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PDEs and combinatorial problems Label patterns On-line selection Urn models Occurrences of exactly solvable PDEs in combinatorial problems Alois Panholzer Institut f ur Diskrete Mathematik und Geometrie TU Wien, Austria

  1. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Problems for random trees/mappings, where labels play essential rˆ ole 3 7 6 9 1 1 17 8 14 11 10 2 16 5 9 3 15 5 13 19 12 4 7 14 2 11 6 18 12 13 8 10 4 Occurrence and avoidance of label-patterns in trees/mappings: Runs Records Alternating mappings 6 / 25

  2. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings n -mappings: functions f from set [ n ] := { 1 , 2 , . . . , n } into itself: f : [ n ] → [ n ] Functional digraph G f of f : G f = ( V f , E f ), with V f = [ n ] and E f = { ( i , f ( i )) , i ∈ [ n ] } Simple structure: connected components of G f : cycles of trees 7 / 25

  3. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings n -mappings: functions f from set [ n ] := { 1 , 2 , . . . , n } into itself: f : [ n ] → [ n ] Functional digraph G f of f : G f = ( V f , E f ), with V f = [ n ] and E f = { ( i , f ( i )) , i ∈ [ n ] } 3 9 6 1 17 8 11 10 2 16 15 5 13 19 7 14 18 12 4 Simple structure: connected components of G f : cycles of trees 7 / 25

  4. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings n -mappings: functions f from set [ n ] := { 1 , 2 , . . . , n } into itself: f : [ n ] → [ n ] Functional digraph G f of f : G f = ( V f , E f ), with V f = [ n ] and E f = { ( i , f ( i )) , i ∈ [ n ] } 3 9 6 1 17 8 11 10 2 16 15 5 13 19 7 14 18 12 4 Simple structure: connected components of G f : cycles of trees 7 / 25

  5. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Interesting example for avoiding a label patterns”: Alternating mappings: labels on each iteration path are up-down alternating sequence: i = f 0 ( i ) < f 1 ( i ) > f 2 ( i ) < f 3 ( i ) > · · · or i = f 0 ( i ) > f 1 ( i ) < f 2 ( i ) > f 3 ( i ) < · · · f 2 ( i ) − f ( i ) � � Equivalently: · ( f ( i ) − i ) < 0 for all i Corresponding quantity for permutations (“labelled line”): alternating permutations enumerated by zig-zag numbers (tangent and secant numbers) 8 / 25

  6. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Interesting example for avoiding a label patterns”: Alternating mappings: labels on each iteration path are up-down alternating sequence: i = f 0 ( i ) < f 1 ( i ) > f 2 ( i ) < f 3 ( i ) > · · · or i = f 0 ( i ) > f 1 ( i ) < f 2 ( i ) > f 3 ( i ) < · · · f 2 ( i ) − f ( i ) � � Equivalently: · ( f ( i ) − i ) < 0 for all i Corresponding quantity for permutations (“labelled line”): alternating permutations enumerated by zig-zag numbers (tangent and secant numbers) 8 / 25

  7. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Interesting example for avoiding a label patterns”: Alternating mappings: labels on each iteration path are up-down alternating sequence: i = f 0 ( i ) < f 1 ( i ) > f 2 ( i ) < f 3 ( i ) > · · · or i = f 0 ( i ) > f 1 ( i ) < f 2 ( i ) > f 3 ( i ) < · · · f 2 ( i ) − f ( i ) � � Equivalently: · ( f ( i ) − i ) < 0 for all i Corresponding quantity for permutations (“labelled line”): alternating permutations enumerated by zig-zag numbers (tangent and secant numbers) 8 / 25

  8. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Interesting example for avoiding a label patterns”: Alternating mappings: labels on each iteration path are up-down alternating sequence: i = f 0 ( i ) < f 1 ( i ) > f 2 ( i ) < f 3 ( i ) > · · · or i = f 0 ( i ) > f 1 ( i ) < f 2 ( i ) > f 3 ( i ) < · · · f 2 ( i ) − f ( i ) � � Equivalently: · ( f ( i ) − i ) < 0 for all i Corresponding quantity for permutations (“labelled line”): alternating permutations enumerated by zig-zag numbers (tangent and secant numbers) 8 / 25

  9. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Structure: zig-zag property on each path in functional graph 3 18 11 2 12 8 7 15 1 13 9 17 10 6 5 4 16 19 14   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 f : ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓   5 17 2 5 4 4 17 15 17 19 15 10 4 19 10 17 10 1 16 local minima alternate with local maxima : avoiding { up-up, down-down } 9 / 25

  10. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Structure: zig-zag property on each path in functional graph 3 18 11 2 12 8 7 15 1 13 9 17 10 6 5 4 16 19 14   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 f : ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓   5 17 2 5 4 4 17 15 17 19 15 10 4 19 10 17 10 1 16 local minima alternate with local maxima : avoiding { up-up, down-down } 9 / 25

  11. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Structure: zig-zag property on each path in functional graph 3 18 11 2 12 8 7 15 1 13 9 17 10 6 5 4 16 19 14   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 f : ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓   5 17 2 5 4 4 17 15 17 19 15 10 4 19 10 17 10 1 16 local minima alternate with local maxima : avoiding { up-up, down-down } 9 / 25

  12. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Combinatorial decomposition: w.r.t. largest labelled node n T 1 T 2 T ′ T r T 1 n C : or T r C ′ n ⇒ treatment requires auxiliary parameter: number of local minima ⇒ require also treatment of quantity for rooted labelled trees (Cayley trees) 10 / 25

  13. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Combinatorial decomposition: w.r.t. largest labelled node n T 1 T 2 T ′ T r T 1 n C : or T r C ′ n ⇒ treatment requires auxiliary parameter: number of local minima ⇒ require also treatment of quantity for rooted labelled trees (Cayley trees) 10 / 25

  14. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Combinatorial decomposition: w.r.t. largest labelled node n T 1 T 2 T ′ T r T 1 n C : or T r C ′ n ⇒ treatment requires auxiliary parameter: number of local minima ⇒ require also treatment of quantity for rooted labelled trees (Cayley trees) 10 / 25

  15. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Corresponding problem for trees: Alternating trees (= intransitive trees): enumerative studies: Postnikov [1997] n T n = 1 � n � � k n − 1 number of rooted alternating trees: 2 n k k =0 occurs in combinatorial studies of hyperplane arrangements: Postnikov and Stanley [2000] Kuba and P. [2010]: studies of alternating tree families via combinatorial decomposition w.r.t. largest node using auxiliary parameter “number of local minima” 11 / 25

  16. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Corresponding problem for trees: Alternating trees (= intransitive trees): enumerative studies: Postnikov [1997] n T n = 1 � n � � k n − 1 number of rooted alternating trees: 2 n k k =0 occurs in combinatorial studies of hyperplane arrangements: Postnikov and Stanley [2000] Kuba and P. [2010]: studies of alternating tree families via combinatorial decomposition w.r.t. largest node using auxiliary parameter “number of local minima” 11 / 25

  17. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Generating functions treatment: trees: quasi-linear first order PDE F z ( z , u ) = uF u ( z , u ) e F ( z , u ) + u connected mappings: linear first order PDE C z ( z , u ) = uC u ( z , u ) e F ( z , u ) + uF u ( z , u ) e F ( z , u ) mappings: set of connected mappings M ( z , u ) = e C ( z , u ) 12 / 25

  18. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Generating functions treatment: trees: quasi-linear first order PDE F z ( z , u ) = uF u ( z , u ) e F ( z , u ) + u connected mappings: linear first order PDE C z ( z , u ) = uC u ( z , u ) e F ( z , u ) + uF u ( z , u ) e F ( z , u ) mappings: set of connected mappings M ( z , u ) = e C ( z , u ) 12 / 25

  19. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Generating functions treatment: trees: quasi-linear first order PDE F z ( z , u ) = uF u ( z , u ) e F ( z , u ) + u connected mappings: linear first order PDE C z ( z , u ) = uC u ( z , u ) e F ( z , u ) + uF u ( z , u ) e F ( z , u ) mappings: set of connected mappings M ( z , u ) = e C ( z , u ) 12 / 25

  20. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Generating functions treatment: trees: quasi-linear first order PDE F z ( z , u ) = uF u ( z , u ) e F ( z , u ) + u connected mappings: linear first order PDE C z ( z , u ) = uC u ( z , u ) e F ( z , u ) + uF u ( z , u ) e F ( z , u ) mappings: set of connected mappings M ( z , u ) = e C ( z , u ) 12 / 25

  21. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Theorem (P., 2012) Generating function solution: ( e F + 1) 2 F = z 1 + e F M ( z ) = 2 e F (1 + (1 − F ) e F ) , with 2 Enumeration formula: for M n number of alternating n-mappings n +1 1 � n + 1 � � ( k − 1) n M n = 2 n +1 k k =0 Asymptotics: √ 2 √ ρ + 2 � n � n ρ := 2 LambertW (1 M n ∼ · , e ) ≈ 0 . 556929 . . . 4 e ρ 13 / 25

  22. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Theorem (P., 2012) Generating function solution: ( e F + 1) 2 F = z 1 + e F M ( z ) = 2 e F (1 + (1 − F ) e F ) , with 2 Enumeration formula: for M n number of alternating n-mappings n +1 1 � n + 1 � � ( k − 1) n M n = 2 n +1 k k =0 Asymptotics: √ 2 √ ρ + 2 � n � n ρ := 2 LambertW (1 M n ∼ · , e ) ≈ 0 . 556929 . . . 4 e ρ 13 / 25

  23. PDEs and combinatorial problems Label patterns On-line selection Urn models Label patterns in trees/mappings Theorem (P., 2012) Generating function solution: ( e F + 1) 2 F = z 1 + e F M ( z ) = 2 e F (1 + (1 − F ) e F ) , with 2 Enumeration formula: for M n number of alternating n-mappings n +1 1 � n + 1 � � ( k − 1) n M n = 2 n +1 k k =0 Asymptotics: √ 2 √ ρ + 2 � n � n ρ := 2 LambertW (1 M n ∼ · , e ) ≈ 0 . 556929 . . . 4 e ρ 13 / 25

  24. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Hiring problem: introduced by Broder et al [SODA, 2008]; independently by Krieger et al [Ann. Prob. 2007] (unknown number of) candidates arrive sequentially each candidate is interviewed and gets a score (absolute or relative) decision whether to hire current candidate or not must be made instantly, depending on scores of candidates seen so far Important class of selection strategies: Quality of recruited staff increases whenever candidate is hired 14 / 25

  25. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Hiring problem: introduced by Broder et al [SODA, 2008]; independently by Krieger et al [Ann. Prob. 2007] (unknown number of) candidates arrive sequentially each candidate is interviewed and gets a score (absolute or relative) decision whether to hire current candidate or not must be made instantly, depending on scores of candidates seen so far Important class of selection strategies: Quality of recruited staff increases whenever candidate is hired 14 / 25

  26. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Hiring problem: introduced by Broder et al [SODA, 2008]; independently by Krieger et al [Ann. Prob. 2007] (unknown number of) candidates arrive sequentially each candidate is interviewed and gets a score (absolute or relative) decision whether to hire current candidate or not must be made instantly, depending on scores of candidates seen so far Important class of selection strategies: Quality of recruited staff increases whenever candidate is hired 14 / 25

  27. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Hiring problem: introduced by Broder et al [SODA, 2008]; independently by Krieger et al [Ann. Prob. 2007] (unknown number of) candidates arrive sequentially each candidate is interviewed and gets a score (absolute or relative) decision whether to hire current candidate or not must be made instantly, depending on scores of candidates seen so far Important class of selection strategies: Quality of recruited staff increases whenever candidate is hired 14 / 25

  28. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Hiring problem: introduced by Broder et al [SODA, 2008]; independently by Krieger et al [Ann. Prob. 2007] (unknown number of) candidates arrive sequentially each candidate is interviewed and gets a score (absolute or relative) decision whether to hire current candidate or not must be made instantly, depending on scores of candidates seen so far Important class of selection strategies: Quality of recruited staff increases whenever candidate is hired 14 / 25

  29. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) ⇒ 5 hired candidates 15 / 25

  30. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) ⇒ 5 hired candidates 15 / 25

  31. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 ⇒ 5 hired candidates 15 / 25

  32. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 ⇒ 5 hired candidates 15 / 25

  33. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 ⇒ 5 hired candidates 15 / 25

  34. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 ⇒ 5 hired candidates 15 / 25

  35. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 ⇒ 5 hired candidates 15 / 25

  36. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 ⇒ 5 hired candidates 15 / 25

  37. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 ⇒ 5 hired candidates 15 / 25

  38. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 1 ⇒ 5 hired candidates 15 / 25

  39. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 1 7 ⇒ 5 hired candidates 15 / 25

  40. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 1 7 ⇒ 5 hired candidates 15 / 25

  41. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 1 7 6 ⇒ 5 hired candidates 15 / 25

  42. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies “Hiring above the median”-strategy: candidate is recruited iff its score is better than median score of already hired staff Assume s 1 < s 2 < · · · < s k are scores of already hired candidates ⇒ new candidate is hired iff score is higher than s ⌊ k +1 2 ⌋ (lower median) 3 2 5 4 1 7 6 ⇒ 5 hired candidates 15 / 25

  43. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Basic question: consider random sequence of n candidates how many candidates h n will be hired? Krieger et al [2007]: Expectation: E ( h n ) √ n → c . It seems impossible to determine c analytically. Helmi and P. [2012]: analytic expression for c Gaither and Ward [2012]: analysis of more general strategies 16 / 25

  44. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Basic question: consider random sequence of n candidates how many candidates h n will be hired? Krieger et al [2007]: Expectation: E ( h n ) √ n → c . It seems impossible to determine c analytically. Helmi and P. [2012]: analytic expression for c Gaither and Ward [2012]: analysis of more general strategies 16 / 25

  45. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Basic question: consider random sequence of n candidates how many candidates h n will be hired? Krieger et al [2007]: Expectation: E ( h n ) √ n → c . It seems impossible to determine c analytically. Helmi and P. [2012]: analytic expression for c Gaither and Ward [2012]: analysis of more general strategies 16 / 25

  46. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Basic question: consider random sequence of n candidates how many candidates h n will be hired? Krieger et al [2007]: Expectation: E ( h n ) √ n → c . It seems impossible to determine c analytically. Helmi and P. [2012]: analytic expression for c Gaither and Ward [2012]: analysis of more general strategies 16 / 25

  47. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Evolution of median of hired staff during “hiring process”: n candidates interviewed; median of hired staff is ℓ -th largest ( n + 1)-th candidate with certain score arrives Markov chain with states ( n , ℓ ) odd and ( n , ℓ ) even : ℓ P = 1 − ℓ ℓ P = 1 − P = n +1 n +1 n +1 ( n , ℓ ) �→ ( n + 1 , ℓ ) ( n , ℓ ) �→ ( n + 1 , ℓ ) ( n , ℓ ) �→ ( n + 1 , ℓ ) 2 1 ℓ P = n +1 ( n , ℓ ) �→ ( n + 1 , ℓ + 1) Linear first order PDE for suitable g.f. of prob. P { h n = k } : zu − u − u 2 z 2 � � z (1 − z ) F z ( z , u ) + F u ( z , u ) − zF ( z , u ) = 0 1 − z 17 / 25

  48. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Evolution of median of hired staff during “hiring process”: n candidates interviewed; median of hired staff is ℓ -th largest ( n + 1)-th candidate with certain score arrives Markov chain with states ( n , ℓ ) odd and ( n , ℓ ) even : ℓ P = 1 − ℓ ℓ P = 1 − P = n +1 n +1 n +1 ( n , ℓ ) �→ ( n + 1 , ℓ ) ( n , ℓ ) �→ ( n + 1 , ℓ ) ( n , ℓ ) �→ ( n + 1 , ℓ ) 2 1 ℓ P = n +1 ( n , ℓ ) �→ ( n + 1 , ℓ + 1) Linear first order PDE for suitable g.f. of prob. P { h n = k } : zu − u − u 2 z 2 � � z (1 − z ) F z ( z , u ) + F u ( z , u ) − zF ( z , u ) = 0 1 − z 17 / 25

  49. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Evolution of median of hired staff during “hiring process”: n candidates interviewed; median of hired staff is ℓ -th largest ( n + 1)-th candidate with certain score arrives Markov chain with states ( n , ℓ ) odd and ( n , ℓ ) even : ℓ P = 1 − ℓ ℓ P = 1 − P = n +1 n +1 n +1 ( n , ℓ ) �→ ( n + 1 , ℓ ) ( n , ℓ ) �→ ( n + 1 , ℓ ) ( n , ℓ ) �→ ( n + 1 , ℓ ) 2 1 ℓ P = n +1 ( n , ℓ ) �→ ( n + 1 , ℓ + 1) Linear first order PDE for suitable g.f. of prob. P { h n = k } : zu − u − u 2 z 2 � � z (1 − z ) F z ( z , u ) + F u ( z , u ) − zF ( z , u ) = 0 1 − z 17 / 25

  50. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Theorem (Helmi and P., 2011) “Hiring above lower median”: Number h n of hired candidates is distributed as follows: � n − 1 −⌊ k ( n − ℓ ℓ − 1 ) 2 ⌋  � ℓ ) , for k = 2 ℓ − 1 odd ,  ⌈ k ( n 2 ⌉− 1  P { h n = k } = = � n ( n − ℓ ℓ − 2 ) � ⌈ k ℓ − 1 ) , for k = 2 ℓ − 2 even . 2 ⌉  ( n  Theorem Expectation of h n satisfies: E ( h n ) = √ π √ n + O (1) . √ h n asymptotically Rayleigh distributed with parameter σ = 2 , ( d ) → ˆ R, where ˆ h n i.e., √ n − − R has density function f ( x ) = x 2 e − x 2 ˆ 4 , for x > 0 . 18 / 25

  51. PDEs and combinatorial problems Label patterns On-line selection Urn models Analysis of on-line selection strategies Theorem (Helmi and P., 2011) “Hiring above lower median”: Number h n of hired candidates is distributed as follows: � n − 1 −⌊ k ( n − ℓ ℓ − 1 ) 2 ⌋  � ℓ ) , for k = 2 ℓ − 1 odd ,  ⌈ k ( n 2 ⌉− 1  P { h n = k } = = � n ( n − ℓ ℓ − 2 ) � ⌈ k ℓ − 1 ) , for k = 2 ℓ − 2 even . 2 ⌉  ( n  Theorem Expectation of h n satisfies: E ( h n ) = √ π √ n + O (1) . √ h n asymptotically Rayleigh distributed with parameter σ = 2 , ( d ) → ˆ R, where ˆ h n i.e., √ n − − R has density function f ( x ) = x 2 e − x 2 ˆ 4 , for x > 0 . 18 / 25

  52. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models P´ olya-Eggenberger urn model: Urn with 2 type of balls: n white balls, m black balls � a b � Transition matrix: M = c d Urn evolution process: choose ball at random examine color put ball back into urn insert balls according transition matrix: if white ball chosen: add a white and b black balls if black ball chosen: add c white and d black balls 19 / 25

  53. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models P´ olya-Eggenberger urn model: Urn with 2 type of balls: n white balls, m black balls � a b � Transition matrix: M = c d Urn evolution process: choose ball at random examine color put ball back into urn insert balls according transition matrix: if white ball chosen: add a white and b black balls if black ball chosen: add c white and d black balls 19 / 25

  54. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models P´ olya-Eggenberger urn model: Urn with 2 type of balls: n white balls, m black balls � a b � Transition matrix: M = c d Urn evolution process: choose ball at random examine color put ball back into urn insert balls according transition matrix: if white ball chosen: add a white and b black balls if black ball chosen: add c white and d black balls 19 / 25

  55. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls 20 / 25

  56. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls p yellow = 7/13 p black = 6/13 20 / 25

  57. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls Inspected color: yellow 20 / 25

  58. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls 2 x 1 x 20 / 25

  59. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls 20 / 25

  60. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls p yellow = 9/16 p black = 7/16 20 / 25

  61. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls Inspected color: black 20 / 25

  62. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls 1 x -1 x 20 / 25

  63. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Example: � 2 1 � ball replacement matrix M = 1 − 1 initial configuration: n = 7 yellow (white) balls and m = 6 black balls 20 / 25

  64. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Diminishing urn models: P´ olya-Eggenberger urn model with ball replacement matrix M in addition: set of absorbing states A ⊂ N 0 × N 0 . urn evolves according to matrix M until absorbing state ( i , j ) ∈ A is reached consider only well defined urns: urn always ends in absorbing state of A 21 / 25

  65. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Diminishing urn models: P´ olya-Eggenberger urn model with ball replacement matrix M in addition: set of absorbing states A ⊂ N 0 × N 0 . urn evolves according to matrix M until absorbing state ( i , j ) ∈ A is reached consider only well defined urns: urn always ends in absorbing state of A 21 / 25

  66. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Diminishing urn models: P´ olya-Eggenberger urn model with ball replacement matrix M in addition: set of absorbing states A ⊂ N 0 × N 0 . urn evolves according to matrix M until absorbing state ( i , j ) ∈ A is reached consider only well defined urns: urn always ends in absorbing state of A 21 / 25

  67. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models Diminishing urn models: P´ olya-Eggenberger urn model with ball replacement matrix M in addition: set of absorbing states A ⊂ N 0 × N 0 . urn evolves according to matrix M until absorbing state ( i , j ) ∈ A is reached consider only well defined urns: urn always ends in absorbing state of A 21 / 25

  68. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: proposed by Knuth and McCarthy [1991] in a bottle there are n small pills and m large pills a large pill is equivalent to two small pills every day a person chooses a pill at random if a small pill is chosen, it is eaten up if a large pill is chosen it is broken into two halves: one half is eaten and the other half, which is now considered to be a small pill, is returned to the bottle Main question: What is the number of small pills X m , n remaining when all large pills have been consumed? 22 / 25

  69. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: proposed by Knuth and McCarthy [1991] in a bottle there are n small pills and m large pills a large pill is equivalent to two small pills every day a person chooses a pill at random if a small pill is chosen, it is eaten up if a large pill is chosen it is broken into two halves: one half is eaten and the other half, which is now considered to be a small pill, is returned to the bottle Main question: What is the number of small pills X m , n remaining when all large pills have been consumed? 22 / 25

  70. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: proposed by Knuth and McCarthy [1991] in a bottle there are n small pills and m large pills a large pill is equivalent to two small pills every day a person chooses a pill at random if a small pill is chosen, it is eaten up if a large pill is chosen it is broken into two halves: one half is eaten and the other half, which is now considered to be a small pill, is returned to the bottle Main question: What is the number of small pills X m , n remaining when all large pills have been consumed? 22 / 25

  71. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (6,1) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  72. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) 6/7 (6,1) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  73. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) 6/7 2/7 (5,1) (6,1) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  74. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) 6/7 2/7 (5,1) (6,1) 1/6 (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  75. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) 6/7 2/7 (4,1) (5,1) (6,1) 1/6 5/5 (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  76. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) 6/7 2/7 (4,1) (5,1) (6,1) 1/5 1/6 5/5 (4,0) (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  77. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) 6/7 2/7 (4,1) (3,1) (5,1) (6,1) 1/5 1/6 4/4 5/5 (4,0) (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  78. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (5,2) (2,2) 6/7 2/7 3/4 (4,1) (3,1) (5,1) (6,1) 1/5 1/6 4/4 5/5 (4,0) (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  79. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (1,3) 2/3 (5,2) (2,2) 6/7 2/7 3/4 (4,1) (3,1) (5,1) (6,1) 1/5 1/6 4/4 5/5 (4,0) (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  80. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (1,3) 2/3 3/4 (5,2) (2,2) (1,2) 6/7 2/7 3/4 (4,1) (3,1) (5,1) (6,1) 1/5 1/6 4/4 5/5 (4,0) (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

  81. PDEs and combinatorial problems Label patterns On-line selection Urn models P´ olya-Eggenberger urn models The pill’s problem: � − 1 0 � ball replacement matrix M = 1 − 1 absorbing states A = { (0 , n ) | n ∈ N 0 } start with 6 large pills and one small pill (1,3) 2/3 3/4 (5,2) (2,2) (1,2) 6/7 2/7 2/3 3/4 (4,1) (3,1) (5,1) (6,1) (1,1) 1/5 1/6 4/4 5/5 (4,0) (5,0) ⇒ The state (0 , 2) ∈ A is reached. 23 / 25

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