20.500.12556/RUL-184467 Gaussovi binomski koeficienti delo diplomskega seminarja Gaussian binomial coefficients V delu obravnavamo Gaussove binomske koeficiente oziroma $q$-binomske koeficiente kot posplošitev navadnih binomskih koeficientov. Predstavimo nekaj osnovnih definicij $q$-kombinatorike in zapišemo nekaj lastnosti in identitet Gaussovih binomskih koeficientov, skupaj s $q$-binomskim izrekom. Velik del je namenjen kombinatoričnim interpretacijam Gaussovih binomskih koeficientov. Pokažemo njihovo povezavo s particijami in Youngovimi diagrami, mrežnimi potmi, inverzijami permutacij ter binarnimi nizi. Nato s pomočjo teh kombinatoričnih interpretacij izpeljemo rekurzivni formuli za Gaussove binomske koeficiente in nekatere druge klasične identitete. Ukvarjamo pa se tudi z algebraičnim pomenom Gaussovih binomskih koeficientov in dokažemo, da preštevajo $k$-dimenzionalne podprostore končnega vektorskega prostora $\mathbb{F}_q^n$. To interpretacijo povežemo tudi s stolpčnimi kanoničnimi oblikami matrik. Nazadnje predstavimo končno projektivno geometrijo in njene lastnosti ter pokažemo, kako primer $q=1$ vodi do običajnih binomskih koeficientov in Boolove algebre podmnožic. In this work, we study Gaussian binomial coefficients, also known as $q$-binomial coefficients, as a generalization of the ordinary binomial coefficients. We introduce some basic definitions from $q$-combinatorics and present several properties and identities of Gaussian binomial coefficients, together with the $q$-binomial theorem. A large part is devoted to combinatorial interpretations of Gaussian binomial coefficients. We establish their connections with partitions and Young diagrams, lattice paths, permutation inversions, and binary words. Using these combinatorial interpretations, we derive the recurrence relations for Gaussian binomial coefficients as well as several other classical identities. We also investigate the algebraic significance of Gaussian binomial coefficients and prove that they enumerate the $k$-dimensional subspaces of the finite vector space $\mathbb{F}_q^n$. Furthermore, we relate this interpretation to echelon matrices. Finally, we introduce finite projective geometry and some of its properties, and show how the case $q=1$ leads to the ordinary binomial coefficients and the Boolean algebra of subsets. Gaussovi binomski koeficienti vektorski podprostor projektivna geometrija Gaussian binomial coefficients vector subspace projective geometry true false false Slovenski jezik Angleški jezik Diplomsko delo/naloga 2026-07-08 08:15:07 2026-07-08 08:15:17 2026-07-17 10:23:06 0000-00-00 00:00:00 2026 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.1 160608 285209859 22037.pdf 22037.pdf 1 19AD147BABFAE4871894655D76E59D4F 04e614b5712971bf16f923312bf2c5839d072af243f12888f29360a3125e674b 45178df2-7a94-11f1-9b0d-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=237296 Fakulteta za matematiko in fiziko 0 0 0