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DOI: 10.1007/s00453-006-0103-y

Algorithmica (2006) 46: 493–503 Algorithmica

©2006 Springer Science+Business Media, Inc.

Asymptotics of Largest Components

in Combinatorial Structures

Mohamed Omar,1Daniel Panario,2Bruce Richmond,1and Jacki Whitely3

Abstract. Given integers mand n, we study the probability that structures of size nhave all components of

size at most m. The results are given in term of a generalized Dickman function of n/m.

Key Words. Decomposable combinatorial structures, Largest components, Permutations, Polynomials over

ﬁnite ﬁelds, Dickman function.

1. Introduction. We consider a class of decomposable combinatorial objects Asuch

that each object has a unique decomposition over a sub-class of the original class C, called

irreducible or connected components. We examine structures such as general graphs or

graphs with certain properties and their components, monic polynomials over ﬁnite ﬁelds

viewed as the product of irreducible polynomials, permutations viewed as set of cycles,

and so on.

There is a notion of size related to these combinatorial objects and their components.

We let Aidenote the number of structures of size iand let Cibe the number of those

that are connected. We let A(x)denote the generating function for the objects and C(x)

denote the generating function for the connected objects or components. It is well known

that if the objects are labelled one uses the exponential generating function and

A(x)=exp(C(x)),

and furthermore if the objects are unlabelled the ordinary generating function is appro-

priate and

A(x)=exp

k≥1

C(xk)

k.

We let An,mbe the number of structures of size nwhose largest component has size

at most m. For a general discussion over both labelled and unlabelled structures let

an=

An

n!,if the structures are labelled,

An,if the structures are unlabelled,

1Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L

3G1. momar@student.math.uwaterloo.ca; lbrichmond@math.uwaterloo.ca.

2School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6.

daniel@math.carleton.ca.

31944 Saint Margaret’s Bay Road, Timberlea, Nova Scotia, Canada B3T 1C3. jackiw@gmail.com.

Received November 30, 2004; revised January 19, 2006. Communicated by P. Jacquet, D. Panario, and

W. Szpankowski. Online publication October 28, 2006.

494 M. Omar, D. Panario, B. Richmond, and J. Whitely

cn=

Cn

n!in the labelled case,

Cn,in the unlabelled case,

an,m=

An,m

n!,in the labelled case,

An,m,in the unlabelled case.

We suppose throughout this paper that the objects satisfy the following asymptotic

conditions:

ci∼a

iRi,ai∼ia−1KRi,

where a,K, and Rare positive constants. These conditions are satisﬁed by many struc-

tures, the best known labelled example is permutations (the components are cycles), and

the best known unlabelled example is polynomials over a ﬁnite ﬁeld (the components

are the irreducible polynomials). See Section 3 for a discussion of these examples and

others. We also suppose that m≥εnfor εa positive constant in our main theorem

(Theorem 1 below).

In Section 2 we ﬁrst show that under certain restrictions almost all objects of size n

have only one component of size iwhen nand itend to inﬁnity (Lemma 1). We then

study the proportion of objects of size nwith the largest component equal to mwhen

m≥εn, for εa positive constant (Theorem 1). Our results are expressed in terms of

a generalized Dickman function ρa(u)(the Dickman function corresponds to ρ1(u)).

This problem has been considered by Gourdon [5]. In the range of applicability of our

theorem, our results simplify and extend Gourdon’s results. Our proof is elementary and

in line with the Bender et al. [1] study of smallest components. We also give bounds

for the generalized Dickman function (Theorem 2). Finally, in Section 3 we give several

examples, including one where Gourdon’s results do not apply.

2. Results and Proofs. We start by giving the following auxiliary result, of indepen-

dent interest, which covers many important combinatorial problems like permutations,

polynomials over ﬁnite ﬁelds, and square-free polynomials over ﬁnite ﬁelds.

LEMMA 1. Suppose a =1. Almost all objects of size n have at most one component

of size i as n,i→∞.Moreover,the fraction with more than one component of size i

is O(i−2).

PROOF. In the labelled case the number of objects, An,i, with no components of size i

has the exponential generating function

exp ∞

=1

cx−cixi=e−cixiA(x).

So

An,i

n!=an−an−ici+an−2ic2

i

1

2! −an−3ic3

i

1

3! +···.

Asymptotics of Largest Components in Combinatorial Structures 495

As n−i→∞,wehavefor≥3,

an−ic

i

!∼KRn−i1

iRi

!∼an(1/i)

!

≤an(1/i)3(1/i)−3

( −3)!.

So as i→∞,

¯

An,i

n!=KRn−KRn1

i+1

2! 1

i2

KRn−···

=an1−1

i+1

2! 1

i2

+Oe1/i

i3

=an1−1

i+O1

i2+Oe1/i

i3.

The number of objects of size nwith two or more components of size iis smaller than

or equal to

n

iCin−i

iCiAn−2i/2∼n!c2

ian−2i

2

∼n!K1

i2

R2iRn−2i/2∼n!KRn

2i2=n!anO(i−2).

If n−iis bounded, we must replace the asymptotic equalities in the last two equations

by O-estimates; the arguments then hold. It follows that the number of objects of size n

with one component of size iis

an1

i+O(i−2)=an

1

i1+O1

i=cn1+O1

i.

The unlabelled case is similar; the radius of convergence of ∞

k=2C(xk)/kis easily

seen to be √R, so from Schur’s lemma (see page 62 of [2]), we have

[xn]A(x)=[xn]exp(C(x)) ·exp

k≥2

C(xk)

k∼α·[xn]exp(C(x)),

where αis a constant. We may now follow the proof for the labelled case, the difference

is only the factor α.

This lemma is one way to see that frequently the combinatorial objects we are con-

sidering behave like the primes even though ciis large while there is at most one prime

of size i.

496 M. Omar, D. Panario, B. Richmond, and J. Whitely

We now suppose m≥εn, for a positive number ε. The proof of our main result uses

a modiﬁcation of the Buchstab Identity (see [6]). Let

ψ(x,y)=|{n≤x,where the largest prime factor of xis ≤y}|.

Buchstab’s identity is deﬁned as

ψ(x,y)=ψ(x,z)−

y<p≤z

ψx

p,p,

where the summation is over primes. We use the following modiﬁcations in the labelled

case since n

iAn−i,iCiovercounts structures with largest components:

An,m≥An,M−

m<i≤Mn

iAn−i,iCi=An,M−

m<i≤M

n!an−i,ici

or

an,m≥an,M−

m<i≤M

an−i,ici.

Furthermore,

An,m≤An,M−

m<i≤Mn

iCi¯

An−i,i+1

since this sum only counts structures with exactly one component of size i. Thus

an,m≤an,M−

m<i≤M

cian−i,i+11+O1

i.

(These inequalities for an,mhold in both the labelled and unlabelled cases.)

We wish, by analogy with the integers ≤nwhich have largest prime factor ≤m,to

deﬁne a function ρa(n/m)so that

an,m∼anρan

m+O1

m.(1)

Thus ρa(n/m)is the probability that the largest component of the structures of size n

is ≤m. It is well known in number theory that the probability that an integer ≤nhas

largest prime factor ≤mis ∼ρ1(log n/log m)where ρa(u)is deﬁned next.

DEFINITION 1. Let ρa(u)=1 for 0 <u≤1. We recursively deﬁne ρa(u)for k<u≤

k+1, k≥1by

ρa(u)=ρa(k)−au

k

ρa(v −1)(v −1)a−1

vadv.

Asymptotics of Largest Components in Combinatorial Structures 497

We now derive a few properties of ρa(u)which satisﬁes the differential-difference

equation

uaρ

a(u)+aρa(u−1)(u−1)a−1=0,

so

uaρa(u)=aua−1ρa(u)−a(u−1)a−1ρa(u−1),

and we have

uaρa(u)=au

u−1

va−1ρa(v) dv.(2)

It now follows that ρa(u)>0, and hence ρ

a(u)<0.Furthermore, for u>1,

ρa(u+δ) −ρa(u)=−au+δ

u

ρa(v −1)(v −1)a−1

vadv,

and since ρa(u)is decreasing and <1wehaveforδ≥0,

0≥ρa(u+δ) −ρa(u)≥−au+δ

u

(1−v−1)a−1

vdv.

It follows that for u≥2 there is an absolute constant Cso that

|ρa(u+δ) −ρa(u)|≤C|δ|.(3)

Note that ρais inﬁnitely differentiable except at u=1 where its right and left derivatives

of all orders exist.

Furthermore, if a=1 then for 1 ≤u<2, ρ1(u)=1−ln u, for 2 ≤u<3,

ρ1(u)=1−ln 2 −u

2

1−ln v

vdv

and so on. The case a=1 gives the classical Dickman’s function (for example, see [6]).

THEOREM 1. Let m ≥εn,where εis a positive constant.Let ρa(u)be deﬁned as in

Deﬁnition 1. Then

an,m

an=ρan

m+O1

m.

PROOF. The proof that (1) holds is an induction on kwhere

n

k+1<m≤n

k.

If m>nthen an,m=anso

an,m

an=1=ρn

m,(4)

498 M. Omar, D. Panario, B. Richmond, and J. Whitely

for m>n, that is, n/m<1. If n/2<m≤n, that is, 1 ≤n/m<2, then

an,m≥an,n−

m<i≤n

an−i,ici

and

an,m≤an,n−

m<i≤n

an−i,i+1ci.(5)

Now n−i<n/2so(n−i)/i=−1+n/i≤1. Also (n−i)/(i+1)<(n−i)/i≤1,

so an,m

an≥1−

m<i≤n

an−i,i

an

ci

and by (1) this is equal to

1−

m<i≤n

ρan

i−1cian−i

an1+O1

i.

Our assumptions on anand ciimply that this is equal to

1−a

m<i≤n

ρn

i−1(1−i/n)a−1

i1+O1

i.

We estimate this sum by an integral. We may use the estimate derived in [1] to do this or

we may routinely apply the Euler–Maclaurin summation estimates. Either way the error

in estimating the sum by an integral is O(1/m).Weﬁnd

m<i≤n

ρan

i−1(1−i/n)a−1

i=n

m

ρan

x−1(1−x/n)a−1

xdx +O1

m.(6)

If we let v=n/xor x=n/v,weﬁnddv=−n/x2dx,dx =−x2/ndv=−ndv/v2

so this is

1

n/m

ρa(v −1)1−1

va−1v

n−n2

v21

ndv+O1

m

=−

1

n/m

ρa(v −1)(v −1)a−1

vadv+O1

m

=n/m

1

ρa(v −1)(v −1)a−1

vadv+O1

m.

Now since n/m<2, 0 <v−1<1, ρa(v −1)≤1, and v−a≤1 so this expression is

smaller than 1 +O(1/m).

Thus using (6) in our ﬁrst inequality for an,m/angives

an,m

an≥1−an/m

1

ρa(v −1)(v −1)a−1

vadv+O1

m.

Asymptotics of Largest Components in Combinatorial Structures 499

Furthermore, from the second inequality, (5), we get

an,m

an≤1−

m<i≤n

ρan

i+1−1+1

i+1(n−i)a−1

na−1

a

i.

Now n/i−n/(i+1)=n/i(i+1)≤n/(εn)(εn+1)≤(1/ε)/m. Thus n/(i+1)−1+

1/(i+1)=n/i−1+O(1/m)and from (3) it follows that ρa(n/(i+1)−1+1/(i+1)) =

ρa(−1+n/i)+O(1/m)and we can apply the same reasoning with an extra term O(1/m)

in (6). Thus

an,m

an=1−an/m

1

ρa(v −1)(v −1)a−1

vadv+O1

m.

We have our asymptotic estimate for n/2<m≤n. If we suppose (1) holds for k−1<

n/m≤kthen since (n−i)/i<k+1−1=kand since (n−i)/(i+1)<kthe

argument above for the case k=2 is easily modiﬁed to apply and one obtains that for

n/(k+1)<m<n/k,

an,m

an=ρa(k)−an/m

k

ρa(v −1)(v −1)a−1

vadv+O1

m.

Thus by induction (1) holds for m≥εn.

REMARK 1. When a=1, ρ1(u)is the original Dickman function.

REMARK 2. Theorem 1 may be compared with one of Gourdon’s results in his Ph.D.

thesis on the largest component of a random structure [5]. Our assumptions here are

slightly weaker as we show by example later, and so we extend his results. However,

when the Flajolet–Odlyzko singularity analysis applies, we have the same results as

Gourdon. On the other hand, our elementary results only hold for m≥εnwhereas

Gourdon proves that

an,m

an=ρan

m+O1

mfor 2 ≤m≤n.

Gourdon does not determine the asymptotic behaviour of ρa(n/m). However, ρ1(u)

is the classical Dickman function and it is well known that ρ1(u)=O(1/(u)).It

follows that when a=1, Gourdon’s leading term is large compared with O(1/m)when

m>n/(log n). Our result only holds for m>εn, of course, but perhaps it is surprising

that our elementary result is so close to the best known result as far as the range of mis

concerned. An analysis similar to that of Tenenbaum [6] gives the asymptotic behaviour

of ρa(u)for all a. As far as we know, such an analysis has not been published and it

would be much more intricate than the one we now give.

We now bound ρa(u).

THEOREM 2. There is a constant C1such that for u >1and a <1,

ρa(u)≤C1au,

500 M. Omar, D. Panario, B. Richmond, and J. Whitely

while for a ≥1there is a constant C2so that

ρa(u)≤Cu

2

(u+1).

PROOF. Let u>1. If 0 <a<1, (u−1)a−1≥ua−1so (u−1)a−1ρa(u−1)≥

ua−1ρa(u−1)≥ua−1ρa(u). We have already seen that

uaρa(u)=aua−1ρa(u)−a(u−1)a−1ρa(u−1)

so uaρa(u)is a decreasing function. Furthermore, from (2)

uaρa(u)=au

u−1

va−1ρa(v) dv=au

u−1

1

vvaρa(v) dv

<au

u−1

vaρa(v) dv<a(u−1)aρa(u−1),

so

ρa(u)< a

ua(u−1)aρa(u−1).

Therefore, it follows by induction that

ρa(u)≤C1au,

for some constant C1and the ﬁrst part of the theorem is proved.

Suppose now a≥1. We have

uaρa(u)≤ρa(u−1)u

u−1

ava−1dv

=ρa(u−1)(ua−(u−1)a)≤ρa(u−1)C2ua−1,

where C2is a constant, so

ρa(u)≤C2

uρa(u−1).

By induction therefore

ρa(u)≤Cu

2

(u+1)

and the second part of the theorem is proved.

3. Examples. The following examples are from [4] and appear also in [5].

•Permutations. The exponential generating function for cycles in permutations is

C(x)=log(1/(1−x)). Therefore Cn=(n−1)! and hence cn=(n−1)!/n!=1/n.

On the other hand, the radius of convergence of the function log(1/(1−x)) is 1.

Consequently, the theorem applies with a=K=1. This gives rise to ρ1.

Asymptotics of Largest Components in Combinatorial Structures 501

•Polynomials. The ordinary generating function for monic polynomials over a ﬁnite

ﬁeld Fqis 1/(1−qx)and we have the well-known approximation for Cn, the number

of irreducible polynomials of degree n,

Cn=qn

n+O(qn/2).

From this approximation we can ﬁnd R, the radius of convergence of C(x), as follows:

R=lim sup

n→∞ |Cn|−1/n=lim sup

n→∞

qn

n+O(qn/2)

−1/n

=q−1.

From our deﬁnition cn=Cn=qn/n+O(qn/2)in the case of unlabelled objects. We

now compute the term 1/(nRn):

1

nRn=R−1n

n=qn

n.

Hence a=K=1 and the theorem applies. The case of squarefree polynomials has

the same asymptotics for an,m/an.

•2-regular graphs. The exponential generating function for labelled 2-regular graphs

is

e−x/2−x2/4

(1−x)1/2=exp 1

2log 1

1−x−x

2−x2

4.

Hence the exponential generating function for the components is

1

2log 1

1−x−x

2−x2

4.

Extracting coefﬁcients yield in C1=C2=0 and

Cn=(n−1)!

2.

Hence cn=1/2nand this gives rise to ρ1/2.

The unlabelled case has the same asymptotics for an,m/an.

•Labelled functional digraphs.IfT(x)denotes the exponential generating function

for labelled rooted trees (deﬁned by T(x)=xexp(T(x))), then we have for cycles

of labelled trees the generating function

log 1

1−T(x)=1

2log 1

1−ex +O(√1−ex).

This is the generating function for labelled funtional digraphs. This gives an example

with a=1

2,K=1.

•Unlabelled functional digraphs. This is also an example with a=1

2,K=1. If

one lets A(x)denote the ordinary generating function for unlabelled rooted trees then

A(x)=xexp(A(xk/k)) and the generating function for cycles of these trees is

∞

k=1

ϕ(k)

klog 1

1−A(xk),

502 M. Omar, D. Panario, B. Richmond, and J. Whitely

where ϕ(k)is Euler’s function. This is the generating function for unlabelled functional

digraphs.

•Coloured permutations. Now we give an example where Gourdon’s results do not

apply but ours do. Colour the integers {1,2,...,n}red and green and consider per-

mutations of them where the colours do not change under the mapping. We can think

of these permutations as being composed of red and green cycles. Let us add the

following reﬁnement; consider a cycle

12··· n

i1i2··· 1,

where as usual is mapped to iand add the restriction that i1=2. There are

clearly (n−2)! such cycles. Let us now consider permutations such that when n=

4,8,...,2i,...,i≥2, no cycles

12··· n

2i2··· 1,

are allowed. Let cndenote the number of such cycles and let anbe the number of

permutations constructed from these cycles. Then

∞

n

anxn=exp

n≥0

cnxn

n!=exp 2

i≥1

xi

i−2

i≥2

x2i

2i(2i−1).

Clearly ci∼2(i−1)!. We have that

ai∼iexp −2

i≥2

1

2i(2i−1)=Ci,

where Cis a constant. We have

∞

n=0

anxn=1

(1−x)2exp −2

i≥2

x2i

2i(2i−1).

An elementary Abelian argument shows that

∞

n=0

anxn∼C

(1−x)2as x→1.

Now

exp(−2i≥1(x2i/2i(2i−1)))

1−x=exp

i≥1

xi

i−2

i≥2

x2i

2i(2i−1)=

i≥2

bixi,

where bi≥0. Thus the anabove are monotonic increasing. From the Hardy–

Karamata–Littlewood Tauberian theorem [3]

an∼Cn.

Asymptotics of Largest Components in Combinatorial Structures 503

We observe that singularity analysis does not apply since

i≥2

x2i

2i(2i−1)

is a lacunary series, so |x|=1 is a natural boundary for it and therefore for ∞

n=0anxn.

We have that k-coloured permutations give examples where a=k,K=1/(k−1)!.

References

[1] Bender, E., Mashatan, A., Panario, D., and Richmond, B. Asymptotics of combinatorial structures with

large smallest component. Journal of Combinatorial Theory,Series A,197 (2004), 117–125.

[2] Burris, S.N. Number Theoretic Density and Logical Limit Laws. Mathematical Surveys and Monographs,

vol. 86, American Mathematical Society, Providence, RI (2000).

[3] Feller, W. An Introduction to Probability Theory and Its Applications. Wiley, New York, 1966

[4] Flajolet, P., and Soria, M. Gaussian limiting distribution for the number of components in combinatorial

structures. Journal of Combinatorial Theory,Series A,53 (1990), 165–182.

[5] Gourdon, X. Combinatoire, algorithmique et g´eom´etrie des polynˆomes. Ph.D. thesis, ´

Ecole Polytech-

nique, Paris, 1996.

[6] Tenenbaum, G. Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press,

Cambridge, 1995.