Proof of a conjecture of Kl\xf8ve on permutation codes under the Chebychev distance

Abstract

Let d be a positive integer and x a real number. Let A_{d, x} be a dtimes 2d matrix with its entries ai,j=xfor1⩽j⩽d+1-i,1ford+2-i⩽j⩽d+i,0ford+1+i⩽j⩽2d.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_{i,j}=\\left\\{ \\begin{array}{ll} x\\ \\ &{} \\quad \\text{ for } \\ 1\\leqslant j\\leqslant d+1-i,\\\\ 1\\ \\ &{} \\quad \\text{ for } \\ d+2-i\\leqslant j\\leqslant d+i,\\\\ 0\\ \\ &{} \\quad \\text{ for } \\ d+1+i\\leqslant j\\leqslant 2d. \\end{array} \\right. \\end{aligned}$$\\end{document}Further, let R_d be a set of sequences of integers as follows: Rd=(ρ1,ρ2,…,ρd)|1⩽ρi⩽d+i,1⩽i⩽d,andρr≠ρsforr≠s.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} R_d=\\left\\{ (\\rho _1, \\rho _2,\\ldots , \\rho _d)|1\\leqslant \\rho _i\\leqslant d+i, 1\\leqslant i \\leqslant d,\\ \\text{ and }\\ \\rho _r\\ne \\rho _s\\ \\quad \\text{ for }\\ r\\ne s\\right\\} . \\end{aligned}$$\\end{document}and define Ωd(x)=∑ρ∈Rda1,ρ1a2,ρ2…ad,ρd.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Omega _d(x)=\\sum _{\\rho \\in R_d}a_{1,\\rho _1}a_{2, \\rho _2}\\ldots a_{d,\\rho _d}. \\end{aligned}$$\\end{document}In order to give a better bound on the size of spheres of permutation codes under the Chebychev distance, Kløve introduced the above function and conjectured that Ωd(x)=∑m=0ddm(m+1)d(x-1)d-m.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Omega _d(x)=\\sum _{m=0}^d{d\\atopwithdelims ()m}(m+1)^d(x-1)^{d-m}. \\end{aligned}$$\\end{document}In this paper, we settle down this conjecture positively.