Advanced Problems: 6514-6517

Integral factorial ratios: irreducible examples with height larger than 1

“…We will make repeated use of these observations below. Indeed, these observations immediately rule out the possibility that both b and c are binomial coefficients 7. Completing the proof of theorem 1.2 We have already shown that, apart from finitely many exceptions, all primitive lists with height 2 and norm at most 1/3 + δ must lie in one of the 28 families catalogued in §3.…”

“…Finally, let us mention some other examples of integral factorial ratios with height larger than 1. In a Monthly problem, Askey [7] gives the two parameter, height 2 family [3…”

Integral factorial ratios: Irreducible examples with height larger than 1

“…Finally, let us mention some other examples of integral factorial ratios with height larger than 1. In a Monthly problem, Askey [1] gives the two parameter, height 2 family…”

“…This forces N(a 1 ) + N(a 2 ) ≤ 1/6 + 2δ, which implies that a 1 and a 2 must be [1] or [1, −2]. Since a has height 2, we are further forced to have a 1 = a 2 = [1], but now we must have x 1 = −x 2 in order to have s(a) = 0, and the resulting a has height 0. Therefore s(a i ) = 0 for i = 1, 2, 3, and the proof of the proposition is complete.…”

A Proof of the $G_2 $ Case of Macdonald’s Root System-Dyson Conjecture

“…Since F of the theorem is obviously with integer coefficients, our theorem implies the not entirely obvious fact that C(m, n) is an integer, thus solving Askey's problem [2].…”