Delannoy Orthants of Legendre Polytopes

Delannoy numbers and 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 quadrant. We thus provide a geometric interpr...

Central 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...

Central 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.

Links We Almost Missed Between Delannoy Numbers and Legendre Polynomials

Shifted 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...