Part-products of 1-free integer compositions
Caroline Shapcott∗
Department of Mathematics
Drexel University
Philadelphia, PA 19104, U.S.A.
cshapcott@drexel.edu
Submitted: Jun 24, 2011; Accepted: Sep 23, 2011; Published: Dec 19, 2011
Mathematics Subject Classifications: 05A16, 60C05, 60F05, 60G40
Abstract
If ~λ is a composition of the positive integer n, define B(~λ) to be the product
of the parts of ~λ. We present a modified version of Hitczenko’s stopped sequence
construction that leads to a proof of the asymptotic lognormality of B for random
1-free compositions (compositions containing no parts of size 1).
1
Introduction
A composition of n is a sequence of positive integers whose terms, referred to as parts,
sum to n. In recent years, there has been considerable interest in compositions with
restrictions on the sizes of the parts. One set of restricted compositions that has been
studied extensively is that of Carlitz compositions, whose adjacent part sizes cannot be
equal [8, 13, 15]. A generalization of Carlitz compositions is introduced by Bender and
Canfield in [2, 3, 4] where the authors study locally restricted compositions, whose part
sizes are restricted to a certain set depending on the position of the part. A variety of
other restricted compositions have been studied, including complete and gap-free compositions, whose part sizes form an interval [10]; compositions with distinct part sizes [14];
compositions with the largest part in the first position [16]; compositions with no parts
of size 2 [6]; and compositions whose parts sizes are restricted to a general set [1, 5].
Problems that are straightforward for unrestricted compositions can become quite
complicated when such restrictions are imposed. In [18] it is proven that, for unrestricted
compositions, the product of parts is asymptotically lognormal. The proof relies heavily on a construction, devised by Hitczenko and others, that models compositions with
Supported in part by N.S.A. grant H98230-09-1-0062. Portions of this work will appear in the author’s
doctoral dissertation, written under the direction of Eric Schmutz at Drexel University.
∗
the electronic journal of combinatorics 18 (2011), #P235
1
stopped sequences of geometric random variables. However, the proof does not extend
immediately to any restricted part-size setting of which we are aware. In particular, one
might expect the proof to extend to compositions with no parts of size 1, since those
parts do not contribute to the product. However, non-trivial adaptations are needed to
make Hitczenko’s technique usable in this setting. The main result in this manuscript is
Theorem 1.
Theorem 1. Let Pn be the uniform probability measure on the set of 1-free compositions
∞
∞
√
P
P
log k
k log k
of n, and let φ = 1+2 5 be the golden ratio. Define the constants µ =
,
ν
=
,
k
φ
φk
k=2
k=2
∞
2 2 P
(log k)2
φ
2
µ̄ =
, and γ = 1 + 5 µ − 1+φ2 µν + µ̄. Then
φk
k=2

log B −
Pn  q
nµ
1+φ2
nγ 2
1+φ2

1
≤ x = √
2π
The rate of convergence is uniform for |x| ≤
Zx
−t2 /2
e
dt + O
−∞
√
n
log n
2/3
log
√n
n
1/3 .
.
We present the new construction used for modeling 1-free compositions, as well as
some preliminary results, before presenting the proof of the main theorem. Throughout
this manuscript, we denote the set of 1-free compositions of n as Λn and an individual
composition having κn parts as ~λ = (λ1 , . . . , λκn ). Pn denotes the uniform probability
measure on Λn , and En denotes the expected value with respect to Pn . If F is a formal
power series in x, then we write [xn ]F to denote the coefficient of xn in F .
2
Expected value of B and log B
Moments of B can be computed using the following lemma. Let ΛSn denote the set of all
compositions of n with parts from some set of positive integers S, and let ΛSn,d denote the
set of all such compositions having d parts.
P
Lemma 1. For a ∈ C, define αn =
B(~λ)a . The generating function for the sequence
~
λ∈ΛS
n
αn is
X
αn xn =
n
Proof. Let Ga (x) =
P
1−
1
P
k a xk
.
k∈S
k a xk . Then the coefficient of xn in Ga (x)d is the sum of the a-th
k∈S
power of the part-products over all compositions with d parts:
X Y
X
[xn ]Ga (x)d = [xn ]
λai xλi = [xn ]
B(~λ)a xn .
~λ∈ΛS
n,d
i
the electronic journal of combinatorics 18 (2011), #P235
~λ∈ΛS
n,d
2
Summing on d we get
n
αn = [x ]
n
X
Ga (x)d = [xn ]
d=1
1
.
1 − Ga (x)
Lemma 1 is a new generalization of the following widely known generating function
for the number of compositions with parts from a general set [7].
Corollary 1. The generating function for the number of compositions of n with parts
from the set S is
X
1
P k.
|ΛSn |xn =
1−
x
n
k∈S
Corollary 2. If ~λ is chosen from a uniform distribution on the set of compositions of n
into parts from the set S, then the expected value of the random variable B is
1
1
n
P k [xn ]
P k.
[x ]
1−
kx
1−
x
k∈S
k∈S
In [18], a probabilistic proof is given to show that the average product over all unrestricted compositions of n is F2n /2n−1 , where F2n is the 2n-th Fibonacci number. An
alternative approach is to cite Corollary 2 with S = Z+ . It is interesting that the Fibonacci numbers also appear in two special cases that are important for this manuscript.
The most prominent occurrence is in the probabilistic construction that will be presented
in the next section. In addition, Corollary 1 with S = {k : k ≥ 2} gives the following
known result.
Corollary 3. The number of 1-free compositions of n is Fn−1 .
Corollary 2 also yields an asymptotic formula for the expected value of B for 1-free
compositions.
Theorem 2. Let ρ1 , ρ2 , and ρ3 be the distinct real roots of the function x3 − x2 − 2x + 1.
If ~λ is chosen from a uniform distribution on the set of 1-free compositions of n, the
expected value of the product of parts is
En (B) =
1
Fn−1
3
X
(1 − ρi )2 ρ−n
i
.
2
2ρ
+
2ρ
−
3ρ3i
i
i
i=1
Asymptotically, the expected value is
En (B) ∼
√
5
φn−1
·
(1−ρ1 )2 ρ−n
1
2ρ1 +2ρ21 −3ρ31
= (1.09 . . . )(1.3887 . . . )n .
As a result of Theorem 1, we know that the expected value of the log product of parts
nµ
is approximately 1+φ
This agrees with results from [18] where the following precise
2.
estimate is obtained.
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3
Theorem 3. If ~λ is chosen from a uniform distribution on the set of 1-free compositions
of n, the expected value of the log product of parts is
nµ
φ
ν
n log n
En (log B) =
+
+1 µ−
+O
.
1 + φ2
5
1 + φ2
φn
The proof of Theorem 3 requires the probabilistic construction presented in Section
3 as well as a detailed consideration of the distribution of the last part λκn . The proof
is omitted because it involves lengthy calculations similar to those in the proof of the
asymptotic lognormality of B.
3
Stopped sequence construction
We begin the new construction by defining an auxiliary space Λ∗n that contains all 1free compositions of n as well as all compositions of n that are 1-free except for the last
part. Let Ωn be the set of all n-length sequences ~ω = (ω1 , . . . , ωn ) with ωi ≥ 2, and let
τn = min{t ≥ 1 : ω1 · · · + ωt ≥ n} be a stopping time associated with the sequence ~ω . We
can then construct random compositions of n by defining the function
!
τX
n −1
Hn (~ω ) = ω1 , · · · , ωτn −1 , n −
ωi .
i=1
Notice that Hn does not always return 1-free compositions of n, since it is possible for the
last part to be a 1. Moreover, if we define a probability measure on the space Λ∗n in the
manner of [10, 11, 12], then Hn does not select compositions uniformly.
Formally, if we select the ωi ’s independently with probability Qn (ωi = k) = φ1k for
k ≥ 2 and define a probability measure on Ωn by
Qn (~ω ) =
1
φω1 +···+ωn
,
then we can define a probability measure Qn ◦ Hn−1 on Λ∗n , where Hn−1 is described as
follows. For a composition ~λ with κn = k parts and last part j,
(
{~ω ∈ Ωn : ωi = λi , i = 1, . . . , k − 1}
if j = 1
Hn−1 (~λ) =
{~ω ∈ Ωn : ωi = λi , i = 1, . . . , k − 1 and ωk ≥ 2} if j > 1.
Although Qn ◦ Hn−1 is not uniform, it is conditionally uniform given the size of the last
part:
k−1
Q 1

1

= φn−1
if j = 1

φλ i
i=1
−1 ~
Qn (Hn (λ)) = k−1
(1)
∞
Q 1 P

1
1

if j > 1.

λ
i =
n−2
i=1
φ
i
i=j
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φ
φ
4
For ease of use, we let Pn∗ = Qn ◦ Hn−1 and denote the expected value with respect to Pn∗
as En∗ . We denote the expected value with respect to Qn as EnQ .
The following observation will complete the new construction so that it is useful for
our analysis of the product of parts. If X ∗ is any random variable on Λ∗n , then there is a
corresponding random variable X on Ωn defined in the obvious way, X(~ω ) = X ∗ (Hn (~ω )),
and having the same distribution as X ∗ :
Lemma 2. If X ∗ is any random variable on Λ∗n and if X = X ∗ ◦ Hn , then for all real
numbers y,
Pn∗ (X ∗ = y) = Qn (X = y).
4
Number of parts
For random unrestricted compositions, the number of parts is a binomial random variable
that has an expected value of n+1
. A local limit theorem is proved in [1] for the number
2
of parts of a composition with parts from a general set, but more specific results for 1-free
compositions are needed for the proof of Theorem 1.
Theorem 4. If ~λ is chosen from a uniform distribution on the set of 1-free compositions
of n, the expected number of parts is
n
φ
n
En (κn ) =
+ +O
.
1 + φ2 5
φ2n
Proof. Observe that the total number of parts over all compositions in ΛSn is
|ΛSn |En (κn )
=
n
X
k=1
k|ΛSn,k |
=
n X
n
X
k=1 d=k
|ΛSn,d|.
Using the fact that the coefficient of xn in G0 (x)d is equal to |ΛSn,d|, as in the proof of
Lemma 1, we have
n X
n
X
k=1 d=k
|ΛSn,d| = [xn ]
n X
n
X
k=1 d=k
Letting S = {k : k ≥ 2}, so that G0 (x) =
obtain
En (κn ) =
G0 (x)d = [xn ]
∞
P
G0 (x)
.
(1 − G0 (x))2
(2)
xk , and combining (2) with Corollary 3, we
k=2
1
G0 (x)
[xn ]
Fn−1
(1 − G (x))2
0
1
3
1
Fn
=
n
Fn−1 − Fn +
Fn−1
5
5
5
n
φ
n
=
+ +O
.
2
1+φ
5
φ2n
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5
A pertinent part of the lognormality proof is to show that the number of parts of a
composition is concentrated around its mean. For unrestricted compositions, Chernoff
bounds can be applied directly to obtain an exponentially small upper bound. For 1-free
compositions, we present the following result.
Lemma 3. There exists a constant C1 > 0 such that
√
C1
n Qn τn − 1+φ2 > n log n ≤ 1+φ2 .
n
Proof. For positive integers α, define Sα =
α
P
ωi . Then for t > 0,
i=1
Qn (τn ≤ α) = Qn (Sα ≥ n) = Qn (etSα ≥ etn ).
By Markov’s inequality and the independence of the ωi ’s,
Qn (etSα ≥ etn ) ≤
EnQ (etω1 )α
EnQ (etSα )
=
.
etn
etn
We compute the moment generating function of ω1 to get, for 0 < t < log φ,
t 2
e
∞
∞
k
t
X
X e
φ
e2t
Q tω1
tk
En (e ) =
e Qn (ω1 = k) =
=
t =
t −1 .
φ
1 − eφ
1 − eφ−1
k=2
k=2
If we let u =
et −1
,
φ−1
then
Qn (τn ≤ α) ≤
e2t
1−u
α
etn
=
e2tα
eα log(1−u) etn
= e−tn+2tα−α log(1−u) .
(3)
Now √
it remains to estimate the exponent in the right-hand side of (3). Define α =
n
⌊ 1+φ
−
n log n⌋ and let t = √1n . Then, with an application of the Taylor expansion
2
1
1
1
√1 + 1
log(1 − u) = − φ−1
− 2n(φ−1)
2 + O n3/2 , we have
2n
n
√
−tn = − n ,
√
2 n
√1
2tα =
−
2
log
n
+
O
,
n
1 + φ2
√
n
log n
−α log(1 − u) =
−
+ O(1).
(1 + φ2 )(φ − 1) φ − 1
(4)
(5)
(6)
Now (4), (5), and (6) can be used to make√ substitutions in the exponent of (3). When
like terms are combined, the coefficient of n is
−1 +
2
1
+
=0
2
2
1+φ
(1 + φ )(φ − 1)
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6
and the coefficient of log n is
−2 −
1
= −(1 + φ2 ).
φ−1
Since the remaining terms are O(1), we have for some C1 > 0,
√
n
C1
2
Qn τn ≤
− n log n ≤ e−(1+φ ) log n+O(1) ≤ 1+φ2 .
2
1+φ
n
√
n
A similar calculation using β = ⌈ 1+φ
n log n⌉ gives the inequality in the other direc2 +
tion.
Corollary 4. There exists a constant C2 > 0 such that
√
C2
n Pn κn − 1+φ
n log n ≤ 1+φ2 .
2 >
n
∗
Proof. By Lemma 2, Pn (κn ≥ j) = Qn (τ
the inequality in Lemma 3 remains
n ≥ j).n Thus
√
unchanged for Pn∗ . Let A be the event κn − 1+φ
≥
n log n. Then
2
Pn∗ (A) =
>
and
5
1
|A
φn−1
1
|A
φn−2
and λκn = 1| +
and λκn ≥ 2| =
1
|A and
φn−2
Fn−1
P (A)
φn−2 n
λκn ≥ 2|
√
C
φn−2 ∗
C2
1
5
Pn (A) ≤
Pn (A) ≤ φ + O φ12n
.
2 ≤
1+φ
1+φ
Fn−1
n
n 2
Asymptotic normality of log B
The following lemma is proved in [18] with techniques from [9] and [17] and Lemma 3
from this manuscript. After the lemma, we will begin the proof of Theorem 1. In this
section, we use Φ(x) to denote the standard normal distribution.
Lemma 4. For all x,

log B −
Pn∗  q
nµ
1+φ2
nγ 2
1+φ2

1
≤ x = √
2π
Zx
−t2 /2
e
−∞
dt + O
log
√n
n
1/3
+
x√
log n
n
.
Proof of Theorem 1. For any integer t ≥ 1,
Pn∗ (B = t) = Pn∗ (B = t and λκn ≥ 2) + Pn∗ (B = t and λκn = 1)
1
1
= n−2 |Bn = t and λκn ≥ 2| + n−1 |Bn = t and λκn = 1|
φ
φ
1
1
= n−2 |Bn = t and all λi ≥ 2| + n−1 |Bn−1 = t and all λi ≥ 2|
φ
φ
Fn−1
Fn−2
= n−2 Pn (B = t) + n−1 Pn−1 (B = t)
φ
φ
the electronic journal of combinatorics 18 (2011), #P235
(7)
(8)
(9)
7
where the step from (7) to (8) is made by noting the bijection between the set of compositions of n − 1 and the set
q of compositions of n with last part 1. Summing over t on
both sides and letting ξn =
Pn∗ (log B ≤ ξn ) =
nγ 2
x
1+φ2
+
nµ
,
1+φ2
we have
Fn−1
Fn−2
Pn−1 (log B ≤ ξn ) + n−2 Pn (log B ≤ ξn ).
n−1
φ
φ
(10)
We would like torewrite Pn−1 (log
B ≤ ξn ) in terms of Pn−1 (log B ≤ ξn−1), so we let
q
1
sn,x = 1 + n−1
and make the following calculation:
x + √ µ/γ2 √
1+φ
n
Pn−1 (log B ≤ ξn ) = Pn−1 log B ≤

log B −
= Pn−1  q
nµ
1+φ2
+
(n−1)µ
1+φ2
(n−1)γ 2
1+φ2
q
nγ 2
x
1+φ2

≤ sn,x 

= Pn−1 (log B ≤ ξn−1) + Pn−1 x <
As a lower bound, we can drop the second term to get
log B −
q
(n−1)µ
1+φ2
(n−1)γ 2
1+φ2

≤ sn,x  .
Pn−1 (log B ≤ ξn ) ≥ Pn−1 (log B ≤ ξn−1 ) .
As an upper bound, we derive from (9) the inequality
Pn∗(B = t) ≥
Fn−1
Pn (B = t)
φn−2
so that
√
φn−2 ∗
Pn (B = t) = φ5 + O φ12n
Pn∗ (B = t).
Fn−1
Then we can obtain an upper bound as follows, with the use of Lemma 4:




√
(n−1)µ
log B − (n−1)µ
log
B
−
5 ∗ 
1
1+φ2
1+φ2



Pn−1 x < q
≤ sn,x ≤
P
x< q
≤ sn,x + O
(n−1)γ 2
(n−1)γ 2
φ n−1
φ2n
1+φ2
1+φ2
√
1/3
5
log
n
x√
log n
√
(Φ(sn,x ) − Φ(x)) + O
=
+ n
n
φ


√
Zsn,x
1/3
5 1
log
n
x√
log n
−t2 /2 
√
√
=
e
dt + O
+ n
n
φ
2π
x
√
1/3
5
log
n
x
log
n
√
≤ √ (sn,x − x) + O
+ √n .
n
φ 2π
Pn (B = t) ≤
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8
√
1 + t = 1 + 2t + O(t2), we can show that
sn,x − x = O √1n + nx .
With use of the Taylor expansion
Therefore,
Pn−1 (log B ≤ ξn ) ≤ Pn−1 (log B ≤ ξn−1) + O
log
√n
n
1/3
+
x√
log n
n
.
Going back to equation (10), we have
Pn∗ (log B ≤ ξn ) =
Letting an =
Fn−1
Pn (log B
φn
Fn−1
Fn−2
Pn−1 (log B ≤ ξn−1 ) + n−2 Pn (log B ≤ ξn )
n−1
φ
φ
1/3
log n
log
√n
+ x√
.
+O
n
n
≤ ξn ) and bn = Pn∗ (log B ≤ ξn ), we rewrite this as
2
bn = φ an + an−1 + O
log
√n
n
1/3
+
x√
log n
n
.
Solving for an gives
an =
1
b
φ2 n
−
1
a
φ2 n−1
+O
log
√n
n
1/3
+
x√
log n
n
.
Iterating ⌊n/2⌋ times, we obtain
an =
1
φ2
n
X
k=⌈n/2⌉
n
X
−1
φ2
+
k=⌈n/2⌉
n−k
−1
φ2
bk +
n−k+1
−1
φ2
·O
⌊n/2⌋+1
log
√k
k
a⌈n/2⌉−1
1/3
+
x√
log k
k
= T1 + T2 + T3 .
We now look at each term separately:
−1 ⌊n/2⌋+1
|T2 | = φ2
a⌈n/2⌉−1 −1 ⌊n/2⌋+1 F⌊n/2⌋−2 ≤ φ2
· ⌊n/2⌋−1 · 1
φ
= O φ1n
the electronic journal of combinatorics 18 (2011), #P235
(11)
9
n 1/3
X −1 n−k+1
x√
log k log
√k
|T3 | = ·
O
+
2
φ
k
k
k=⌈n/2⌉
! n
1/3
X n−k x
log
n
log
n
−1
√
+√
≤ C ·
φ2
⌈n/2⌉
⌈n/2⌉
k=⌈n/2⌉
1/3
log
log n
√n
=O
+ x√
.
n
n
(12)
With use of Lemma 4, we have
T1 =
1
φ2
n
X
−1
φ2
k=⌈n/2⌉
n−k
Φ(x) +
1
φ2
n
X
−1
φ2
k=⌈n/2⌉
= T1,a + T1,b .
n−k
·O
log
√k
k
1/3
+
x√
log k
k
We again look at each term separately:
T1,b = O(T3 )
T1,a =
1
φ2
n
X
−1
φ2
k=⌈n/2⌉
=
Φ(x) φ12
⌊n/2⌋ X
k=0
=
(13)
n−k
−1
φ2
Φ(x)
k
Φ(x)
1
+
O
.
φn
1 + φ2
(14)
Putting the estimates from (11), (12), (13), and (14) together, we get
1/3
Φ(x)
log
n
x
log
n
√
an =
+O
+ √n .
n
1 + φ2
Recalling that an =
Finally,
Fn−1
Pn (log B
φn
≤ ξn ), we therefore have
√
n
Pn (log B ≤ ξn ) = Fφn−1 an = φ 5 + O φ12n
an .
√
1/3
φ 5
log
n
x
log
n
√
Pn (log B ≤ ξn ) = Φ(x)
+O
+ √n
n
1 + φ2
1/3
log
n
x
log
n
√
= Φ(x) + O
+ √n .
n
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10
Therefore the rate of convergence is uniform as long as
√ 2/3
|x| ≤ lognn
.
|x| log n
√
n
≤
log
√n
n
1/3
, i.e. as long as
References
[1] Cyril Banderier and Pawel Hitczenko.
Enumeration and asymptotics of restricted compositions having the same number of parts. Preprint available at
http://www.math.drexel.edu/∼ phitczen/BaHi5.pdf, 2011.
[2] Edward A. Bender and E. Rodney Canfield. Locally restricted compositions I. Restricted adjacent differences. Electron. J. Combin., 12:R57, 2005.
[3] Edward A. Bender and E. Rodney Canfield. Locally restricted compositions II.
General restrictions and infinite matrices. Electron. J. Combin., 16(1):R108, 2009.
[4] Edward A. Bender and E. Rodney Canfield. Locally restricted compositions III.
Adjacent-part periodic inequalities. Electron. J. Combin., 17(1):R145, 2010.
[5] Miklós Bóna and Arnold Knopfmacher. On the probability that certain compositions
have the same number of parts. Ann. Comb., 14(3):291–306, 2010.
[6] Phyllis Chinn and Silvia Heubach. Integer sequences related to compositions without
2’s. J. Integer Seq., 6(2):Article 03.2.3, 2003.
[7] Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge, 2009.
[8] William M. Y. Goh and Pawel Hitczenko. Average number of distinct part sizes in a
random Carlitz composition. European J. Combin., 23(6):647–657, 2002.
[9] Allan Gut and Svante Janson. The limiting behaviour of certain stopped sums and
some applications. Scand. J. Statist., 10(4):281–292, 1983.
[10] Pawel Hitczenko and Arnold Knopfmacher. Gap-free compositions and gap-free samples of geometric random variables. Discrete Math., 294(3):225–239, 2005.
[11] Pawel Hitczenko and Guy Louchard. Distinctness of compositions of an integer: a
probabilistic analysis. Random Structures Algorithms, 19(3-4):407–437, 2001. Analysis of algorithms (Krynica Morska, 2000).
[12] Pawel Hitczenko and Carla D. Savage. On the multiplicity of parts in a random
composition of a large integer. SIAM J. Discrete Math., 18(2):418–435, 2004.
[13] Boris L. Kheyfets. The number of part sizes of a given multiplicity in a random
Carlitz composition. Adv. in Appl. Math., 35(3):335–354, 2005.
[14] A. Knopfmacher and B. Richmond. Compositions with distinct parts. Aequationes
Math., 49(1-2):86–97, 1995.
[15] Arnold Knopfmacher and Helmut Prodinger. On Carlitz compositions. European J.
Combin., 19(5):579–589, 1998.
the electronic journal of combinatorics 18 (2011), #P235
11
[16] Arnold Knopfmacher and Neville Robbins. Compositions with parts constrained by
the leading summand. Ars Combin., 76:287–295, 2005.
[17] A. Rényi. On the asymptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar., 8:193–199, 1957.
[18] Caroline Shapcott. Dissertation (in preparation). PhD thesis, Drexel University.
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