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PDEs and combinatorial problems Label patterns On-line selection Urn models
Occurrences of exactly solvable PDEs incombinatorial problems
Alois Panholzer
Institut fur Diskrete Mathematik und GeometrieTU Wien, Austria
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 problemsstudy behaviour of parametersenumeration requires auxiliary quantities
Evolution of random structuresgrowth of parametersinformation on quantity at “discretetime”
Discrete stochastic processes
1
1
2
1
32
1
2
3
1 1 1
2 3
4
2 3
4
2 3 4
1
2
3
2
3 4
1 1
2 4
3
4
p = 1
p = 1/2p = 1/2
p = 1/3
p = 1/3p = 1/3p = 1/3 p = 1/3
p = 1/3
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 problemsstudy behaviour of parametersenumeration requires auxiliary quantities
Evolution of random structuresgrowth of parametersinformation on quantity at “discretetime”
Discrete stochastic processes
1
1
2
1
32
1
2
3
1 1 1
2 3
4
2 3
4
2 3 4
1
2
3
2
3 4
1 1
2 4
3
4
p = 1
p = 1/2p = 1/2
p = 1/3
p = 1/3p = 1/3p = 1/3 p = 1/3
p = 1/3
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 problemsstudy behaviour of parametersenumeration requires auxiliary quantities
Evolution of random structuresgrowth of parametersinformation on quantity at “discretetime”
Discrete stochastic processes
1
1
2
1
32
1
2
3
1 1 1
2 3
4
2 3
4
2 3 4
1
2
3
2
3 4
1 1
2 4
3
4
p = 1
p = 1/2p = 1/2
p = 1/3
p = 1/3p = 1/3p = 1/3 p = 1/3
p = 1/3
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 problemsstudy behaviour of parametersenumeration requires auxiliary quantities
Evolution of random structuresgrowth of parametersinformation on quantity at “discretetime”
Discrete stochastic processes
1
1
2
1
32
1
2
3
1 1 1
2 3
4
2 3
4
2 3 4
1
2
3
2
3 4
1 1
2 4
3
4
p = 1
p = 1/2p = 1/2
p = 1/3
p = 1/3p = 1/3p = 1/3 p = 1/3
p = 1/3
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,mz
num
−→ 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,mz
num
−→ 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,mz
num
−→ 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,mz
num
−→ 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
often yield explicit solutions
exact formulæasymptotics 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
often yield explicit solutions
exact formulæasymptotics 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
often yield explicit solutions
exact formulæasymptotics 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
often yield explicit solutions
exact formulæasymptotics 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 playessential role
6
810
12
11
35
141
7
2
9
4
13
5
1
18
13
3
9
15
118
12
2
17
10
6
4
1619
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 playessential role
6
810
12
11
35
141
7
2
9
4
13
5
1
18
13
3
9
15
118
12
2
17
10
6
4
1619
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 playessential role
6
810
12
11
35
141
7
2
9
4
13
5
1
18
13
3
9
15
118
12
2
17
10
6
4
1619
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
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
118
12
2
17
10
6
4
1619
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
118
12
2
17
10
6
4
1619
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) > · · · or
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) > · · · or
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) > · · · or
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) > · · · or
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
27
1
18
136
3
9
15
11812
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
27
1
18
136
3
9
15
11812
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
27
1
18
136
3
9
15
11812
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
orC ′
T1 T2
TrT ′
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
orC ′
T1 T2
TrT ′
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
orC ′
T1 T2
TrT ′
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
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
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
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
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 . . .
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 . . .
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 . . .
13 / 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 bemade 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 bemade 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 bemade 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 bemade 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 bemade 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4 1
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4 1 7
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4 1 7
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4 1 7 6
⇒ 5 hired candidates15 / 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 ofalready hired staff
Assume s1 < s2 < · · · < sk are scores of already hired candidates
⇒ new candidate is hired iff score is higher than sb k+12c (lower median)
3 2 5 4 1 7 6
⇒ 5 hired candidates15 / 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 `-th largest
(n + 1)-th candidate with certain score arrives
Markov chain with states (n, `)odd and (n, `)even:
2 1
P = 1− `n+1
(n, `) 7→ (n + 1, `)
P = `n+1
(n, `) 7→ (n + 1, `)
P = `n+1
(n, `) 7→ (n + 1, `+ 1)
P = 1− `n+1
(n, `) 7→ (n + 1, `)
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 `-th largest
(n + 1)-th candidate with certain score arrives
Markov chain with states (n, `)odd and (n, `)even:
2 1
P = 1− `n+1
(n, `) 7→ (n + 1, `)
P = `n+1
(n, `) 7→ (n + 1, `)
P = `n+1
(n, `) 7→ (n + 1, `+ 1)
P = 1− `n+1
(n, `) 7→ (n + 1, `)
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 `-th largest
(n + 1)-th candidate with certain score arrives
Markov chain with states (n, `)odd and (n, `)even:
2 1
P = 1− `n+1
(n, `) 7→ (n + 1, `)
P = `n+1
(n, `) 7→ (n + 1, `)
P = `n+1
(n, `) 7→ (n + 1, `+ 1)
P = 1− `n+1
(n, `) 7→ (n + 1, `)
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 candidatesis distributed as follows:
P{hn = k} =
(n−1−b k2c
d k2e−1
)( nd k2e) =
(n−``−1)(n`)
, for k = 2`− 1 odd,
(n−``−2)
( n`−1)
, for k = 2`− 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)−−→ R, where R has density function
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 candidatesis distributed as follows:
P{hn = k} =
(n−1−b k2c
d k2e−1
)( nd k2e) =
(n−``−1)(n`)
, for k = 2`− 1 odd,
(n−``−2)
( n`−1)
, for k = 2`− 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)−−→ R, where R has density function
f (x) =x
2e−
x2
4 , for x > 0.
18 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-Eggenberger urn models
Polya-Eggenberger urn model:
Urn with 2 type of balls: n white balls, m black balls
Transition matrix: M =(a bc 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
Polya-Eggenberger urn models
Polya-Eggenberger urn model:
Urn with 2 type of balls: n white balls, m black balls
Transition matrix: M =(a bc 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
Polya-Eggenberger urn models
Polya-Eggenberger urn model:
Urn with 2 type of balls: n white balls, m black balls
Transition matrix: M =(a bc 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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −1
)initial configuration:
n = 7 yellow (white) balls and m = 6 black balls
pyellow = 7/13
pblack = 6/13
20 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −1
)initial configuration:
n = 7 yellow (white) balls and m = 6 black balls
pyellow = 9/16
pblack = 7/16
20 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Example:
ball replacement matrix M =(2 11 −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
Polya-Eggenberger urn models
Diminishing urn models:
Polya-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
Polya-Eggenberger urn models
Diminishing urn models:
Polya-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
Polya-Eggenberger urn models
Diminishing urn models:
Polya-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
Polya-Eggenberger urn models
Diminishing urn models:
Polya-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
Polya-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 iseaten and the other half, which is now considered to be asmall pill, is returned to the bottle
Main question:
What is the number of small pills Xm,n remaining whenall large pills have been consumed?
22 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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 iseaten and the other half, which is now considered to be asmall pill, is returned to the bottle
Main question:
What is the number of small pills Xm,n remaining whenall large pills have been consumed?
22 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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 iseaten and the other half, which is now considered to be asmall pill, is returned to the bottle
Main question:
What is the number of small pills Xm,n remaining whenall large pills have been consumed?
22 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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
Polya-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
Polya-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)
6/72/7
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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)
6/72/7
1/6
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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
1/65/5
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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)
6/72/7
1/65/51/5
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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)
6/72/7
1/65/51/5
4/4
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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
1/65/51/5
4/4
3/4
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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)
6/72/7
1/65/51/5
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
Polya-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) 6/72/7
1/65/51/5
4/4
3/4
2/33/4
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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)
6/72/7
1/65/51/5
4/4
3/4
2/33/4
2/3
⇒ The state (0, 2) ∈ A is reached.
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-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)
6/72/7
1/65/51/5
4/4
3/4
2/33/4
2/31/2
(0,2)
⇒ the state (0, 2) ∈ A is reached
23 / 25
PDEs and combinatorial problems Label patterns On-line selection Urn models
Polya-Eggenberger urn models
Analysis of pill’s problem urn M =(−1 0
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 functionhm,n(v) :=
∑k P{Xm,n = k}vk :
hm,n(v) = mv
∫ 1
0(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
Polya-Eggenberger urn models
Analysis of pill’s problem urn M =(−1 0
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 functionhm,n(v) :=
∑k P{Xm,n = k}vk :
hm,n(v) = mv
∫ 1
0(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
Polya-Eggenberger urn models
Analysis of pill’s problem urn M =(−1 0
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 functionhm,n(v) :=
∑k P{Xm,n = k}vk :
hm,n(v) = mv
∫ 1
0(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
Polya-Eggenberger urn models
Analysis of pill’s problem urn M =(−1 0
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 functionhm,n(v) :=
∑k P{Xm,n = k}vk :
hm,n(v) = mv
∫ 1
0(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, indistribution to an exponentially distributed r.v. X withparameter λ = 1, i.e.,
Xm,nnm + 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, aftersuitable 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