Normal ordering coefficients in q-deformed generalized Ore algebras are interpreted as mixed rook and file placements on staircase and Laguerre boards, with extensions to polynomial Weyl algebras introducing associated q-deformed Stirling and Lah numbers.
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Rook theory, normal ordering in the $q$-deformed Ore algebra and the polynomial generalization
Normal ordering coefficients in q-deformed generalized Ore algebras are interpreted as mixed rook and file placements on staircase and Laguerre boards, with extensions to polynomial Weyl algebras introducing associated q-deformed Stirling and Lah numbers.