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U interesu za istraživanjem nedovoljno istraženog, pogotovo na području Republike Hrvatske, u radu se iznosi tematika i obrađuje problematika jednog starog, ali još uvijek intrigantnog problema. Točnije, u radu je predstavljen i obrađen Dudeneyjev Haberdasherov problem kao problem dijeljenja jednakostraničnog trokuta na najmanji broj dijelova pri čemu se bez preokretanja dijelova slagalice može oblikovati kvadrat. Unatoč nebrojenim pokušajima rješavanja ovog problema tijekom 120 godina njegova postojanja, jedino do sada poznato valjano rješenje rješenje je njegova autora dano davne 1903. godine, godinu dana nakon predstavljanja problema. U radu se uz opis konstrukcije i opravdanje valjanosti Dudeneyjeva rješenja posebno razmatra i jedno rješenje koje se potkrijepljeno matematičkim dokazom pokazalo pogrešnim te se zapravo radi samo o dobroj aproksimaciji kvadrata. Razmotrena je i klasa rješenja dijeljenja trokuta na dijelove koji se daju oblikovati u pravokutnik među kojima je u jednom posebnom slučaju i kvadrat. Nadalje, navode se razlozi i primjeri zašto su ovakvi i slični problemi korisni i prikladni za korištenje na svim razinama obrazovanja. Dio rada posvećen je istraživanju temeljenom na Haberdasherovom problemu. Iznose se rezultati i detaljna analiza prikupljenih podataka provedenog istraživanja kojim su se ispitivali različiti aspekti Haberdasherova problema kao instrumenta. Tijekom istraživanja oblikovane su različite varijante zadatka prilagođene različitim grupama sudionika, In the interest of researching what is insufficiently researched, especially in the territory of the Republic of Croatia, this thesis paper presents and deals with the issue of an old but still intriguing problem. More precisely, in the thesis Dudeney's Haberdasher's problem is presented and processed as a problem of dividing an equilateral triangle into the smallest number of parts, whereby a square can be formed without turning over the pieces of the puzzle. Despite countless attempts to solve this problem during the 120 years of its existence, the only valid solution known so far is the solution given by its author back in 1903, one year after the problem was presented. In addition to the description of the construction and the justification of the validity of Dudeney's solution, the thesis paper also considers one solution which, supported by a mathematical proof, turned out to be wrong, and in fact it is only a good approximation of the square. A class of solutions for dividing a triangle into parts that can be shaped into a rectangle, including a square in one special case, was also considered. Furthermore, there are reasons and examples given for why these and similar problems are practical and suitable for use at all levels of education. Part of this thesis paper is devoted to research based on Haberdasher's problem. The conducted research examined different aspects of Haberdasher's problem as an instrument. The results and detailed analysis of the collected data are presented by different variants of the task that was adapted to separate groups of participants.

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued., 17 pages

Let Δ = { δ 1 , δ 2 , … , δ m } be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function D Δ ( z , u 1 , u 2 , … , u m ) , which counts polygon dissections and marks subgraph copies of δ i with the variable u i . We prove that this is always algebraic, through an explicit combinatorial decomposition depending on Δ . The decomposition also gives a defining system for D Δ ( z , 0 ) , which encodes polygon dissections that avoid these patterns as subgraphs. In this way, we are able to extract normal limit laws for the patterns when they are encoded, and perform asymptotic enumeration of the resulting classes when they are avoided. The results can be transferred to the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.

The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

In this paper we develop Gray codes for two families of geometric objects: non-crossing partitions and dissections of a convex polygon by means of non-crossing diagonals.

We show that certain canonical realizations of the complexes Hom(G,H) and Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of such projected Hom-complexes: the dissections of a convex polygon into k-gons, Postnikov's generalized permutohedra, staircase triangulations, the complex dual to the lower faces of a cyclic polytope, and the graph of weak compositions of an integer into a fixed number of summands., 23 pages, 5 figures; improved exposition; accepted for publication in JCTA

The (type-A) associahedron is a polytope related to polygon dissections which arises in several mathematical subjects. We propose a B-analogue of the associahedron. Our original motivation was to extend the analogies between type-A and type-B noncrossing partitions, by exhibiting a simplicial polytope whose h-vector is given by the rank-sizes of the type-B noncrossing partition lattice, just as the h-vector of the (simplicial type-A) associahedron is given by the Narayana numbers. The desired polytope Q^B"n is constructed via stellar subdivisions of a simplex, similarly to Lee's construction of the associahedron. As in the case of the (type-A) associahedron, the faces of Q^B"n can be described in terms of dissections of a convex polygon, and the f-vector can be computed from lattice path enumeration. Properties of the simple dual Q^B^*"n are also discussed and the construction of a space tessellated by Q^B^*"n is given. Additional analogies and relations with type A and further questions are also discussed.

For each value of k⩾2, we determine the number pn of ways of dissecting a polygon in the projective plane into n subpolygons with k+1 sides each. In particular, if k=2 we recover a result of Edelman and Reiner (1997) on the number of triangulations of the Möbius band having n labelled points on its boundary. We also solve the problem when the polygon is dissected into subpolygons of arbitrary size. In each case, the associated generating function ∑pnzn is a rational function in z and the corresponding generating function of plane polygon dissections. Finally, we obtain asymptotic estimates for the number of dissections of various kinds, and determine probability limit laws for natural parameters associated to triangulations and dissections.

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}., Comment: To appear in Adv. Appl. Math

We say that the sequence (a"n) is quasi-polynomial in n if there exist polynomials P"0,...,P"s"-"1 and an integer n"0 such that, for all n>=n"0, a"n=P"i(n) where i=n(mod s). We present several families of combinatorial objects with the following properties: Each family of objects depends on two or more parameters, and the number of isomorphism types of objects is quasi-polynomial in one of the parameters whenever the values of the remaining parameters are fixed to arbitrary constants. For each family we are able to translate the problem of counting isomorphism types of objects into the problem of counting integer points in a union of parametrized rational polytopes. The families of objects to which this approach is applicable include combinatorial designs, linear and unrestricted codes, and dissections of regular polygons.