42 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
Theorems, Problems and Conjectures
These notes are designed to offer some (perhaps new) codicils to related
work, a list of problems and conjectures seeking (preferably) combinatorial
proofs. The main items are Eulerian polynomials and hook/contents of Young
diagram, mostly on the latter. The new additions include items on Frobenius
theorem and multi-core partitions; most recently, some problems on (what we
call) colored overpartitions. Formulas analogues to or in the spirit of works
by Han, Nekrasov-Okounkov and Stanley are distributed throughout. Concluding
remarks are provided at the end in hopes of directing the interested
researcher, properly.Comment: 14 page
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 of the form for some integer , and are parametrized by a binary word that
encodes some periodicity conditions at infinity. Aztec diamond and pyramid
partitions correspond respectively to and to the limit case
. For each word 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
Rademacher-Type Formulas for Partitions and Overpartitions
A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of Ï4_0 | Ï_â1 is presented
Rademacher-Type Formulas for Restricted Partition and Overpartition Functions
A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes
Combinatorial and Asymptotic Statistical Properties of Partitions and Unimodal Sequences
Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit shapes, based in part on a method of Petrov. We also mention a few notable corollaries---for example, we obtain a limit shape for so-called overpartitions\u27\u27 by a simple DeSalvo-Pak-type transfer.
To aid in the proof of these limit shapes, we prove an asymptotic formula for the number of partitions of the integer n into distinct parts where the largest part is at most t times the square root of n for fixed t. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres\u27 asymptotic formula for distinct parts partitions when instead the number of parts is bounded by t times the square root of n. The probabilistic approach is equivalent to a circle method/saddle-point method calculation, but it makes some of the steps more intuitive and even predicts the shape of the asymptotic formula, to some degree.
Finally, motivated by certain problems concerning Rogers-Ramanujan-type identities, we give combinatorial proofs of three families of inequalities among certain types of integer partitions
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
Asymptotics, Equidistribution and Inequalities for Partition Functions
This thesis consists of three research projects on asymptotics, equidistribution properties and inequalities for partition and overpartition functions. We start by proving that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for large enough, the two quantities are different, and that which of the two is bigger depends on the parity of By doing so, we answer a conjecture formulated by Bringmann and Mahlburg (2012). We continue by placing this problem in a broader context and by proving that the same results are true for partitions into any powers. For this, we invoke an estimate on Gauss sums found by Banks and Shparlinski (2015) using the effective lower bounds on center density from the sphere packing problem established by Cohn and Elkies (2003). Finally, we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions, and we show that the number of overpartitions of with rank congruent to modulo is equidistributed with respect to as for any In addition, we prove some inequalities between ranks of overpartitions recently conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018)