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 nth 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 geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago (Good in Proc. Camb. Philos. Soc. 54:39–42, 1958; Lawden in Math. Gaz. 36:193–196, 1952; Moser and Zayachkowski in Scr. Math. 26:223–229, 1963). The polytopes we construct are closely related to the root polytopes introduced by Gelfand et al. (Arnold–Gelfand mathematical seminars: geometry and singularity theory, pp. 205–221. Birkhauser, Boston, 1996).

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