We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients in the special case of Kostka numbers.

Let di(m) denote the coefficients of the Boros-Moll polynomials. Moll’s minimum conjecture states that the sequence {i(i+1)(di (m)−di−1(m)di+1(m))}1≤i≤m attains its minimum at i = m with 2−2mm(m + 1) ( 2m m )2 . This conjecture is stronger than the log-concavity conjecture proved by Kauers and Paule. We give a proof of Moll’s conjecture by utilizing the spiral property of the sequence {di(m)}0≤...

We establish approximate log-concavity for a wide family of combinatorially defined integer-valued functions. Examples include the number of non-negative integer matrices (contingency tables) with prescribed row and column sums (margins), as a function of the margins and the number of integer feasible flows in a network, as a function of the excesses at the vertices. As a corollary, we obtain a...