Egecioglu, ii. and J.B. Remmel, Brick tabloids and the connection matrices between bases of symmetric functions, Discrete Applied Mathematics 34 (1991) 107-120. Let H, denote the space of symmetric functions, homogeneous of degree n. In this paper we introduce a new set of combinatorial objects called I-brick tabloids and its variants, which we use *to give combinatorial interpretations of the ...

We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N , which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M .

Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph is independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

For any set representation (permutation representation) of the symmetric group Sn, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.

Finding Nash and Bayes-Nash equilibria in games is a hard problem both analytically and computationally. We restrict out attention to symmetric Bayes-Nash equilibria in auctions and propose a computational method that takes advantage of the symmetry of equilibria and structure of auction games. The method is iterated best-response where all players move simultaneously. We present experimental r...

We characterize the “best” model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct “best” model geometries in an appropriate sense–these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The sec...

In [9], the present authors and Richard O’Malley showed that in order for a function be universally polygonally approximable it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ − (1− ε) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = S∞ n=1 En, such that each Ei is closed and 1-sy...

We study random unitary channels which reproduce the action of the twirling channel corresponding to the representation of the symmetric group on an n-fold tensor product. We derive upper and lower bounds on the randomness cost of implementing such a map which depend exponentially on the number of systems. Consequently, symmetric twirling can be regarded as a reasonable Shannontheoretic protoco...

In this paper we construct the analogue of Dedekind η−function on the moduli space of polarized CY manifolds. We prove that the L norm of η(τ ) is the regularized determinants of the Laplacians of the CY metric on (0, 1) forms. We construct the analogue of the Baily-Borel Compactification of the moduli space of polarized CY and prove that it has the same properties as the Baily-Borel compactifi...

Let hλ, eλ, and mλ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition λ respectively. We give combinatorial interpretations for the coefficients that arise in expanding mλ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of cer...