Hook formulas for skew shapes III. Multivariate and product formulas

Abstract

We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series [MPP1, MPP2]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.

On the largest Kronecker and Littlewood–Richardson coefficients

“…We refer to a survey article [AR] for exact formulas (cf. [MPP2]), and to [MPP1] for the upper and lower bounds. 7.2.…”

Complexity Problems in Enumerative Combinatorics

“…is the superfactorial [158, A000178]. This is a special case of two large series of shapes recently discovered in [98] and [128]. One can also view SYT(λ) as linear extensions of the corresponding poset, and in this case it also can be derived from the Selberg integral [98].…”

Asymptotics of principal evaluations of Schubert polynomials for layered permutations

“…By analogy with the tilings, one can ask if u(n) satisfies some sort of super-multiplicativity property. Formally, let w ⊗ 1 c denote the Kronecker product permutation of size cn, whose permutation matrix equals the Kronecker product of the permutation matrix P w and the identity I c (see [MPP1]).…”

“…In fact, Prop. 6.5 in [MPP1] gives explicit constructions of large families of permutations w ∈ S n , for which log Υ w = Θ(n). These permutations are very far from being layered (in the transposition distance), suggesting that if true, proving Conjecture 1.4 might not be easy.…”

“…2.4.7]. The first constructive lower bound was given by the authors in [MPP1,§6], where the asymptotics of Υ w was computed for several families of permutations. Notably, for a permutation…”