Delannoy numbers and Legendre polytopes
نویسنده
چکیده
We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the n-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. Résumé. No construisons un polytope de dimension n dont le complexe de bord est comprimé et dont les nombres de faces dans toute triangulation “en tirant des sommets” sont les coefficients des puissances de (x − 1)/2 dans le n-ième polynôme de Legendre. Nous montrons que les nombres centraux de Delannoy comptent toutes les faces dans la triangulation “en tirant des sommets” en ordre lexicographique qui contiennent un point dans un certain quadrant ouvert. Ainsi nous produisons une interprétation géometrique d’une rélation entre les nombres de Delannoy centraux et les polynômes de Legendre, notée il y a 50 ans. Nos polytopes sont reliés intimément aux polytopes de racines introduits par Gelfand, Graev, et Postnikov.
منابع مشابه
Delannoy Orthants of Legendre Polytopes
We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the n-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open generalized orthant. We thus provide a geomet...
متن کاملCentral Delannoy Numbers and Balanced Cohen- Macaulay Complexes
We introduce a new join operation on colored simplicial complexes that preserves the Cohen-Macaulay property. An example of this operation puts the connection between the central Delannoy numbers and Legendre polynomials in a wider context.
متن کاملCentral Delannoy numbers, Legendre polynomials, and a balanced join operation preserving the Cohen-Macaulay property
We introduce a new join operation on colored simplicial complexes that preserves the CohenMacaulay property. An example of this operation puts the connection between the central Delannoy numbers and Legendre polynomials in a wider context. Résumé. Nous introduisons une nouvelle opération qui joint des complexes simpliciaux équilibrés d’une telle manière que la propriété de Cohen-Macaulay est pr...
متن کاملShifted Jacobi Polynomials and Delannoy Numbers
We express a weighted generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [12, 17, 19], to all Delannoy numbers and certain Jacobi polynomials. Another specialization provides a weighted lattice path enumeration model for shifte...
متن کامل