Occurrences of exactly solvable PDEs in combinatorial problems - - PowerPoint PPT Presentation

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 Alois.Panholzer@tuwien.ac.at CanaDAM, Memorial University of Newfoundland, Saint John’s,

• 11. 6. 2013

1 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Typical situations yielding multivariate recursive descriptions: Combinatorial enumeration problems

study behaviour of parameters enumeration requires auxiliary quantities

Evolution of random structures

growth of parameters information on quantity at “discrete time”

Discrete stochastic processes

2 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Typical situations yielding multivariate recursive descriptions: Combinatorial enumeration problems

study behaviour of parameters enumeration requires auxiliary quantities

Evolution of random structures

growth of parameters information on quantity at “discrete time”

Discrete stochastic processes

2 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Typical situations yielding multivariate recursive descriptions: Combinatorial enumeration problems

study behaviour of parameters enumeration requires auxiliary quantities

Evolution of random structures

growth of parameters information on quantity at “discrete time”

Discrete stochastic processes

2 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Typical situations yielding multivariate recursive descriptions: Combinatorial enumeration problems

study behaviour of parameters enumeration requires auxiliary quantities

Evolution of random structures

growth of parameters information on quantity at “discrete time”

Discrete stochastic processes

2 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

− → Recursive description of bi-(multi-)variate quantity

Fn,m

Bivariate generating functions:

F(z, u) =

n

• m Fn,mznum

− → functional equation for F(z, u) Interested in problems yielding first order (quasi-)linear PDE:

a(z, u, F) · Fz + b(z, u, F) · Fu = c(z, u, F)

3 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

− → Recursive description of bi-(multi-)variate quantity

Fn,m

Bivariate generating functions:

F(z, u) =

n

• m Fn,mznum

− → functional equation for F(z, u) Interested in problems yielding first order (quasi-)linear PDE:

a(z, u, F) · Fz + b(z, u, F) · Fu = c(z, u, F)

3 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

− → Recursive description of bi-(multi-)variate quantity

Fn,m

Bivariate generating functions:

F(z, u) =

n

• m Fn,mznum

− → functional equation for F(z, u) Interested in problems yielding first order (quasi-)linear PDE:

a(z, u, F) · Fz + b(z, u, F) · Fu = c(z, u, F)

3 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

− → Recursive description of bi-(multi-)variate quantity

Fn,m

Bivariate generating functions:

F(z, u) =

n

• m Fn,mznum

− → functional equation for F(z, u) Interested in problems yielding first order (quasi-)linear PDE:

a(z, u, F) · Fz + b(z, u, F) · Fu = c(z, u, F)

3 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Solution method for solving such PDEs − → “Method of characteristics” Way to find transformation

ξ = ξ(z, u) η = η(z, u)

˜ F(ξ, η) reduces to linear first order ODE − → solution of generating function Difficulty: not always possible to state transformation explicitly

4 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Solution method for solving such PDEs − → “Method of characteristics” Way to find transformation

ξ = ξ(z, u) η = η(z, u)

˜ F(ξ, η) reduces to linear first order ODE − → solution of generating function Difficulty: not always possible to state transformation explicitly

4 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Solution method for solving such PDEs − → “Method of characteristics” Way to find transformation

ξ = ξ(z, u) η = η(z, u)

˜ F(ξ, η) reduces to linear first order ODE − → solution of generating function Difficulty: not always possible to state transformation explicitly

4 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

Solution method for solving such PDEs − → “Method of characteristics” Way to find transformation

ξ = ξ(z, u) η = η(z, u)

˜ F(ξ, η) reduces to linear first order ODE − → solution of generating function Difficulty: not always possible to state transformation explicitly

4 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

PDEs occurring in combinatorial problems

• ften yield explicit solutions

exact formulæ asymptotics often “automatically”

if solutions “very nice” → think about simpler proof (bijective, etc.) Illustrating examples: Label quantities in trees/mappings (with Marie-Louise Bruner; Helmut Prodinger) On-line selection strategies under uncertainty (with Ahmed Helmi and Conrado Martinez) Urn models (with Markus Kuba and Hsien-Kuei Hwang)

5 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

PDEs occurring in combinatorial problems

• ften yield explicit solutions

exact formulæ asymptotics often “automatically”

if solutions “very nice” → think about simpler proof (bijective, etc.) Illustrating examples: Label quantities in trees/mappings (with Marie-Louise Bruner; Helmut Prodinger) On-line selection strategies under uncertainty (with Ahmed Helmi and Conrado Martinez) Urn models (with Markus Kuba and Hsien-Kuei Hwang)

5 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

PDEs occurring in combinatorial problems

• ften yield explicit solutions

exact formulæ asymptotics often “automatically”

if solutions “very nice” → think about simpler proof (bijective, etc.) Illustrating examples: Label quantities in trees/mappings (with Marie-Louise Bruner; Helmut Prodinger) On-line selection strategies under uncertainty (with Ahmed Helmi and Conrado Martinez) Urn models (with Markus Kuba and Hsien-Kuei Hwang)

5 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

PDEs and combinatorial problems

PDEs occurring in combinatorial problems

• ften yield explicit solutions

exact formulæ asymptotics often “automatically”

if solutions “very nice” → think about simpler proof (bijective, etc.) Illustrating examples: Label quantities in trees/mappings (with Marie-Louise Bruner; Helmut Prodinger) On-line selection strategies under uncertainty (with Ahmed Helmi and Conrado Martinez) Urn models (with Markus Kuba and Hsien-Kuei Hwang)

5 / 25

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

• le

6 8 10 12 11 3 5 14 1 7 2 9 4 13 5 1 18 13 3 9 15 11 8 12 2 17 10 6 4 16 19 7 14

Occurrence and avoidance of label-patterns in trees/mappings: Runs Records Alternating mappings

6 / 25

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

• le

6 8 10 12 11 3 5 14 1 7 2 9 4 13 5 1 18 13 3 9 15 11 8 12 2 17 10 6 4 16 19 7 14

Occurrence and avoidance of label-patterns in trees/mappings: Runs Records Alternating mappings

6 / 25

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

• le

6 8 10 12 11 3 5 14 1 7 2 9 4 13 5 1 18 13 3 9 15 11 8 12 2 17 10 6 4 16 19 7 14

Occurrence and avoidance of label-patterns in trees/mappings: Runs Records Alternating mappings

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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 Gf of f : Gf = (Vf , Ef ), with Vf = [n] and Ef = {(i, f (i)), i ∈ [n]} Simple structure: connected components of Gf : cycles of trees

7 / 25

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 Gf of f : Gf = (Vf , Ef ), with Vf = [n] and Ef = {(i, f (i)), i ∈ [n]}

5 1 18 13 3 9 15 11 8 12 2 17 10 6 4 16 19 7 14

Simple structure: connected components of Gf : cycles of trees

7 / 25

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 Gf of f : Gf = (Vf , Ef ), with Vf = [n] and Ef = {(i, f (i)), i ∈ [n]}

5 1 18 13 3 9 15 11 8 12 2 17 10 6 4 16 19 7 14

Simple structure: connected components of Gf : cycles of trees

7 / 25

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) > · · ·

• r

i = f 0(i) > f 1(i) < f 2(i) > f 3(i) < · · · Equivalently:

• f 2(i) − f (i)
• · (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

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) > · · ·

• r

i = f 0(i) > f 1(i) < f 2(i) > f 3(i) < · · · Equivalently:

• f 2(i) − f (i)
• · (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

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) > · · ·

• r

i = f 0(i) > f 1(i) < f 2(i) > f 3(i) < · · · Equivalently:

• f 2(i) − f (i)
• · (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

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) > · · ·

• r

i = f 0(i) > f 1(i) < f 2(i) > f 3(i) < · · · Equivalently:

• f 2(i) − f (i)
• · (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

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

17 10 16 19 5 4 2 7 1 18 13 6 3 9 15 11 8 12 14

f :   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 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

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

17 10 16 19 5 4 2 7 1 18 13 6 3 9 15 11 8 12 14

f :   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 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

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

17 10 16 19 5 4 2 7 1 18 13 6 3 9 15 11 8 12 14

f :   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 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

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 C :

n n

• r

C ′ T1 T2 Tr T ′ T1 Tr ⇒ treatment requires auxiliary parameter: number of local minima ⇒ require also treatment of quantity for rooted labelled trees (Cayley trees)

10 / 25

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 C :

n n

• r

C ′ T1 T2 Tr T ′ T1 Tr ⇒ treatment requires auxiliary parameter: number of local minima ⇒ require also treatment of quantity for rooted labelled trees (Cayley trees)

10 / 25

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 C :

n n

• r

C ′ T1 T2 Tr T ′ T1 Tr ⇒ treatment requires auxiliary parameter: number of local minima ⇒ require also treatment of quantity for rooted labelled trees (Cayley trees)

10 / 25

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] number of rooted alternating trees: Tn = 1 2n

n

• k=0

n k

• kn−1
• ccurs 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

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] number of rooted alternating trees: Tn = 1 2n

n

• k=0

n k

• kn−1
• ccurs 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

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 Fz(z, u) = uFu(z, u)eF(z,u) + u connected mappings: linear first order PDE Cz(z, u) = uCu(z, u)eF(z,u) + uFu(z, u)eF(z,u) mappings: set of connected mappings M(z, u) = eC(z,u)

12 / 25

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 Fz(z, u) = uFu(z, u)eF(z,u) + u connected mappings: linear first order PDE Cz(z, u) = uCu(z, u)eF(z,u) + uFu(z, u)eF(z,u) mappings: set of connected mappings M(z, u) = eC(z,u)

12 / 25

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 Fz(z, u) = uFu(z, u)eF(z,u) + u connected mappings: linear first order PDE Cz(z, u) = uCu(z, u)eF(z,u) + uFu(z, u)eF(z,u) mappings: set of connected mappings M(z, u) = eC(z,u)

12 / 25

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 Fz(z, u) = uFu(z, u)eF(z,u) + u connected mappings: linear first order PDE Cz(z, u) = uCu(z, u)eF(z,u) + uFu(z, u)eF(z,u) mappings: set of connected mappings M(z, u) = eC(z,u)

12 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

Label patterns in trees/mappings

Theorem (P., 2012) Generating function solution: M(z) = (eF + 1)2 2eF(1 + (1 − F)eF), with F = z 1 + eF 2 Enumeration formula: for Mn number of alternating n-mappings Mn = 1 2n+1

n+1

• k=0

n + 1 k

• (k − 1)n

Asymptotics: Mn ∼ √ 2√ρ + 2 4 · n eρ n , ρ := 2LambertW (1 e ) ≈ 0.556929 . . .

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PDEs and combinatorial problems Label patterns On-line selection Urn models

Label patterns in trees/mappings

Theorem (P., 2012) Generating function solution: M(z) = (eF + 1)2 2eF(1 + (1 − F)eF), with F = z 1 + eF 2 Enumeration formula: for Mn number of alternating n-mappings Mn = 1 2n+1

n+1

• k=0

n + 1 k

• (k − 1)n

Asymptotics: Mn ∼ √ 2√ρ + 2 4 · n eρ n , ρ := 2LambertW (1 e ) ≈ 0.556929 . . .

13 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

Label patterns in trees/mappings

Theorem (P., 2012) Generating function solution: M(z) = (eF + 1)2 2eF(1 + (1 − F)eF), with F = z 1 + eF 2 Enumeration formula: for Mn number of alternating n-mappings Mn = 1 2n+1

n+1

• k=0

n + 1 k

• (k − 1)n

Asymptotics: Mn ∼ √ 2√ρ + 2 4 · n eρ n , ρ := 2LambertW (1 e ) ≈ 0.556929 . . .

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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

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

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

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

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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 s1 < s2 < · · · < sk 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

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 hn will be hired? Krieger et al [2007]: Expectation: E(hn) √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

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 hn will be hired? Krieger et al [2007]: Expectation: E(hn) √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

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 hn will be hired? Krieger et al [2007]: Expectation: E(hn) √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

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 hn will be hired? Krieger et al [2007]: Expectation: E(hn) √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

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:

2 1 P = 1 −

ℓ n+1

(n, ℓ) → (n + 1, ℓ) P =

ℓ n+1

(n, ℓ) → (n + 1, ℓ) P =

ℓ n+1

(n, ℓ) → (n + 1, ℓ + 1) P = 1 −

ℓ n+1

(n, ℓ) → (n + 1, ℓ)

Linear first order PDE for suitable g.f. of prob. P{hn = k}: z(1 − z)Fz(z, u) +

• zu − u − u2z2

1 − z

• Fu(z, u) − zF(z, u) = 0

17 / 25

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:

2 1 P = 1 −

ℓ n+1

(n, ℓ) → (n + 1, ℓ) P =

ℓ n+1

(n, ℓ) → (n + 1, ℓ) P =

ℓ n+1

(n, ℓ) → (n + 1, ℓ + 1) P = 1 −

ℓ n+1

(n, ℓ) → (n + 1, ℓ)

Linear first order PDE for suitable g.f. of prob. P{hn = k}: z(1 − z)Fz(z, u) +

• zu − u − u2z2

1 − z

• Fu(z, u) − zF(z, u) = 0

17 / 25

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:

2 1 P = 1 −

ℓ n+1

(n, ℓ) → (n + 1, ℓ) P =

ℓ n+1

(n, ℓ) → (n + 1, ℓ) P =

ℓ n+1

(n, ℓ) → (n + 1, ℓ + 1) P = 1 −

ℓ n+1

(n, ℓ) → (n + 1, ℓ)

Linear first order PDE for suitable g.f. of prob. P{hn = k}: z(1 − z)Fz(z, u) +

• zu − u − u2z2

1 − z

• Fu(z, u) − zF(z, u) = 0

17 / 25

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 hn of hired candidates is distributed as follows: P{hn = k} = n−1−⌊ k

2 ⌋

⌈ k

2 ⌉−1

• n

⌈ k

2 ⌉

• =

     (n−ℓ

ℓ−1)

(n

ℓ) ,

for k = 2ℓ − 1 odd, (n−ℓ

ℓ−2)

( n

ℓ−1),

for k = 2ℓ − 2 even. Theorem Expectation of hn satisfies: E(hn) = √π√n + O(1). hn asymptotically Rayleigh distributed with parameter σ = √ 2, i.e.,

hn √n (d)

− − → ˆ R, where ˆ R has density function ˆ f (x) = x 2e− x2

4 ,

for x > 0.

18 / 25

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 hn of hired candidates is distributed as follows: P{hn = k} = n−1−⌊ k

2 ⌋

⌈ k

2 ⌉−1

• n

⌈ k

2 ⌉

• =

     (n−ℓ

ℓ−1)

(n

ℓ) ,

for k = 2ℓ − 1 odd, (n−ℓ

ℓ−2)

( n

ℓ−1),

for k = 2ℓ − 2 even. Theorem Expectation of hn satisfies: E(hn) = √π√n + O(1). hn asymptotically Rayleigh distributed with parameter σ = √ 2, i.e.,

hn √n (d)

− − → ˆ R, where ˆ R has density function ˆ f (x) = x 2e− x2

4 ,

for x > 0.

18 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

P´

• lya-Eggenberger urn model:

Urn with 2 type of balls: n white balls, m black balls Transition matrix: M = a b

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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

P´

• lya-Eggenberger urn model:

Urn with 2 type of balls: n white balls, m black balls Transition matrix: M = a b

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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

P´

• lya-Eggenberger urn model:

Urn with 2 type of balls: n white balls, m black balls Transition matrix: M = a b

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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

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PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

pyellow = 7/13 pblack = 6/13

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PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

Inspected color: yellow

20 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

2 x 1 x

20 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

20 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

pyellow = 9/16 pblack = 7/16

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PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

Inspected color: black

20 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

1 x

• 1 x

20 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Example: ball replacement matrix M = 2 1

1 −1

• initial configuration:

n = 7 yellow (white) balls and m = 6 black balls

20 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Diminishing urn models: P´

• lya-Eggenberger urn model with ball replacement matrix M

in addition: set of absorbing states A ⊂ N0 × N0. 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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Diminishing urn models: P´

• lya-Eggenberger urn model with ball replacement matrix M

in addition: set of absorbing states A ⊂ N0 × N0. 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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Diminishing urn models: P´

• lya-Eggenberger urn model with ball replacement matrix M

in addition: set of absorbing states A ⊂ N0 × N0. 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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Diminishing urn models: P´

• lya-Eggenberger urn model with ball replacement matrix M

in addition: set of absorbing states A ⊂ N0 × N0. 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

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-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 Xm,n remaining when all large pills have been consumed?

22 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-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 Xm,n remaining when all large pills have been consumed?

22 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-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 Xm,n remaining when all large pills have been consumed?

22 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1)

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2)

6/7

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1)

6/7 2/7

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0)

6/7 2/7 1/6

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1)

6/7 2/7 1/6 5/5

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0)

6/7 2/7 1/6 5/5 1/5

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0) (3,1)

6/7 2/7 1/6 5/5 1/5 4/4

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0) (3,1) (2,2)

6/7 2/7 1/6 5/5 1/5 4/4 3/4

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0) (3,1) (2,2) (1,3)

6/7 2/7 1/6 5/5 1/5 4/4 3/4 2/3

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0) (3,1) (2,2) (1,3) (1,2)

6/7 2/7 1/6 5/5 1/5 4/4 3/4 2/3 3/4

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0) (3,1) (2,2) (1,3) (1,2) (1,1)

6/7 2/7 1/6 5/5 1/5 4/4 3/4 2/3 3/4 2/3

⇒ The state (0, 2) ∈ A is reached.

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

The pill’s problem: ball replacement matrix M = −1 0

1 −1

• absorbing states A = {(0, n)|n ∈ N0}

start with 6 large pills and one small pill

(6,1) (5,2) (5,1) (5,0) (4,1) (4,0) (3,1) (2,2) (1,3) (1,2) (1,1)

6/7 2/7 1/6 5/5 1/5 4/4 3/4 2/3 3/4 2/3 1/2

(0,2)

⇒ the state (0, 2) ∈ A is reached

23 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Analysis of pill’s problem urn M = −1

1 −1

• Decompose path in urn evolution according to first step

− → recurrence for probabilities P{Xm,n = k} Introducing suitable g.f. − → linear first order PDE (z − z2 − u)Hz(z, u, v) + u(1 − z)Hu(z, u, v) − zH(z, u, v) = uv (1 − vz)2 Explicit formula for probability generating function hm,n(v) :=

k P{Xm,n = k}vk:

hm,n(v) = mv 1 (1 + (v − 1)q)n(1 − q − (v − 1)q log q)m−1dq

24 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Analysis of pill’s problem urn M = −1

1 −1

• Decompose path in urn evolution according to first step

− → recurrence for probabilities P{Xm,n = k} Introducing suitable g.f. − → linear first order PDE (z − z2 − u)Hz(z, u, v) + u(1 − z)Hu(z, u, v) − zH(z, u, v) = uv (1 − vz)2 Explicit formula for probability generating function hm,n(v) :=

k P{Xm,n = k}vk:

hm,n(v) = mv 1 (1 + (v − 1)q)n(1 − q − (v − 1)q log q)m−1dq

24 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Analysis of pill’s problem urn M = −1

1 −1

• Decompose path in urn evolution according to first step

− → recurrence for probabilities P{Xm,n = k} Introducing suitable g.f. − → linear first order PDE (z − z2 − u)Hz(z, u, v) + u(1 − z)Hu(z, u, v) − zH(z, u, v) = uv (1 − vz)2 Explicit formula for probability generating function hm,n(v) :=

k P{Xm,n = k}vk:

hm,n(v) = mv 1 (1 + (v − 1)q)n(1 − q − (v − 1)q log q)m−1dq

24 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

P´

• lya-Eggenberger urn models

Analysis of pill’s problem urn M = −1

1 −1

• Decompose path in urn evolution according to first step

− → recurrence for probabilities P{Xm,n = k} Introducing suitable g.f. − → linear first order PDE (z − z2 − u)Hz(z, u, v) + u(1 − z)Hu(z, u, v) − zH(z, u, v) = uv (1 − vz)2 Explicit formula for probability generating function hm,n(v) :=

k P{Xm,n = k}vk:

hm,n(v) = mv 1 (1 + (v − 1)q)n(1 − q − (v − 1)q log q)m−1dq

24 / 25

PDEs and combinatorial problems Label patterns On-line selection Urn models

Results: Pills problem

Theorem (Hwang, Kuba and P., 2007) If m → ∞ the r.v. Xm,n converges, after suitable scaling, in distribution to an exponentially distributed r.v. X with parameter λ = 1, i.e., Xm,n

n m + log m (d)

− − → X

(d)

= Exp(1), where X has density f (x) = e−x, x ≥ 0. If m is fixed and n → ∞ the r.v. Xm,n converges, after suitable scaling, in distribution to a beta distributed r.v. Bm, i.e., Xm,n n

(d)

− − → Bm

(d)

= Beta(1, m), where Bm has density f (x) = m(1 − x)m−1, 0 ≤ x ≤ 1.

25 / 25