Related papers: A geometric proof of the Brenti--Welker identity

We give a combinatorial proof of an identity that involves Eulerian numbers and was obtained algebraically by Brenti and Welker (2009). To do so, we study alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the…

Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this…

We show that a binomial identity arising in the context of the study of series expansions of $1/\pi$ can be seen as an incarnation of Whipples second theorem for hypergeometric series.

We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon's identity is re-established by constructing an infinite family of non-pure simplicial complexes $\Delta(n)$, indexed by the…

In this paper, we prove that the theta divisor of a smooth hyperelliptic curve has a natural and explicit embedded resolution of singularities using iterated blowups of Brill-Noether subvarieties. We also show that the Brill-Noether…

We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.

We give another proof for the (-1)-enumeration of self-complementary plane partitions with at least one odd side-length by specializing a certain Schur function identity. The proof is analogous to Stanley's proof for the ordinary…

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to…

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional…

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…

Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular…

For any odd $k$, a connection is established between the dihedral and supersymmetric extensions of the Tremblay-Turbiner-Winternitz Hamiltonians $H_k$ on a plane. For this purpose, the elements of the dihedral group $D_{2k}$ are realized in…

Identities between Whittaker and modified Bessel functions are derived for particular complex orders. Certain polynomials appear in such identities, which satisfy a fourth order differential equation (not of hypergeometric type), and they…

We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.

For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in…

The N=2 supersymmetric extension of the Schr\"odinger-Hamiltonian with 1/r-potential in d dimension is constructed. The system admits a supersymmetrized Laplace-Runge-Lenz vector which extends the rotational SO(d) symmetry to a hidden…

We construct the (enhanced Rogers) dilogarithm function from the spin Chern-Simons invariant of C*-connections. This leads to geometric proofs of basic dilogarithm identities and a geometric context for other properties, such as the…

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}(k[\Delta])$ of the Stanley-Reisner ring $k[\Delta]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the…

In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.

In a recent paper (arXiv:2301.09744), Erickson and Hunziker consider partitions in which the arm-leg difference is an arbitrary constant $m$. In previous works, these partitions are called $(-m)$-asymmetric partitions. Regarding these…