Unlabeled Motzkin numbers - Canadian Mathematical … · Unlabeled Motzkin numbers Consider a circle with n equally spaced points, which we will call vertices. A set of noncrossing

Unlabeled Motzkin numbers

Max Alekseyev

Dept. Computer Science and Engineering

2013

Max Alekseyev Unlabeled Motzkin numbers

Catalan numbers

Catalan numbers can be defined by the explicit formula:

Cn =1

n + 1

(2n

n

)=

(2n)!

n!(n + 1)!

and the ordinary generating function:

C(x) =∞∑n=0

Cn · xn =1−√

1− 4x

2x.

Catalan numbers for n = 0, 1, . . . form the sequence (A000108 in OEIS):

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, . . .


Chord configurations

We are particularly interested in the combinatorial interpretation of Cn asthe number of expressions containing n properly embedded pairs ofparentheses. For n = 3, these expressions are:

(

1

(

2

)

3

(

4

)

5

)

6

(

1

(

2

(

3

)

4

)

5

)

6

(

1

)

2

(

3

(

4

)

5

)

6

(

1

)

2

(

3

)

4

(

5

)

6

(

1

(

2

)

3

)

4

(

5

)

6

which can be further interpreted as the number of configurations of nnoncrossing chords connecting 2n labeled points on a circle:

6

5

4

3

2

1

6

5

4

3

2

1

6

5

4

3

2

1

6

5

4

3

2

1

6

5

4

3

2

1


Motzkin numbers

Catalan number Cn represents the number of configurations of nnoncrossing chords connecting 2n labeled points on a circle.

Motzkin number Mn represents the number of configurations of (anynumber of) noncrossing chords connecting n labeled points on a circle.

We can easily expressed Motzkin numbers in terms of Catalan numbers:

Mn =

bn/2c∑k=0

(n

2k

)Ck .

The generating functions of Motzkin numbers is:

M(x) =∞∑n=0

Mn · xn =1

1− x· C(

x2

(1− x)2

)=

1− x −√

1− 2x − 3x2

2x2.


Unlabeled Motzkin numbers

Consider a circle with n equally spaced points, which we will call vertices.A set of noncrossing chords connecting vertices is called a chordconfiguration.

We define two types of unlabeled Motzkin numbers counting the numberof chord configurations on unlabeled vertices. Namely, we define cyclic anddihedral Motzkin numbers counting the number of chord configurations upto the action of cyclic and dihedral groups, respectively.


Cyclic and dihedral Motzkin numbers

Cyclic Motzkin number MCn represents the number of chord configurations

on n vertices under the action of the cyclic group (of rotations) Cn.Burnside lemma allows us to give the following expression for MC

n .

MCn =

1

n

∑c∈Cn

Hc , (1)

where Hc is the number of chord configurations invariant w.r.t. c .

Similarly, dihedral Motzkin number MDn represents the number of chord

configurations on n vertices under the action of the dihedral group Dn.Viewing elements of Dn as n rotations, forming the cyclic subgroup Cn,and n reflections, forming a set Rn, we compute MD

n as follows:

MDn =

1

2n

∑c∈Cn

Hc +∑r∈Rn

H r

=1

2MC

n +1

2n

∑r∈Rn

H r . (2)


Periods and special configurations

We define the period of a chord configuration S as the smallest positiveinteger p such that S is invariant w.r.t. rotation of the circle by the anglep · 2πn .Clearly, the period of any chord configuration on n vertices divides n.

A chord configuration on n vertices is called special if it contains a chordconnecting two diametrically opposite vertices.

Special configurations exist only for even n.

Period of a special configuration can be only n or n/2.

The number of special configurations of period n/2 equals Mn/2−1.

The number of special configurations of period n is(Mn/2−1

2

).


Chord configurations and periods

Below we list of all configurations of chords connecting n (2 ≤ n ≤ 6)vertices and specify their periods p.

p = 1

p = 1 p = 3p = 3 p = 3

p = 3

p = 2 p = 2

p = 2

p = 4

p = 5 p = 5 p = 5

p = 6 p = 6

p = 5

p = 6 p = 6 p = 6 p = 6

n = 4:

n = 5:

n = 6:

n = 3: p = 1 p = 3n = 2: p = 1p = 1

p = 1


Nonspecial chord configurations

A chord c in a nonspecial chord configuration partition the vertices otherthan the endpoints of c into two subsets formed by vertices laying at thesame side of c .We define the span of c as the smaller of these subsets together with theendpoints of c.

For a configuration of period m < n, the span of each chord does notexceed m.

A chord is called maximal if its endpoints do not reside within the span ofany other chord. It is easy to see that all chords in a chord configurationreside within the spans of the maximal chords.


Nonspecial configurations of fixed period

Let b(n,m) be the number of nonspecial configurations on n vertices,whose period equals m. Clearly, b(n,m) can be non-zero only if m dividesn.

Rotation of a chord configuration of period m by the angle m · 2πntranslates maximal chords into maximal chords. Therefore, the number ofmaximal chords in such configuration must be a multiple of n/m.

For m | n, define b(n,m, k) as the number of chord configurations on nvertices of period m with k · n/m maximal chords.Similarly, let c(n,m, k) be the number of such configurations with alabeled maximal chord.

Clearly,

b(n,m) =∑k≥0

b(n,m, k).

So our goal is to find b(n,m, k).Max Alekseyev Unlabeled Motzkin numbers

Formula for b(n,m, k)

Theorem

For m | n, m < n, and k ≥ 1, c(n,m, k) equals the coefficient of xmy k in(1− yM(x)

x2

1− x

)−1

=

(1− y

1− x −√

1− 2x − 3x2

2(1− x)

)−1

.

Lemma

For m | n and k ≥ 1,

c(n,m, k) =∑

d|(m,k)

b(n, m/d, k/d) · k/d.

LemmaFor m | n, we have b(n,m, 0) = [m = 1] and for k ≥ 1,

b(n,m, k) =1

k

∑d|(m,k)

µ(d) · c(n, m/d, k/d),

where µ(·) is Mobius function.


Proof of theorem

Proof.Consider an arbitrary chord configuration on n vertices with period dividing m andcontaining k · n/m maximal chords, one of which is labeled. Let P be set of m consecutivevertices on the circle that starts with the counterclockwise endpoint of the labeledmaximal chord and goes clockwise. Then P contains the spans of k maximal chords.Let ti (1 ≤ i ≤ k) be the size of the span of the i-th (counting clockwise) maximalchord in P. Then the number of chord configurations within this span is Mti−2.Let zi (1 ≤ i ≤ k) be the number of vertices between the endpoints of i-th and (i + 1)-thmaximal chords (or the end of P for i = k) so that the total number of vertices is

t1 + z1 + t2 + z2 + · · ·+ tk + zk = m.

Then c(n,m, k) as the total number of chords configurations within P equals

∑t1+z1+···+tk+zk=m

Mt1−2·Mt2−2 · · ·Mtk−2 = [xm−2k ]M(x)k(1−x)−k = [xm]

(M(x)

x2

1− x

)k

.

We multiply this by y k and sum over k ≥ 0 to get

c(n,m, k) = [xmy k ](

1− yM(x) x2

1−x

)−1

.


Generating function for b(n,m)

We define the following functions:

T (x) = − ln(1− x − x2 · M(x)

),

B(x) =∞∑q=1

µ(q)

q· T (xq) = ln

∞∏q=1

(1− xq − x2q · M(xq)

)−µ(q)/q,

F(x) =∞∑q=1

ϕ(q)

q· T (xq) = ln

∞∏q=1

(1− xq − x2q · M(xq)

)−ϕ(q)/q,

where ϕ(·) is Euler totient function.

Lemma

For positive integers m | n with m < n,

b(n,m) = [xm] B(x).


Configurations of fixed period

Let b′(n,m) be the number of chord configurations (both special andnonspecial) whose period equals m. Clearly, b′(n,m) can be non-zero onlyif m divides n, Mn =

∑m|n b′(n,m) ·m, and MC

n =∑

m|n b′(n,m).

Lemma

For m|n, we have b′(n,m) = b(n,m) if m < n/2.Furthermore, b′(n, n/2) = b(n, n/2) + Mn/2−1 if n is even, and

b′(n, n) =1

n

Mn −∑m|nm<n

b′(n,m) ·m

.


Cyclic Motzkin numbers

Theorem

The generating function for the number of asymmetric chord configurations b′(n, n) is

∞∑n=0

b′(n, n) · xn =1− (1− x) · M(x)− x2 · M(x2)

2+ ln(M(x)) + B(x)− T (x)

=1− (1− x) · M(x)− x2 · M(x2)

2+ ln(M(x)) + ln

∏q≥2

(1− xq − x2q · M(xq)

)−µ(q)/q.

Theorem∞∑n=0

MCn · xn =

1− (1− x) · M(x) + x2 · M(x2)

2+ ln(M(x)) + F(x)− T (x)

=1− (1− x) · M(x) + x2 · M(x2)

2+ ln(M(x)) + ln

∏q≥2

(1− xq − x2q · M(xq)

)−ϕ(q)/q

.


Dihedral Motzkin numbers: computing H r

Lemma

For an even n = 2m and a fixed reflection r ∈ Rn about a diameter connecting centersof two arcs of the circle, we have

H r = [xm]M(x)

1− x · M(x).

Lemma

For an even n = 2m and a fixed reflection r ∈ Rn about a diameter connecting two ofthe vertices, we have

H r = [xm]1

1− x · M(x)+ Mm−1 = [xm]

(1

1− x · M(x)+ x · M(x)

).

LemmaFor an odd n = 2m + 1 and a fixed reflection r ∈ Rn, we have

H r = [xm]M(x)

1− x · M(x).


Dihedral Motzkin numbers

LemmaFor even n = 2m, we have∑r∈Rn

H r = [xm] m·(

1 +M(x)

1− x · M(x)+ x · M(x)

)= [xn]

n

2

(1 +M(x2)

1− x2 · M(x2)+ x2 · M(x2)

).

For odd n = 2m + 1,∑r∈Rn

H r = [xm] (2m + 1)M(x)

1− x · M(x)= [xn] n

x · M(x2)

1− x2 · M(x2).

TheoremFor any integer n ≥ 0,

1

2n

∑r∈Rn

H r = [xn]

(1 + (2x + 1) · M(x2)

4(1− x2 · M(x2))+

x2 · M(x2)

4

).


Dihedral Motzkin numbers

Using formula MDn = 1

2MCn + 1

2n

∑r∈Rn

H r , we deduce the generatingfunction for dihedral Motzkin numbers:

∞∑n=0

MDn · xn =

1− (1− x) · M(x) + 2x2 · M(x2)

4

+ln(M(x)) + F(x)− T (x)

2

+1 + (2x + 1) · M(x2)

4(1− x2 · M(x2))


Unlabeled Motzkin number in OEIS

Currently there are three sequences in the Online Encyclopedia of IntegerSequences (OEIS) http://oeis.org related to unlabeled Motzkinnumbers:

A175954 Cyclic Motzkin numbers

A175955 Number of asymmetric (w.r.t. rotations) chordconfigurations b′(n, n)

A185100 Dihedral Motzkin numbers


http://oeis.org

Acknowledgements

Dr. Glenn Tesler, University of California, San Diego

National Science Foundation grant no. IIS-1253614


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Unlabeled Motzkin numbers - Canadian Mathematical … · Unlabeled Motzkin numbers Consider a circle with n equally spaced points, which we will call vertices. A set of noncrossing