213 research outputs found
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
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
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
The Selberg integral and Young books
The Selberg integral is an important integral first evaluated by Selberg in
1944. Stanley found a combinatorial interpretation of the Selberg integral in
terms of permutations. In this paper, new combinatorial objects "Young books"
are introduced and shown to have a connection with the Selberg integral. This
connection gives an enumeration formula for Young books. It is shown that
special cases of Young books become standard Young tableaux of various shapes:
shifted staircases, squares, certain skew shapes, and certain truncated shapes.
As a consequence, product formulas for the number of standard Young tableaux of
these shapes are obtained.Comment: 13 pages, 11 figure
Perfect sampling algorithm for Schur processes
We describe random generation algorithms for a large class of random
combinatorial objects called Schur processes, which are sequences of random
(integer) partitions subject to certain interlacing conditions. This class
contains several fundamental combinatorial objects as special cases, such as
plane partitions, tilings of Aztec diamonds, pyramid partitions and more
generally steep domino tilings of the plane. Our algorithm, which is of
polynomial complexity, is both exact (i.e. the output follows exactly the
target probability law, which is either Boltzmann or uniform in our case), and
entropy optimal (i.e. it reads a minimal number of random bits as an input).
The algorithm encompasses previous growth procedures for special Schur
processes related to the primal and dual RSK algorithm, as well as the famous
domino shuffling algorithm for domino tilings of the Aztec diamond. It can be
easily adapted to deal with symmetric Schur processes and general Schur
processes involving infinitely many parameters. It is more concrete and easier
to implement than Borodin's algorithm, and it is entropy optimal.
At a technical level, it relies on unified bijective proofs of the different
types of Cauchy and Littlewood identities for Schur functions, and on an
adaptation of Fomin's growth diagram description of the RSK algorithm to that
setting. Simulations performed with this algorithm suggest interesting limit
shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints
present in v2
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.Comment: figure added, plus several minor change
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