11,201 research outputs found
Bottom Schur functions
We give a basis for the space V spanned by the lowest degree part
\hat{s}_\lambda of the expansion of the Schur symmetric functions s_\lambda in
terms of power sums, where we define the degree of the power sum p_i to be 1.
In particular, the dimension of the subspace V_n spanned by those
\hat{s}_\lambda for which \lambda is a partition of n is equal to the number of
partitions of n whose parts differ by at least 2. We also show that a symmetric
function closely related to \hat{s}_\lambda has the same coefficients when
expanded in terms of power sums or augmented monomial symmetric functions.
Proofs are based on the theory of minimal border strip decompositions of Young
diagrams.Comment: 16 pages, 13 figures To appear in the Electronic Journal of
Combinatoric
Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
We study asymptotics of perfect matchings on a large class of graphs called
the contracting square-hexagon lattice, which is constructed row by row from
either a row of a square grid or a row of a hexagonal lattice. We assign the
graph periodic edge weights with period , and consider the
probability measure of perfect matchings in which the probability of each
configuration is proportional to the product of edge weights. We show that the
partition function of perfect matchings on such a graph can be computed
explicitly by a Schur function depending on the edge weights. By analyzing the
asymptotics of the Schur function, we then prove the Law of Large Numbers
(limit shape) and the Central Limit Theorem (convergence to the Gaussian free
field) for the corresponding height functions. We also show that the
distribution of certain type of dimers near the turning corner is the same as
the eigenvalues of Gaussian Unitary Ensemble, and that in the scaling limit
under the boundary condition that each segment of the bottom boundary grows
linearly with respect the dimension of the graph, the frozen boundary is a
cloud curve whose number of tangent points to the bottom boundary of the domain
depends on the size of the period, as well as the number of segments along the
bottom boundary
Row-strict quasisymmetric Schur functions
Haglund, Luoto, Mason, and van Willigenburg introduced a basis for
quasisymmetric functions called the quasisymmetric Schur function basis,
generated combinatorially through fillings of composition diagrams in much the
same way as Schur functions are generated through reverse column-strict
tableaux. We introduce a new basis for quasisymmetric functions called the
row-strict quasisymmetric Schur function basis, generated combinatorially
through fillings of composition diagrams in much the same way as Schur
functions are generated through row-strict tableaux. We describe the
relationship between this new basis and other known bases for quasisymmetric
functions, as well as its relationship to Schur polynomials. We obtain a
refinement of the omega transform operator as a result of these relationships.Comment: 17 pages, 11 figure
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