286 research outputs found
Plane overpartitions and cylindric partitions
Generating functions for plane overpartitions are obtained using various
methods such as nonintersecting paths, RSK type algorithms and symmetric
functions. We extend some of the generating functions to cylindric partitions.
Also, we show that plane overpartitions correspond to certain domino tilings
and we give some basic properties of this correspondence.Comment: 42 pages, 11 figures, corrected typos, revised parts, figures
redrawn, results unchange
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
Random sampling of plane partitions
This article presents uniform random generators of plane partitions according
to the size (the number of cubes in the 3D interpretation). Combining a
bijection of Pak with the method of Boltzmann sampling, we obtain random
samplers that are slightly superlinear: the complexity is in
approximate-size sampling and in exact-size sampling
(under a real-arithmetic computation model). To our knowledge, these are the
first polynomial-time samplers for plane partitions according to the size
(there exist polynomial-time samplers of another type, which draw plane
partitions that fit inside a fixed bounding box). The same principles yield
efficient samplers for -boxed plane partitions (plane partitions
with two dimensions bounded), and for skew plane partitions. The random
samplers allow us to perform simulations and observe limit shapes and frozen
boundaries, which have been analysed recently by Cerf and Kenyon for plane
partitions, and by Okounkov and Reshetikhin for skew plane partitions.Comment: 23 page
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
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
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