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- W4213087551 abstract "A new object is introduced into the theory of partitions that generalizes plane partitions: cylindric partitions. We obtain the generating function for cylindric partitions of a given shape that satisfy certain row bounds as a sum of determinants of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-binomial coefficients. In some special cases these determinants can be evaluated. Extending an idea of Burge (J. Combin. Theory Ser. A <bold>63</bold> (1993), 210–222), we count cylindric partitions in two different ways to obtain several known and new summation and transformation formulas for basic hypergeometric series for the affine root system <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A overTilde Subscript r> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>widetilde A_{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we provide new and elementary proofs for two <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A overTilde Subscript r> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>widetilde A_{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> basic hypergeometric summation formulas of Milne (Discrete Math. <bold>99</bold> (1992), 199–246)." @default.
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- W4213087551 date "1997-01-01" @default.
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- W4213087551 title "Cylindric Partitions" @default.
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- W4213087551 doi "https://doi.org/10.1090/s0002-9947-97-01791-1" @default.
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