Related papers: Catalan numbers and Schubert polynomials for $w=1(…

The solution to the general univariate polynomial equation has been sought for centuries. It is well known there is no general solution in radicals for degrees five and above. The hyper-Catalan numbers $C[m_2,m_3,m_4,\ldots]$ count the ways…

It is well known that the number of tilting modules over a path algebra of type A_n coincides with the Catalan number C(n). Moreover, the number of support tilting modules of type A_n is the Catalan number C(n+1). We show that the convex…

We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types B, C, D which naturally extends the family of Lascoux of Schutzenberger in type A. These polynomials satisfy positivity, orthogonality, and stability properties,…

We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials, and more generally the $q$-Schr\"oder polynomials. By interpreting them as $\mathfrak{sl}_2$-characters, we show that the…

We give a direct straightforward proof that there is an isomorphism between the algebra of the Catalan monoid C_n that is, the monoid of all order-preserving, weakly increasing self-maps f of [n] = {1,...,n}, over any commutative ring with…

We give a new formula for double Grothendieck polynomials based on Magyar's orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials $\mathfrak S_w(\mathbf x;\mathbf y)$. We also prove a…

We show that the Eulerian-Catalan numbers enumerate Dyck permutations. We provide two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analogue of…

We prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ${n \choose 2}!$ for both the code weights and the Chevalley weights. We also define weights which…

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic:…

Weighted Catalan numbers are a class of weighted sums over Dyck paths. Well-studied for their arithmetic properties and applications to enumerative combinatorics, these numbers were recently generalized to the setting of $k$-dimensional…

Combining the "method of restriction equations" of Rim\'anyi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities $A_3: ({\bf C}^{\bullet},0)\to…

We give a simple formula for some determinants, and an analogous formula for pfaffians, both of which are polynomial identities. The second involve some expressions that interpolate between determinants and pfaffians. We give several…

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial…

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we…

We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded…

For the Schur polynomials bounded and unbounded generalizations of the Cauchy identities are found.

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…

Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate…

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron…

Many $W$-algebras (e.g. the $W_N$ algebras) are consistent for all values of the central charge except for a discrete set of exceptional values. We show that such algebras can be contracted to new consistent degenerate algebras at these…