Hook formulas for skew shapes III. Multivariate and product formulas

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 "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. 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.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317) where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10 added, resembles published versio

Hook formulas for skew shapes

International audienceThe celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction

On the qq-Enumeration of Barely Set-Valued Tableaux and Plane Partitions

Barely set-valued tableaux are a variant of Young tableaux in which one box contains two numbers as its entry. It has recently been discovered that there are product formulas enumerating certain classes of barely set-valued tableaux. We give some q-analogs of these product formulas by introducing a version of major index for these tableaux. We also give product formulas and q-analogs for barely set-valued plane partitions. The proofs use several probability distributions on the set of order ideals of a poset, depending on the real parameter q > 0, which we think could be of independent interest.Comment: 38 pages, 6 tables, 3 figure

On the Okounkov-Olshanski formula for standard tableaux of skew shapes

The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two $q$-analogues for reverse plane partitions, which complements work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.Comment: 37 pages, 7 figures, v2 has a shorter proof of Lemma 8.10 and updated reference