÷ƒ’À; è TeX output 2006.04.17:1555‹ ÿÿÿÿ ¢K ýEÌ&‘:9ïcolor push Blackïhtml:ïcolor push gray 0ï color popï html:Ž’ÕÁGï color popŽŽ œ3Ú‘:9 ý|Ì&ïhtml:ï html:Ÿ.¸M‘Tvkïcolor push Blackï color popŽŽ‘ZVgï5PSfile=logo129.eps llx=0 lly=0 urx=99 ury=16 rwi=2880ŽŽŽŽŸA©4‘/9ïcolor push Blackï color popŽŽ‘5þóÂÖN ½q cmbx12»A–Ç
Relation“Bet–ÿZw“een–Ç
Restricted“andŽŸº9‘#µUnrestricted–Ç
W‘þ=eigh›ÿZted“Motzkin“P˜athsŽŸ.sç’ ¾ÒóX«Q ff cmr12¼W–þÓ4en-jin‘³/W“oanŽ¤ ’ ±DßHo•›¼w“ard‘³/Univ“ersit“yŽ¡’ £¨UW‘þÓ4ashington,–³/DC“20059Ž¡’ ÜdUSAŽ¡’ Çï+html:ïcolor push cmyk 0 1 0 0ó߆µT ff cmtt12½wwoan@fac.howard.eduï html:ï color popŽŸ(C’ Ìévïcolor push Blackï color popŽŽ’ ÒÉróò"V
ó3
cmbx10¿AbstractŽŸƒ‘lÏïcolor push Black‘ðï color popŽŽ‘-Ì»óKñ`y
ó3
cmr10¾W‘ÿee–·jconsider“those“lattice“paths“that“use“the“steps“\up",–»«\levš²!el",“and–·j\do˜wn"“withŽ¤
™š‘_ìassigned›¬
w•²!eigh“ts˜ó b>
ó3
cmmi10ÂwKn;–Ó1b;“c¾.‘îÒIn˜probabilit•²!y˜theory‘ÿe,‘wthe˜total˜w“eigh“t˜is˜1.‘îÒIn˜com“binatorics,Ž¡‘_ìwš²!e–[replace“w˜eigh˜t“b˜y“the“n˜um˜bMÞer“of“colors.‘ûöHere“w˜e“giv˜e“a“com˜binatorial“proMÞof“of“aŽ¡‘_ìrelation›¦fbMÞet•²!w“een˜restricted˜and˜unrestricted˜w“eigh“ted˜Motzkin˜paths.ŽŸ9ïhtml:ï html:ŸÞïó*ÂÖN G® cmbx12Õ1Ž‘(ôIn‘ÿuÂtro‘ Š=ductionŽŸb#óX«Q cmr12¹W‘ÿVe–~Oconsider“those“lattice“paths“in“the“Cartesian“plane“starting“from“(0ó+·ág£ cmmi12Ö;‘ÿþ¹0)“that“use“the“stepsŽ¤€ ó,!",š
cmsy10×fÖUò¬;–ÿþL;“DSŽ×g¹,›t#where‘X¤ÖU‘Qk¹=‘‡(1Ö;“¹1),˜an‘X¤up-step,˜ÖL–‡¹=“(1Ö;‘ÿþ¹0),˜a–X¤levš¬rel-step“and“ÖD‘d¹=‘‡(1Ö;‘ÿþ× ¹1)Ö;“¹a“do˜wn-Ž¡step,›ª@with–P»assigned“w•¬reigh“ts–P»ÖwRª;‘ÿþb¹,˜and“Öc¹.‘ kIn“probabilit¬ry“theory‘ÿV,˜the“total“w•¬reigh“t–P»is“1.‘ kInŽ¡com•¬rbinatorics,‘È!w“e–›Öregard“w•¬reigh“t–›Öas“the“n•¬rum“bSŽer–›Öof“colors“and“normalize“b¬ry“setting“Öw‘Õ¹=‘‚æ1.Ž¡Let–C1ÖP‘ä÷¹bšSŽe“a“path.‘BzW‘ÿVe“dene“the“w•¬reigh“t–C1ÖwRª¹(ÖP‘¡Æ¹)“to“b˜e“the“pro˜duct“of“the“w•¬reigh“t–C1of“the“steps.Ž¡Let–9ÖA¹(Ön;‘ÿþkg¹)“bšSŽe“the“set“of“all“w•¬reigh“ted–9lattice“paths“ending“at“the“p˜oin¬rt“(Ön;›ÿþkg¹)Ö;“¹and“let“ÖM‘@ä¹(Ön;˜kg¹)Ž¡bšSŽe–ë¾the“set“of“lattice“paths“in“ÖA¹(Ön;‘ÿþkg¹)“that“nev¬rer“go“b˜elo¬rw“the“Öx¹-axis.‘