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The q-binomial coefficients [ n m ] = ∏m i=1(1 − qn−m+i)/(1 − q), for integers 0 ≤ m ≤ n, are known to be polynomials with non-negative integer coefficients. This readily follows from the q-binomial theorem, or the many combinatorial interpretations of [ n m ] . In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of q-facto...
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We present here a proof that a certain rational function Cn(q, t) which has come to be known as the “q, t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in the J. Algebraic Combin. 5 (1996), no. 3, 191–244, where it is further conjectured that Cn(q, t) is the Hilbert Series of the Diagona...
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We study pfaffian analogues of immanants, which we call pfaffinants. Our main object is the TL-pfaffinants which are analogues of Rhoades and Skandera’s TL-immanants. We show that TL-pfaffinants are positive when applied to planar networks and explain how to decompose products of complementary pfaffians in terms of TL-pfaffinants. We conjecture in addition that TL-pfaffinants have positivity pr...
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ژورنال
عنوان ژورنال: Aequationes mathematicae
سال: 2010
ISSN: 0001-9054,1420-8903
DOI: 10.1007/s00010-010-0055-9