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Naštevanje vzponov in spustov, dolin in vrhov v Catalanovih besedah
No full textWe provide generating functions, formulas, and asymptotic expressions for the number of Catalan words based on the number of runs of ascents (descents), runs of weak ascents (descents), l-valleys, valleys, symmetric valleys, l-peaks, peaks, and symmetric peaks. We also establish some bijections with restricted Dyck paths and ordered trees that transports some statistics
Dvodelni (1,1, k)-mešani grafi
No full textMixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs, in terms of the number of vertices, are presented for small diameters. Moreover, two infinite families of such graphs with diameter k and number of vertices of the order of 2k/2 are proposed, one of them being totally regular (1,1)-mixed graphs. In addition, we present two more infinite families called chordal ring and chordal double ring mixed graphs, which are bipartite and related to tessellations of the plane. Finally, we give an upper bound that improves the Moore bound for bipartite mixed graphs for r = z = 1
