The Catalan number Cn enumerates parenthesizations of x0 * · · · * xn where * is a binary operation. We introduce the modular Catalan number C k,n to count equivalence classes of parenthesizations of x0 * · · · * xn when * satisfies a k-associative law generalizing the usual associativity. This leads to a study of restricted families of Catalan objects enumerated by C k,n with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for C k,n with two different proofs. For each n ≥ 0 we compute the largest size of k-associative equivalence classes and show that the number of classes with this size is a Catalan number.
Formulating a Schubert problem as solutions to a system of equations in either Plücker space or local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's α-theory.
Abstract. We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.
Abstract. We establish a congruence modulo four in the real Schubert calculus on the Grassmannian of m-planes in 2m-space. This congruence holds for fibers of the Wronski map and a generalization to what we call symmetric Schubert problems. This strengthens the usual congruence modulo two for numbers of real solutions to geometric problems. It also gives examples of geometric problems given by fibers of a map whose topological degree is zero but where each fiber contains real points.
Abstract. We describe a large-scale computational experiment studying structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems whose computed numbers of real solutions variously exhibit nontrivial upper bounds, lower bounds, gaps, and a congruence modulo four. We present a family of Schubert problems, one in each Grassmannian, and prove that their real osculating instances have the observed lower bounds and gaps.
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