Finding matchings in sequences - Clemson Universitygoddard/MINI/2010/holliday.pdf · Finding matchings that occur in sequences Sarah Holliday Southern Polytechnic State University

Finding matchings in sequences

Sarah Holliday

Southern Polytechnic State University

Finding matchings that occur in sequences

Sarah Holliday

Southern Polytechnic State University

Sequences in matchings

Sarah Holliday

Southern Polytechnic State University

Story time!

Sarah Holliday

Southern Polytechnic State University

Once upon a time,

• in 12th century Italy

there was a

businessman named

Gugliemo Bonacci of

Pisa. He traveled

around the

Mediterranean

region, buying and

selling.

Story time…

• In 1170, his son

Leonardo was born.

He took Leonardo

with him on his trips.

During longer

voyages, he taught

Leonardo some

mathematics

(arithmetic).

Story time…

• Leonardo made

contact with some

Greek math (Euclid’s

elements) and Arab

Algebra (Al-

Khowarizmi).

Story time…

• Europe was still

using Roman

Numerals, but

Leonardo of Pisa

thought the Hindu-

Arabic numerals with

their fancy place-

system were soooo

cool!

Story time…

• Leo was so excited,

he wrote a book in

1202, called Liber

abaci. He wrote it

primarily to teach

Europeans about the

Hindu-Arabic

numeral system.

Story time…

• He was so proud of

himself (and clearly,

rightly so), that in

1228, he released a

second edition that

included the

following problem:

Story problem:

• A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.

Fibonacci sequences

Sarah Holliday

Southern Polytechnic State University

Fibonacci sequence

• 0

• 1

• 1

• 2

• 3

• 5

• 8

• 13

OEIS

• Online Encyclopedia of Integer Sequences

• Neil Sloane (Pres., OEIS Foundation) : oeis.org

OEIS

• Calls the Fibonacci sequence by A000045, and uses 0,1,1,2,3,5,8,13,…

• In other words, F0=0, F1=1, Fn+2=Fn+1+Fn.

OEIS

OEIS

• Fn+2 = number of binary sequences of length n that have no consecutive 0s.

Forbidden Substrings

Sarah Holliday

Southern Polytechnic State University

Forbidden substrings

• From Schilling, the number of binary sequences without a run of k zeros or ones is given by an = an-1 + an-2 + …+ an-k+1

• In the case of k = 2, we repeat the Fibonacci sequence.

Forbidden substrings

True Fact

Let |B| = r ≤ q = |A|, and S be generated by the k-action of P={σ1, σ2, ... σp} on B, where each σi is a permutation of B and P defines a latin rectangle, then for n ≥ k ≥ 3 the number of S-free strings of length n over A, an, satisfies an=(q-1)an-1 + (q-1)an-2

+ ... + (q-1)an-k+2 + (q-p)an-k+1 + p(q-r)an-k, ai=qi, i=0,1,...,k-1.

OEIS

• Fn+2 = number of binary sequences of length n that have no consecutive 0's.

• Fn+2 = number of subsets of {1,2,...,n} that contain no consecutive integers.

• Fn+1 = number of tilings of a 2 X n rectangle by 2 X 1 dominoes.

OEIS

• Fn+1 = number of matchings in a path graph on n vertices

(Emeric Deutsch)

Getting Fibonacci sequences out of graphs

Sarah Holliday

Southern Polytechnic State University

Definition: Path

• Path on n vertices (not n edges.)

• Our notation is Pn

Definition: Matching

• A matching is a set of disjoint edges.

• The set of all matchings of a graph G is M(G)

OEIS

• Fn+1 = number of matchings in a path graph on n vertices

(Emeric Deutsch)

Number of matchings in a path graph on n vertices

A B C D E

M(P5)={Ø,{AB},{BC},{CD},{DE},{AB,CD},{AB,DE},{BC,DE}}

Five vertices, eight matchings. F5 = 8

Definition: Product

• The cartesian product of graphs G and H is denoted G □ H.

• V(G □ H) = V(G) X V(H)

• E(G □ H) = E(G) X V(H) U V(G) X E(H)

G=

H=G □ H=

OEIS

• Fn+1 is the number of perfect matchings in the ladder graph Ln = P2 □ Pn

(Sharon Sela)

Defintion: Perfect matching

• A matching that uses all of the vertices of a graph.

• The set of all perfect matchings of a graph is PM(G)

OEIS

• Fn+1 is the number of perfect matchings in the ladder graph Ln = P2 □ Pn

(Sharon Sela)

Fn+1 is the number of perfect matchings in the ladder graph

Ln = P2 □ Pn

A B C D E

1

2

PM(L5)={{(A1,A2), (B1,B2), (C1,C2), (D1,D2), (E1,E2)},{(A1,B1), (A2,B2), (C1,C2), (D1,D2), (E1,E2)},{(A1,A2), (B1,C1), (B2,C2), (D1,D2), (E1,E2)},{(A1,A2), (B1,B2), (C1,D1), (C2,D2), (E1,E2)},{(A1,A2), (B1,B2), (C1,C2), (D1,E1), (D2,E2)},{(A1,B1), (A2,B2), (C1,D1), (C2,D2), (E1,E2)},{(A1,B1), (A2,B2), (C1,C2), (D1,E1), (D2,E2)},{(A1,A2), (B1,C1), (B2,C2), (D1,E1), (D2,E2)}}

Lemma

• Pn has the same number of total matchings as Ln=Pn □ P2 has perfect matchings.

• |M(Pn)| = |PM(Ln)|

Lemma

• The argument is constructive; each matching of the M(Pn) corresponds to a perfect matching of the PM(Ln) as follows: For each edge of m of M(Pn), add the two corresponding edges of Ln, and for each unsaturated vertex in m, add the edges of the K2 that use those vertices.

Lemon

• For any graph G, G □ P2 has the same number of perfect matchings as G has total matchings.

• |M(G)| = |PM(G □ P2)|

Lemon

True Fact:

• |PM(G □ P2)| =

|M(G)| + |PM(G)|(|PM(G)| - 1)

|PM(G □ P2)| = |M(G)| + |PM(G)|(|PM(G)| - 1)

• Each matching m in M(G) becomes a perfect matching p of PM(G □ P2) by duplicating the edges of m on both copies of G, and using P2 edges of G □ P2 to match the pairs of vertices unsaturated by m. Each pi of PM(G) can be placed on one of the G of G □ P2 and a distinct pj of PM(G) on t'other G of G □ P2.

Generalized Fibonacci Numbers

Sarah Holliday

Southern Polytechnic State University

Generalized Fibonacci Numbers

• Fn+2 = aFn+1 + bFn, F0 = c, F1 = d.

Generalized Fibonacci Numbers

Generalized Fibonacci Numbers

• Fn+2 = aFn+1 + bFn, F0 = c, F1 = d.

• If c = 0, with a = 1, b = 1, d = 1, these are the traditional Fn Fibonacci numbers.

• If c=1, with a = 1, b = 1, d = 1, these are the Fn+1 Fibonacci numbers.

• If c = 2, with a = 1, b = 1, d = 1, these are the Ln Lucas numbers.

H, Komatsu

• For the generalized Fn+2 = Fn+1 + Fn with F0 = c and F1 = 1, we can count the number of matchings in a “bloated” cycle.

H, Komatsu

• For the generalized Fn+2 = Fn+1 + Fn with F0 = 6 and F1 = 1, we use the cycle with (6-1) edges in the “bloat”.

A B C D E

{Ø, AB1, AB2 , AB3, AB4, AB5, BC,CD,DE,AE, AB1CD, AB2CD, AB3CD, AB4CD, AB5CD, AB1DE, AB2DE, AB3DE, AB4DE, AB5DE, BCDE, BCAE, CDEA}

True Fact

• For the generalized Fn+2 = Fn+1 + Fn with F0 = 6 and F1 = 1, we can use the path with (6) edges in the “bloat”.

A B C D E

{Ø, AB1, AB2 , AB3, AB4, AB5, AB6, BC,CD,DE, AB1CD, AB2CD, AB3CD, AB4CD, AB5CD, AB6CD, AB1DE, AB2DE, AB3DE, AB4DE, AB5DE, AB6DE, BCDE}

False fact

• |M(L’n)| = |PM(Pn □ K2)|, using bloated Pn

|M(L’n)| = |PM(P’n □ K2)|,

using bloated Pn

{vwxyz, AB1ab1xyz, AB2 ab2xyz, AB3ab3xyz, AB4ab4xyz, AB5ab5xyz, AB6ab6xyz, BCbcyz,CDcdwz,DEdevwx, AB1ab1CDcdz, AB2ab2CDcdz, AB3ab3CDcdz, AB4ab4CDcdz, AB5ab5CDcdz, AB6ab6CDcdz, AB1ab1DEdex, AB2ab2DEdex, AB3DEab3dex, AB4ab4DEdex, AB5ab5DEdex, AB6ab6DEdex, BCbcDEdev, AND AB1ab2xyz, AB1ab3xyz, AB1ab4xyz,...}

New Fact

• L’n = P’n □’ K2, using bloated Pn

• We need to define a new type of product operation, □’ in which the first copy P’n is bloated and the second copy is Pn not bloated.

Definition

• L’n = P’n □’ K2

True Fact

• For the generalized Fn+2 = Fn+1 + Fn with F0 = c and F1 = 1, we can count the number of perfect matchings in a “bloated” path “multiplied” against a path.

• |PM(L’n)| = |PM(P’n □’ K2)| = |M(P’n)|

True Fact

• For Fn+2 = Fn+1 + bFn, F0 = 1 F1 = d, we construct a graph as follows: start with a path on n vertices, with each edge inflated b times, but the first edge bloated to b-d+1 edges.

• The total number of matchings of this graph is Fn

True Fact

• Each edge inflated b times, and the first edge bloated to b-d+1 edges.

A different generalization

• Gn = Gn-1 + Gn-2 + Gn-3 + … + Gn-k+1

Definition: Matching

• A matching is a set of disjoint edges.

• A subgraph of Pn that does not contain a P3.

Gn=Gn-1 + Gn-2 + Gn-3 +…+ Gn-k+1

• The number of subgraphs of Pn+1 that do not contain Pk.

Goals

• To identify the family of graphs whose perfect matchings generate the sequences given by:

Gn = a1Gn-1 + a2Gn-2 + a3Gn-3 + … + an-1G1