Forests of chord diagrams with a given number of trees

0.08 MB·English

by  Huseyin Acan

Forests of chord diagrams with a given number of trees
Hu¨seyin Acan
School of Mathematical Sciences
Monash University
5
Melbourne, VIC 3800
1
0 Australia
2 [email protected]

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Abstract
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In this short note we find the number of forests of chord diagrams with a given number of trees and

] a given number of chords.
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Keywords: chord diagram, tree, forest, forest chord diagram.
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h 2010 AMS Subject Classification: Primary: 05A15; Secondary: 05A10
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A chord diagram of size n is a pairing of 2n points. One can take 2n points on a circle, label them
m
with the numbers 1,...,2n in clockwise order, and join the two points in a pair with a chord for a pictorial
[
representation. Given a chord diagram C, the intersection graph G is defined as follows: the vertices of
C
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G correspond to the chords of C and two vertices in G are adjacent if and only if the corresponding
v C C
2 chords cross each other in C; see Figure 1. When GC is a tree or forest, we call C a tree chord diagram or
9 a forest chord diagram, respectively.
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1
3
0 2 4
. (2,9) (3,5)
1 1 5
0
5 10 6
1
: 9 7 (1,8) (7,10) (4,6)
v 8
i
X
Figure 1: A chord diagram with five chords and the corresponding intersection graph.
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Dulucq and Penaud [1] found the number of tree chord diagrams with n chords as 3n−3 /(2n−1) by
n−1
showing a bijection between tree chord diagrams with n chords and ternary plane trees with n−1 internal
(cid:0) (cid:1)
vertices. In this note we find f(n,m), the number of forests of chord diagrams consisting of n chords and
m trees. We also find r(n,m), the number of forest chord diagrams consisting of n chords and m rooted
trees. In particular, we prove the following two theorems.
Theorem 1. For positive integers n and m such that m ≤ n, we have
1 · 2n · 3n−2m−1 , if 1 ≤ m ≤ n−1;
f(n,m)= n−m m−1 n−m−1
(n+11 · (cid:0)2nn , (cid:1) (cid:0) (cid:1) if n = m.
Theorem 2. For positive integers n and m(cid:0) s(cid:1)uch that m ≤ n, we have
1 2n
r(n,m)= Σ +Σ ,
1 2
m m−1
(cid:18) (cid:19)
(cid:0) (cid:1)
1
where
m n−m−1
m m+j −1 2j +k 3n−3m+k−j −1
Σ = (−1)k 2m−k3j
1
k j n−m−j n−m−j −1
k=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
X X
and
m
m n−1
Σ = (−1)k 2m−k3n−m.
2
k n−m
k=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
The generating function G(x) of ternary trees satisfies the equation G(x) = 1+xG(x)3. For a positive
integer k, let t denote the number of tree chord diagrams with k chords and let T(x) = t xk. By
k k≥1 k
the bijection shown by Dulucq and Penaud [1], we have T(x) = xG(x). Hence, using the above relation
P
for G(x), we get
3
T(x) T(x)
= 1+x . (1)
x x
(cid:18) (cid:19)
Before we start our proofs, we define non-crossing partitions and cite a result about such partitions
with given parameters.
Definition. Two blocks of a partition of the set [m] := {1,...,m} cross each other if one block contains
{a,c} and the other block contains {b,d} where a < b < c < d. A partition of [m] is called a non-crossing
partition if no two blocks cross each other.
The relevance of this definition is that the components of a chord diagram with n chords determine a
non-crossing partition of [2n]. (The components of a chord diagram C correspond to the components of
G .) More precisely, each block of the partition consists of the vertices of the chords in a component.
C
In order to count the forests with m trees, we will consider non-crossing partitions of [2n], and for
each such partition we will count the trees in each block. The following theorem about the number of
non-crossing partitions was proved by Kreweras [2]. (See Stanley [3, Exercise 5.35] for a bijective proof.)
Theorem 3 (Kreweras). The number of non-crossing partitions of [n] with s blocks of size j is (n)k−1 ,
j s1!···sn!
where k = s +···+s and (n) = n(n−1)···(n−k+2).
1 n k−1
Proof of Theorem 1. For n = m, we have n non-crossing chords, and the number of such configurations
is given by the nth Catalan number, which is (2n)!/(n!(n+1)!). Now assume 1≤ m < n.
For a component H of a chord diagram, we definethesupport of H as the setof endpoints of its chords.
Let {B ,...,B } be the set of blocks in a non-crossing partition of [2n], where each block has an even
1 m
cardinality. The number of forest chord diagrams such that each B is a tree is t ×···×t . In
j |B1|/2 |Bm|/2
order to find the number of forests with m trees, we need to take a sum over all non-crossing partitions of
[2n] with m blocks of even cardinality.
We say that a non-crossing partition of [2n] has type (s ,...,s ) when there are s blocks of size 2i.
1 n i
In that case we have n is = 2n. By Theorem 3, the number of non-crossing partitions of [2n] of type
i=1 i
(s ,··· ,s ) is (2n)!/(R!s !···s !), where R = 2n+1−(s +···+s ). Hence,
1 n 1 n 1 n
P
(2n)!
f(n,m)= ts1···tsn , (2)
1 n (2n+1−m)!s !···s !
1 n
s1X,...,sn
where the sum is over all nonnegative integers s ,...,s such that
1 n
n n
s = m and is = n. (3)
i i
i=1 i=1
X X
2
We write (2) as
(2n)! ts1 tsn
f(n,m)= 1 ··· n
(2n+1−m)! s ! s !
s1,...,sn 1 n
meeXting(3)
(2n)! T(x)m
= [xn]
(2n+1−m)! m!
1 2n
= [xn]T(x)m. (4)
m m−1
(cid:18) (cid:19)
Finally, we need to find the coefficient of xn in the series T(x)m. By Eq. (1), we have z = x(z+1)3, where
T(x)
z =z(x) = −1. Hence,
x
m
T(x)
[xn]T(x)m = [xn−m] = [xn−m](z+1)m
x
(cid:18) (cid:19)
m m
m m k
= [xn−m]zk = [xn−m−k](x+1)3(n−m), (5)
k k n−m
k=0(cid:18) (cid:19) k=0(cid:18) (cid:19)
X X
where the last equality follows from the Lagrange inversion formula. Thus,
m
1 2n m k 3(n−m)
f(n,m)=
m m−1 k n−m n−m−k
(cid:18) (cid:19)k=0(cid:18) (cid:19) (cid:18) (cid:19)
X
m
1 2n m−1 3(n−m)
=
n−m m−1 k−1 n−m−1−(k−1)
(cid:18) (cid:19)k=1(cid:18) (cid:19)(cid:18) (cid:19)
X
1 2n 3n−2m−1
=
n−m m−1 n−m−1
(cid:18) (cid:19)(cid:18) (cid:19)
as desired.
Proof of Theorem 2. Analogous to Eq. (2), we have
(2n)!
r(n,m) = (1·t )s1(2·t )s2···(n·t )sn , (6)
1 2 n
(2n+1−m)!s !···s !
1 n
s1X,...,sn
since in this case we need to choose a root for each of the trees in a forest. The sum is over all nonnegative
integer tuples (s ,...,s ) satisfying (3). We rewrite (6) as
1 n
(2n)! (1·t )s1 (n·t )sn
1 n
r(n,m) = ··· . (7)
(2n+1−m)! s ! s !
1 n
s1X,...,sn
Let B(n,m) be this sum without the coefficient (2n)!/(2n+1−m)!, i.e.,
(1·t )s1 (n·t )sn
1 n
B(n,m)= ··· .
s ! s !
1 n
s1X,...,sn
3
We have
(1·t )s1 (n·t )sn
B(n,m)xmyn = 1 ··· n xmyn
s ! s !
1 n
nX,m≥1 Xn,ms1X,...,sn
(jt xyj)s
j
=
s!
j≥1s≥0
YX
= exp x jt yj = exR(y),
j
 
j≥1
X
 
where R(y) = nt yn. Thus, B(n,m) is the coefficient of xmyn in the function exR(y), which is the
n≥1 n
same as the coefficient of yn in R(y)m/m!. Hence, using (7)
P
1 2n
r(n,m)= [xn]R(x)m. (8)
m m−1
(cid:18) (cid:19)
Finally, we need to find the coefficient of xn in R(x)m. First note that R(x) = xT′(x), where T(x) is the
generating function for unrooted trees. By (1), T(x) satisfies
xT(x) = x2+T(x)3,
from which it follows
2x−T(x)
R(x)= xT′(x) = x· .
x−3(T(x))2
Then,
m
2x−T(x)
[xn]R(x)m = [xn−m] .
x−3T(x)2
(cid:18) (cid:19)
Writing now z =T(x),
z m 3z2 −m
[xn]R(x)m = [xn−m] 2− 1−
x x
m m (cid:16) (cid:17) (cid:16)z k ∞(cid:17) −m z2 j
=[xn−m] 2m−k(−1)k × (−3)j
k x j x
!
Xk=0(cid:18) (cid:19) (cid:16) (cid:17) Xj=0(cid:18) (cid:19) (cid:18) (cid:19)
m n−m
m −m
= 2m−k(−3)j(−1)k[xn−m+k+j]z2j+k
k j
k=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19)
X X
m n−m−1
m m+j−1 2j +k 3n−3m+k−j −1
= (−1)k 2m−k3j
k j n−m−j n−m−j −1
k=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
X X
m
m n−1
+ (−1)k 2m−k3n−m, (9)
k n−m
k=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
where we use
a 3b−2a−1 , if a < b;
[xb]za = b−a b−a−1
(1, (cid:0) (cid:1) if a = b
in the last equality above (cf. Eq. (5)). Combining (8) and (9), we get
1 2n
r(n,m)= Σ +Σ ,
1 2
m m−1
(cid:18) (cid:19)
(cid:0) (cid:1)
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where
m n−m−1
m m+j −1 2j +k 3n−3m+k−j −1
Σ = (−1)k 2m−k3j
1
k j n−m−j n−m−j −1
k=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
X X
and
m
m n−1
Σ = (−1)k 2m−k3n−m,
2
k n−m
k=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
which finishes the proof.
References
[1] S. Dulucq and J.-G. Penaud, Cordes, arbres et permutations, Discrete Math. 117 (1993), no. 1–3,
89–105.
[2] G. Kreweras, Sur les partitions non crois´ees d’un cycle, Discrete Math. 1 (1972), no. 4, 333-350.
[3] R. P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999.
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