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We provide a bijection between the set of factorizations, that is, ordered (n − 1)-tuples of transpositions in S n whose product is (12...n), and labelled trees on n vertices. We prove a refinement of a theorem of Dénes [3] that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a(More)
A radioallergosorbent assay (RAST) was developed and used to determine the levels of IgE antibodies to soluble antigens of Mycobacterium tuberculosis (BCG vaccine strain) in sera from patients with tuberculosis and leprosy and in healthy control subjects. Total IgE levels in the same sera were quantitated with a commercial radioimmunoassay kit. Patients(More)
We present a randomized approximation algorithm for counting contingency tables, m × n non-negative integer matrices with given row sums R = in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial N O(ln N) complexity, where N = r 1 + · · · + r m = c 1 + · · · + c n. Various classes of margins are smooth, e.g., when(More)
Fulton and Woodward [8] have recently identified the smallest degree of q that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result, and provide an alternative characterization of this smallest degree in terms of the rim hook formula [2] for(More)
Fulton's universal Schubert polynomials give cohomology formulas for a class of degener-acy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear combinations of products of stable Grothendieck polynomials. We prove an explicit combinatorial formula for the(More)
Buch and Fulton [7] conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono [21] proved this conjecture as an immediate consequence of their " component formula ". We present an alternative proof of the component formula by substituting combinatorics for Gröbner degeneration(More)