Enumeration of Standard Young Tableaux

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

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.Comment: Accepted to Advances in Applied Mathematics. Final versio

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

Checkerboard style Schur multiple zeta values and odd single zeta values

We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.Comment: 21 pages. Added Corollary 3.7 and the case (a,b)=(1,2) in Section

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

From Aztec diamonds to pyramids: steep tilings

We introduce a family of domino tilings that includes tilings of the Aztec diamond and pyramid partitions as special cases. These tilings live in a strip of $\mathbb{Z}^2$ of the form $1 \leq x-y \leq 2\ell$ for some integer $\ell \geq 1$, and are parametrized by a binary word $w\in\{+,-\}^{2\ell}$ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to $w=(+-)^\ell$ and to the limit case $w=+^\infty-^\infty$. For each word $w$ and for different types of boundary conditions, we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips, that admits a natural multivariate generalization. The main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism (which we slightly extend in order to handle Littlewood-type identities). In probabilistic terms our tilings map to Schur processes of different types (standard, Pfaffian and periodic). We also introduce a more general model that interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6, new improved proof of Proposition 11