1 Permutations and Combinations
Multinomial
Coefficients
Twelvefold Way
Cycle
Decompositions
2 Inclusion-Exclusion-Principle & Möbius Inversion
PIE
Möbius
Inversion Formula
3 Generating Functions
Ordinary and
Exponential
Newton’s
Binomial Theorem
Reccurence
Relations
4 Partitions
(Non-Crossing)
Partitions of [n]
Ferrer Diagrams
Standard
Young Tableaux
5 Partially Ordered Sets
(Symmetric)
Chain Partitions
Dimension
Posets Between
Two Lines
6 Designs
Existence/
Non-Existence
Steiner Triple
Systems etc.
Latin Squares
1 Permutations and Combinations
Multinomial
Coefficients
n
r1 ,...,rk
=
Twelvefold Way
n!
r1 !·...·rk !
Cycle
Decompositions
where n = r1 + . . . + rk
# n-permutations of multiset M = {r1 · t1 , . . . , rk · tk }
Multinomial Theorem:
(x1 + . . . + xk )n =
P
r1 + . . . + rk = n
n
r1 ,...,rk
xr11
rk
· · · xk
1 Permutations and Combinations
Multinomial
Coefficients
n balls
k boxes
U
L
L
U
L
L
U
U
Twelvefold Way
Cycle
Decompositions
≤ 1 per box
k
≥ 1 per box
n−1
arbitrary
n+k−1
n
k−1
k−1
1
k
sII
k (n)
n
1
n!
Pk
II
s
i=1 i (n)
sII
k (n)k!
pk (n)
kn
Pk
i=1
pi (n)
1 Permutations and Combinations
Multinomial
Coefficients
Cycle
Decompositions
Twelvefold Way
2
π = (123)(5)(46) = 231654 = 1
4
5
3
– unique partition into disjoint cycles
– exactly k cycles −→ sIk (n) permutations
– no trivial cycles −→ derangements
6
2 Inclusion-Exclusion-Principle & Möbius Inversion
Möbius
Inversion Formula
PIE
– properties P1 , . . . , Pm
– N (S) = {x | x has Pi for all i ∈ S}
P
Then
(−1)
S ⊆ [m]
|S|
|N (S)|
S ⊆ [m]
elements have
none of the properties.
Usage:
Define Pi
−→
Bound |N (S)|
−→
Apply PIE
2 Inclusion-Exclusion-Principle & Möbius Inversion
Möbius
Inversion Formula
PIE
Stronger PIE
P
g(A) =
f (S)
⇒
f (A) =
S ⊆ A
f (n) =
d|n
x ≤ y
(−1)|A|−|S| g(S)
S ⊆ A
Möbius Inversion
P
g(n) =
f (d) ⇒
General Posets
P
g(y) =
f (x)
P
P
µ(d)g( nd )
d|n
⇒
f (y) =
P
x ≤ y
µ(x, y)g(x)
3 Generating Functions
Ordinary and
Exponential
F (x) =
P
Newton’s
Binomial Theorem
fn x
n
Reccurence
Relations
unlabeled objects
n ≥ 0
n
P P
A(x) · B(x) =
(
ak bn−k ) xn
n ≥ 0 k = 0
G(x) =
P
n ≥ 0
xn
gn n!
A(x) · B(x) =
n
P P
n ≥ 0 k = 0
labeled objects
n
k
ak bn−k
x
n
n!
3 Generating Functions
Ordinary and
Exponential
Newton’s
Binomial Theorem
Newton’s Binomial Theorem
n
P
n k
n
(1 + x) =
k x
Reccurence
Relations
n ∈ R − {0}
k = 0
2
F (x) = 1 + x · F (x)
⇒
F (x) =
P
n ≥ 0
Catalan Numbers:
Cn =
2n
1
n+1 n
2n
1
n+1 n
xn
3 Generating Functions
Ordinary and
Exponential
reccurence
p(A)f = g
Newton’s
Binomial Theorem
Reccurence
Relations
initial values
f (0) = f0 , . . . , f (k) = fk
homogeneous (g = 0):
p(A) = (A − r)k ⇒ f (n) = c1 rn + · · · + nk−1 ck rn
p(A) = p1 (A) · p2 (A) ⇒ f (n) = f1 (n) + f2 (n)
non-homogeneous (g 6= 0):
f0 (n) gen. sol.
f1 (n) particular sol.
−→
−→ f = f0 + f1
of hom. system
of non-hom. system
4 Partitions
(Non-Crossing)
Partitions of [n]
Bell Numbers
Ferrer Diagrams
Bn =
n
P
Standard
Young Tableaux
sII
k (n)
k = 0
# ways to split n persons in groups
1
Non-Crossing Partitions
2
7
3
counted by Catalan numbers
Cn =
2n
1
n+1 n
6
5
4
4 Partitions
(Non-Crossing)
Partitions of [n]
Standard
Young Tableaux
Ferrer Diagrams
p(n) = # ways to write n as a sum
Pentagonal Numbers ωk
odd
peven
(n)
−
p
d
d (n)
11 = 5 + 3 + 2 + 1
=
(−1)k if n = ωk
0
Thm. podd (n) = pdist (n)
otherwise
4 Partitions
(Non-Crossing)
Partitions of [n]
π = 5264237
7
6
5
4
3
2
1
Ferrer Diagrams
RSC
←→
Standard
Young Tableaux
1 2 3 7
4 6
5 T1
1 3 6 7
2 4
5 T2
Hook Length Formula
t(λ) =
1234567
Q n!
|hi,j |
(i, j)
5 Partially Ordered Sets
(Symmetric)
Chain Partitions
Dimension
Dilworth’s Theorem
Posets Between
Two Lines
Multiset Lattices
partition into w(P ) chains
symmetric chain partition
5 Partially Ordered Sets
(Symmetric)
Chain Partitions
Dimension
Posets Between
Two Lines
dim(P ) = min{k | P = L1 ∩ · · · ∩ Lk }
Thm: dim(P ) ≤ w(P )
dim(Sn ) = w(Sn ) = n
Thm: dim(P ) ≤ 2 ⇔ ∃L ordering all 1 ⊕ 2
5 Partially Ordered Sets
(Symmetric)
Chain Partitions
Dimension
interval order
no 2 ⊕ 2
triangle order
∃L ordering all 2 ⊕ 2
Posets Between
Two Lines
segment order
dim(P ) ≤ 2
curve order
all posets
6 Designs
Existence/
Non-Existence
Steiner Triple
Systems etc.
v points, blocks are k-sets,
every t-tuple of points in λ blocks
t-(v, k, λ) Design:
Thm: # blocks = λ ·
Latin Squares
v
t
/
k
t
repetition of i-sets = λ ·
v−i
t−i
/
k−i
t−i
Thm: # blocks ≥ v ≥ (t + 1)(k − t + 1)
2-(7, 3, 1)-design
6 Designs
Existence/
Non-Existence
Steiner Triple
Systems etc.
Latin Squares
Affine Planes
2-(n2 , n, 1)-designs
n prime power
Steiner Triple Systems
2-(v, 3, 1)-designs
v ∈ {1, 3} (mod 6)
Projective Planes
2-(q 2 + q + 1, q + 1, 1)-designs
q prime power
6 Designs
Steiner Triple
Systems etc.
Existence/
Non-Existence
0
1
3
2
1
2
0
3
3
0
2
1
2
3
1
0
n − 1 MOLS
Thm.
of order n
Latin Squares
n-by-n array filled with Zn
– each row is a permutation
– each column is a permutation
←→
affine plane
2 ←→
of order n
n = pk