13,693 research outputs found
The weighted hook length formula
Based on the ideas in [CKP], we introduce the weighted analogue of the
branching rule for the classical hook length formula, and give two proofs of
this result. The first proof is completely bijective, and in a special case
gives a new short combinatorial proof of the hook length formula. Our second
proof is probabilistic, generalizing the (usual) hook walk proof of
Green-Nijenhuis-Wilf, as well as the q-walk of Kerov. Further applications are
also presented.Comment: 14 pages, 4 figure
The weighted hook-length formula II: Complementary formulas
Recently, a new weighted generalization of the branching rule for the hook
lengths, equivalent to the hook formula, was proved. In this paper, we
generalize the complementary branching rule, which can be used to prove
Burnside's formula. We present three different proofs: bijective, via weighted
hook walks, and via the ordinary weighted branching rule.Comment: 20 pages, 9 figure
Quasimodularity and large genus limits of Siegel-Veech constants
Quasimodular forms were first studied in the context of counting torus
coverings. Here we show that a weighted version of these coverings with
Siegel-Veech weights also provides quasimodular forms. We apply this to prove
conjectures of Eskin and Zorich on the large genus limits of Masur-Veech
volumes and of Siegel-Veech constants.
In Part I we connect the geometric definition of Siegel-Veech constants both
with a combinatorial counting problem and with intersection numbers on Hurwitz
spaces. We introduce modified Siegel-Veech weights whose generating functions
will later be shown to be quasimodular.
Parts II and III are devoted to the study of the quasimodularity of the
generating functions arising from weighted counting of torus coverings. The
starting point is the theorem of Bloch and Okounkov saying that q-brackets of
shifted symmetric functions are quasimodular forms. In Part II we give an
expression for their growth polynomials in terms of Gaussian integrals and use
this to obtain a closed formula for the generating series of cumulants that is
the basis for studying large genus asymptotics. In Part III we show that the
even hook-length moments of partitions are shifted symmetric polynomials and
prove a formula for the q-bracket of the product of such a hook-length moment
with an arbitrary shifted symmetric polynomial. This formula proves
quasimodularity also for the (-2)-nd hook-length moments by extrapolation, and
implies the quasimodularity of the Siegel-Veech weighted counting functions.
Finally, in Part IV these results are used to give explicit generating
functions for the volumes and Siegel-Veech constants in the case of the
principal stratum of abelian differentials. To apply these exact formulas to
the Eskin-Zorich conjectures we provide a general framework for computing the
asymptotics of rapidly divergent power series.Comment: 107 pages, final version, to appear in J. of the AM
Combinatorial families of multilabelled increasing trees and hook-length formulas
In this work we introduce and study various generalizations of the notion of
increasingly labelled trees, where the label of a child node is always larger
than the label of its parent node, to multilabelled tree families, where the
nodes in the tree can get multiple labels. For all tree classes we show
characterizations of suitable generating functions for the tree enumeration
sequence via differential equations. Furthermore, for several combinatorial
classes of multilabelled increasing tree families we present explicit
enumeration results. We also present multilabelled increasing tree families of
an elliptic nature, where the exponential generating function can be expressed
in terms of the Weierstrass-p function or the lemniscate sine function.
Furthermore, we show how to translate enumeration formulas for multilabelled
increasing trees into hook-length formulas for trees and present a general
"reverse engineering" method to discover hook-length formulas associated to
such tree families.Comment: 37 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
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
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