V. Levenshtein　first　proposed　the　sequence　reconstruction　problem　in　2001.　This　problem　studies　the　same　sequence　from　some　set　is　transmitted　over　multiple　channels,　and　the　decoder　receives　the　different　outputs.　Assume　that　the　transmitted　sequence　is　at　distance　d　from　some　code　and　there　are　at　most　r　errors　in　every　channel.　Then　the　sequence　reconstruction　problem　is　to　find　the　minimum　number　of　channels　required　to　recover　exactly　the　transmitted　sequence　that　has　to　be　greater　than　the　maximum　intersection　between　two　metric　balls　of　radius　r,　where　the　distance　between　their　centers　is　at　least　d.　In　this　paper,　we　study　the　sequence　reconstruction　problem　of　permutations　under　the　Hamming　distance.　In　this　model　we　define　a　Cayley　graph　over　the　symmetric　group,　study　its　properties　and　find　the　exact　value　of　the　largest　intersection　of　its　two　metric　balls　for　d=2r.　Moreover,　we　give　a　lower　bound　on　the　largest　intersection　of　two　metric　balls　for　d=2r-1.　©　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature　2024.