Lost in Translation: A Reflection on the Ballot Problem and André's Original Method

Transcription

1 Lost in Trnsltion: A Reflection on the Bllot Prolem nd André's Originl Method Mrc Renult Shippensurg University Presented t MthFest August 5, 2007 The Bllot Prolem (1887) In how mny wys cn upsteps nd downsteps e ordered so tht no step ends on or elow the x-xis? GOOD = 8 = 6 BAD 1

2 The numer of good pths is Joseph Bertrnd Bertrnd sked Is there direct proof? Désiré André (1887) Solves the llot prolem! And mthemticins celerte! Désiré André Tody, the most fmous solution to the llot prolem is André s Reflection Method 2

3 Numer of good pths = Numer of good pths from (1, 1) to T. T Terminl point T hs coordintes (+, -) T Trick: count the numer of d pths from (1,1) to T. T Bd pths from (1,1) to T T All pths from (1, -1) to T 3

4 Numer of good pths from (0,0) = Numer of good pths from (1,1) = [ Totl numer of pths from (1,1) ] - [ Numer of d pths from (1,1) ] = [ Totl numer of pths from (1,1) ] reflection - [ Totl numer of pths from (1,-1) ] Totl # of pths from (1,1): - 1 upsteps downsteps Totl: 1 Totl # of pths from (1,-1): upsteps - 1 downsteps Totl: 1 4

5 Solution to the Bllot Prolem: 1 1 The celerted reflection method of André MthWorld I.P. Goulden nd Luis G. Serrno, Mintining the Spirit of the Reflection Principle when the Boundry hs Aritrry Integer Slope, J. Comintoril Theory (A) 104 (2003) André gve direct geometric ijection etween the suset of d pths nd the set A of ll pths from (1, -1) to (m, n), nd the result then follows immeditely J.H. Vn Lint nd R.M. Wilson, A Course in Comintorics, Cmridge University Press, p. 151: The reflection principle of Fig ws used y the French comintorilist D. André ( ) in his solution of Bertrnd s fmous llot prolem I. Krtzs nd S.E. Shreve, Brownin Motion nd Stochstic Clculus, Springer, They write Here is the rgument of Désiré André nd proceed with the reflection method. H. Buer, Proility Theory, Wlter de Gruyter, Berlin, New York, p. 231: In the literture, this reflection principle is usully ttriuted to D. André ( ). It occurs in the form of such geometric rgument in André [1887]. P. Hilton nd J. Pedersen, Ctln numers, their generliztions, nd their uses, Mth. Intelligencer. 13 (1991) D. Stnton nd D. White, Constructive Comintorics, Springer-Verlg, New York, D. Zeilerger, André s reflection proof generlized to the mny-cndidte llot prolem, Discrete Mthemtics 44 (1983)

6 The celerted reflection method of André MthWorld I.P. Goulden nd Luis G. Serrno, Mintining the Spirit of the Reflection Principle when the Boundry hs Aritrry Integer Slope, J. Comintoril Theory (A) 104 (2003) André gve direct geometric ijection etween the suset of d pths nd the set A of ll pths from (1, -1) to (m, n), nd the result then follows immeditely J.H. Vn Lint nd R.M. Wilson, A Course in Comintorics, Cmridge University Press, p. 151: The reflection principle of Fig ws used y the French comintorilist D. André ( ) in his solution of Bertrnd s fmous llot prolem I. Krtzs nd S.E. Shreve, Brownin Motion nd Stochstic Clculus, Springer, They write Here is the rgument of Désiré André nd proceed with the reflection method. H. Buer, Proility Theory, Wlter de Gruyter, Berlin, New York, p. 231: In the literture, this reflection principle is usully ttriuted to D. André ( ). It occurs in the form of such geometric rgument in André [1887]. P. Hilton nd J. Pedersen, Ctln numers, their generliztions, nd their uses, Mth. Intelligencer. 13 (1991) D. Stnton nd D. White, Constructive Comintorics, Springer-Verlg, New York, D. Zeilerger, André s reflection proof generlized to the mny-cndidte llot prolem, Discrete Mthemtics 44 (1983) The prolem is A Recent Discovery André never used the reflection method! Wht André did: 1. Count # d llot permuttions. 2. Sutrct tht from the totl # of permuttions to get # of good permuttions. How André counted d outcomes 6

7 André s Actul Method Bllots re mrked with A or B. Two ctegories of d llot permuttions: Those tht strt with A Those tht strt with B Next slide Esy: every permuttion strting with B is d. There re ( 1) of these. Clim: # of d permuttions strting with A = # of ll permuttions with A s nd ( 1) B s. A A B B A B A A Given d permuttion strting with A Find the first d B Remove it Exchnge the two prts Done! A A B A B A A A B A A A A B A B A A A A B Now reverse the process 7

8 Clim: # of d permuttions strting with A = # of ll permuttions with A s nd ( 1) B s. Given permuttion with A s nd ( 1) B s A B A A A A B A B A A A A B A A B A B A A Scn from right until A s exceed B s (y 1). Exchnge the two prts Insert B Done! Thus ( 1) A A B B A B A A ds strt with A Bd permuttions: Those tht strt with A Those tht strt with B 2 ( 1) Good llot permuttions: ( 1) 2 No geometry No reflection (trnsposing A s nd B s) 8

9 The Generlized Bllot Prolem Fix positive integer k. How mny pths with 1-unit upsteps nd k-unit downsteps hve no step ending on or elow the x-xis? k k = 3 The reflection method does not generlize. André s originl method does! k = 3. Clssify d pths: B 0, B 1, B 2, B 3. A pth in B 0 A pth in B 1 A pth in B 2 A pth in B 3 9

10 For ritrry k we crete B 0, B 1, B 2,, B k. Fct: These sets ll hve the sme size! By André s find-d-step-remove-it-exchnge-twosides trick, ech set hs size ( 1) Thus, the numer of d pths is Thus, the numer of good pths is ( 1) ( k 1) ( 1) ( k 1) k Concluding Thoughts So where nd when did the reflection method originte? Aely 1923? 1915? When did André strt getting credit for the reflection method? 1950 s? Erlier? Lost (nd Found) in Trnsltion: André s Actul Method nd its Appliction to the Generlized Bllot Prolem 10