402 KB22 KB200820252020Nakamigawa, Tomokiletnik:18številka:2str. 381-391ISSN:1855-3966COBISSID_HOST:42357763URN:URN:NBN:SI:doc-6VFZ6PM1enUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneachord diagramchord expansionGenocchi numberGenocchijevo številoSeidel triangleSeidelov trikotniktetivni diagramtetivni razpletThe expansion of a chord diagram and the Genocchi numbers|A chord diagram ?$E$? is a set of chords of a circle such that no pair of chords has a common endvertex. Let ?$v_1, v_2, \dots, v_{2n}$? be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram ?$\{v_1v_{n+1}, v_2v_{n+2}, \dots, v_nv_{2n}\}$? is called an ?$n$?-crossing and a chord diagram ?$\{v_1v_2, v_3v_4, \dots, v_{2n-1}v_{2n}\}$? is called an ?$n$?-necklace. For a chord diagram ?$E$? having a 2-crossing ?$S = \{x_1x_3, x_2x_4\}$?, the expansion of ?$E$? with respect to ?$S$? is to replace ?$E$? with ?$E_1 = (E \setminus S) \cup \{x_2x_3, x_4x_1\}$? or ?$E_2 = (E \setminus S) \cup \{x_1x_2, x_3x_4\}$?. Beginning from a given chord diagram ?$E$? as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let ?$\mathcal{NCD}(E)$? be the multiset of the leaves. In this paper, the multiplicity of an ?$n$?-necklace in ?$\mathcal{NCD}(E)$? is studied. Among other results, it is shown that the multiplicity of an ?$n$?-necklace generated from an ?$n$?-crossing equals the Genocchi number when ?$n$? is odd and the median Genocchi number when ?$n$? is evenTetivni diagram ?$E$? je množica tetiv kroga, v kateri noben par tetiv nima skupnega krajišča. Naj bo ?$v_1, v_2, \dots, v_{2n}$? zaporedje točk, urejenih v smeri urinega kazalca vzdolž oboda kroga. Tetivni diagram ?$\{v_1v_{n+1}, v_2v_{n+2}, \dots, v_nv_{2n}\}$? se imenuje ?$n$?-križišče, tetivni diagram ?$\{v_1v_2, v_3v_4, \dots, v_{2n-1}v_{2n}\}$? pa je ?$n$?-ogrlica. Naj bo ?$E$? tetivni diagram, ki ima 2-križišče ?$S = \{x_1x_3, x_2x_4\}$?; potem se zamenjava ?$E$? z ?$E_1 = (E \setminus S) \cup \{x_2x_3, x_4x_1\}$? ali z ?$E_2 = (E \setminus S) \cup \{x_1x_2, x_3x_4\}$? imenuje razplet ?$E$? glede na ?$S$?. Če začnemo z danim tetivnim diagramom ?$E$? kot korenom, potem pa delamo tetivne razplete na oba načina, dobimo dvojiško drevo, katerega listi so izključno tetivni diagrami brez križišč. Naj bo ?$\mathcal{NCD}(E)$? mnogotera množica listov tega drevesa. V tem članku preučujemo večkratnost ?$n$?-ogrlice v mnogoteri množici ?$\mathcal{NCD}(E)$?. Poleg drugih rezultatov, ki jih dobimo, pokažemo tudi, da je večkratnost ?$n$?-ogrlice, dobljene iz ?$n$?-križišča, enaka Genocchijevemu številu, če je ?$n$? liho število, in sredinskemu Genocchijevemu številu, če je ?$n$? sodo številoTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije