A q-RIOUS POSITIVITY

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-factorials that happen to be polynomials. The fact that the binomial coefficients (1) ( n m ) = n! (n−m)!m! are integers easily follows from the following arithmetic argument. The order in which a prime p enters n! is given by (2) ordp n! = ⌊ n p ⌋ + ⌊ n p2 ⌋ + ⌊ n p3 ⌋ + · · · , where b · c is the integer-part function. Setting x = (n − m)/p and y = m/p in the inequality bx+ yc − bxc − byc ≥ 0, and summing k over the positive integers, we see that ordp ( n m ) ≥ 0 for any prime p. This obviously implies that ( n m ) ∈ Z. A standard way to establish integrality purely combinatorially amounts to interpreting the factorial ratio in (1) as coefficients in the expansion