 an upper bound for the maximum length was found, at the same time it was verified this result could not be improved. The upper bound is n&(n), where o(n) stands for the functional inverse of the Ackermann function [3]. Later Komjath found a direct construction for the lower bound. Considering more general subwords of type abab . . . of length s + 2 instead of just ababa he proved the lower bound n&(n) [9]. This paper deals with a natural extension of the original problem. We study sequences not containing a given forbidden word (i.e., subsequence) on generally more than two letters. Complete characterization of forbidden words with a linear upper bound on two letters is given. Further results concerning this extension are described in [lo].

Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings

The number conn counts matchings X on {1, 2, ....., 2n}, which are partitions into n two-element blocks, such that the crossing graph of X is connected. Similarly, cron counts matchings whose crossing graph has no isolated vertex. (If it has no edge, Catalan numbers arise.) We apply generating functions techniques and prove, using a more generally applicable criterion, that the sequences (conn) and (cron) are not P-recursive. On the other hand, we show that the residues of conn and cron modulo any fixed power of 2 can be determined P-recursively. We consider also the numbers scon of symmetric connected matchings. Unfortunately, their generating function satisfies a complicated differential equation which we cannot handle.

On trees and noncrossing partitions

Abstract We give a simple and natural proof of (an extension of) the identity P(k, l, n) = P2(k − 1, l − 1, n − 1). The number P(k, l, n) counts noncrossing partitions of {1, 2, …,l} into n parts such that no part contains two numbers x and y, 0