Identities of Rothe-Abel- Schläfli-Hurwitz-type

AbstractSeveral convolution identities, containing many free parameters, are shown to follow in a very simple way from a combinatorial construction. By specialization of the parameters one can find many of the known generalizations or variations of Abel's generalization of the binomial theorem, including those obtained by Rothe, Schläfli, and Hurwitz. A convolution identity related to Mellin's expansion of algebraic functions, proposed recently by Louck (but contained in equivalent form in earlier work by Raney and Mohanty), and a counting formula for labelled trees by rising edges, due to Kreweras, are also shown to follow from the general approach

A Combinatorial Miscellany: Antipodes, Parking Cars, and Descent Set Powers

In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results depend upon the base expansion of both the dimension and the power of these statistics. Finally, we inspect the ƒ-vector of the descent polytope DPv, proving a maximization result using an analogue of the boustrophedon transform

Trees, Forests, and Total Positivity: I. qq-Trees and qq-Forests Matrices

We consider matrices with entries that are polynomials inqarising from naturalq-generalisations of two well-known formulas that count: forests onnvertices withkcomponents; and rooted labelled trees onn+ 1 vertices wherekchildren of the rootare lower-numbered than the root. We give a combinatorial interpretation of thecorresponding statistic on forests and trees and show, via the construction of vari-ous planar networks and the Lindstr ̈om-Gessel-Viennot lemma, that these matricesare coefficientwise totally positive. We also exhibit generalisations of the entriesof these matrices to polynomials ineightindeterminates, and present some conjec-tures concerning the coefficientwise Hankel-total positivity of their row-generatingpolynomials

Approximating Nash social welfare under rado valuations

We consider the problem of approximating maximum Nash social welfare (NSW) while allocating a set of indivisible items to n agents. The NSW is a popular objective that provides a balanced tradeoff between the often conflicting requirements of fairness and efficiency, defined as the weighted geometric mean of the agents' valuations. For the symmetric additive case of the problem, where agents have the same weight with additive valuations, the first constant-factor approximation algorithm was obtained in 2015. Subsequent work has obtained constant-factor approximation algorithms for the symmetric case under mild generalizations of additive, and O(n)-approximation algorithms for subadditive valuations and for the asymmetric case. In this paper, we make significant progress towards both symmetric and asymmetric NSW problems. We present the first constant-factor approximation algorithm for the symmetric case under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems, including as special cases assignment (OXS) valuations and weighted matroid rank functions. Furthermore, our approach also gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant

Exact duality transformations for sigma models and gauge theories

We present an exact duality transformation in the framework of Statistical Mechanics for various lattice models with non-Abelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie group G with global GxG-symmetry (the chiral model) and with variables in coset spaces G/H and a global G-symmetry (for example, the non-linear O(N) or RP^N models) in any dimension d>=1. It is also available for lattice gauge theories with local gauge symmetry in dimensions d>=2 and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory. The duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model. Transformations are available for the partition function, for expectation values of fundamental variables (correlators and generalized Wilson loops) and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition functions (free energies of dislocations, vortices or monopoles). Whereas the original models are formulated in terms of compact Lie groups G and H, coset spaces G/H and integrals over them, the configurations of the dual model are given in terms of representations and intertwiners of G. They are spin networks and spin foams. The partition function of the dual model describes the group theoretic aspects of the strong coupling expansion in a closed form.Comment: 57 pages, 15 figures, LaTeX; v2: references update