THE RAMANUJAN JOURNAL, 9, 63–75, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in the Netherlands.
On Some Sets with Even Valued Partition Function
F. BEN SAID∗ [email protected] des Sciences de Monastir, Avenue de l’environnement, 5019, Monastir, Tunisie
Dedicated to Professor J.-L. Nicolas on the occasion of his 60th birthday
Received June 13, 2002; Accepted April 10, 2003
Abstract. Let IN be the set of positive integers, B = {b1 < · · · < bh} ⊂ IN , N ∈ IN and N ≥ bh .A = A0(B, N ) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sarkozy) such that A ∩ {1, . . . , N } = Band p(A, n) ≡ 0 (mod 2) for n ∈ IN and n > N , where p(A, n) denotes the number of partitions of n with partsin A. Let us denote by σ (A, n) the sum of the divisors of n belonging to A. In a paper jointly written with J.-L.Nicolas, we have recently proved that, for all k ≥ 0, the sequence (σ (A, 2kn))n≥1 mod 2k+1 is periodic with anodd period qk . In this paper, we will characterize for any fixed odd positive integer q, the sets B and the integersN such that q0 = q, and those for which qk = q for all k ≥ 0. Moreover, a set A = A0(B, N ) is constructed withthe property that its period, i.e. the period of (σ (A, n))n≥1 mod 2, is 217, and for which the counting function isasymptotically equal to that of A0({1, 2, 3, 4, 5}, 5) which is a set of period 31.
Key words: even partition functions, periodic sequences, sets defined by partition parity
2000 Mathematics Subject Classification: Primary—11P81, 11P83
1. Introduction
IN will denote the set of positive integers andP(IN ) is the subsets of IN . IfA = {a1, a2, . . .}⊂ IN (where a1 < a2 < · · ·), then p(A, n) denotes the number of partitions of n with partsin A, i.e. the number of solutions of the equation
a1x1 + a2x2 + · · · = n
in non negative integers x1, x2, . . . . As usual, we shall set p(A, 0) = 1.We shall use the generating function
fA(z) =∞∑
n=0
p(A, n)zn =∏
a∈A
1
1 − za. (1.1)
If B = {b1, . . . , bh} �= ∅ (where b1 < · · · < bh), B ⊂ IN , N ∈ IN and N ≥ bh , then thereis (cf. [5]) a unique set A = A0(B, N ) such that
A ∩ {1, . . . , N } = B and p(A, n) ≡ 0 (mod 2) for n ∈ IN , n > N . (1.2)
∗Research supported by MIRA 2002 program no 0203012701, Number Theory, Lyon-Monastir.
64 BEN SAID
Let us recall the construction as described in [5]. The set A = A0(B, N ) will be defined byrecursion. We write An = A ∩ {1, 2, . . . , n} so that
AN = A ∩ {1, 2, . . . , N } = B.
Assuming that n ≥ N + 1 and An−1 has been defined so that p(A, m) is even for N + 1 ≤m ≤ n − 1. Then set
n ∈ A if and only if p(An−1, n) is odd.
It follows from the construction that for n ≥ N + 1 we have
if n ∈ A, p(A, n) = 1 + p(An−1, n)
if n /∈ A, p(A, n) = p(An−1, n)
which shows that p(A, n) is even for n ≥ N + 1.Note that in the same way, any finite set B = {b1, b2, . . . , bk} can be extended to an
infinite set A so that Abk = B and the parity of p(A, n) is given for n ≥ N + 1 (where Nis any integer such that N ≥ bk).
As observed in [6], by the uniqueness of the construction of A = A0(B, N ), if M is anyinteger ≥N and B′ = A ∩ {1, . . . , M}, then
A = A0(B, N ) = A0(B′, M). (1.3)
If A ⊂ IN , let χ (A, n) denotes the characteristic function of A, i.e.,
χ (A, n) ={
1 if n ∈ A0 if n /∈ A,
and for n ≥ 1,
σ (A, n) =∑
d|nχ (A, d)d =
∑
d|n,d∈Ad. (1.4)
It was recently proved (cf. [3]) that for all k ≥ 0 and all A = A0(B, N ), the sequence(σ (A, 2kn))n≥1 mod 2k+1 is periodic with an odd period qk . This property allowed one in[1] to determine the elements of the set A0({1, 2, 3}, 3) which satisfies qk = 7 for all k ≥ 0,and then to obtain an asymptotic for the counting function of this set. Although the methodgiven there can be applied to any set A = A0(B, N ), the calculations may prove difficult forlarger q0. In this paper, we will show out the general nature of the sets A = A0(B, N ), andwe will give a new method capable of solving more easily the above mentioned problemsfor some sets A = A0(B, N ) with large q0.
If A = A0(B, N ), let us define the polynomial PB,N (already considered in [6])
PB,N (z) =∑
0≤n≤J
εnzn, (1.5)
ON SOME SETS WITH EVEN VALUED PARTITION FUNCTION 65
where J is the largest integer such that p(A, J ) is odd (such a J does exist since p(A, 0) =1), and εn is defined by
p(A, n) ≡ εn (mod 2), εn ∈ {0, 1}.
We shall say that PB,N is the characteristic polynomial of A = A0(B, N ) and we will write
PB,N = charA0(B, N ). (1.6)
Note that PB,N is of degree at most N and if IF2[z] is the ring of polynomials with coefficientsin {0, 1}, then PB,N ∈ IF2[z] and PB,N (0) = 1. Let the factorization of PB,N into pairwiserelatively prime irreducible polynomials over IF2[z] be
PB,N = Pα11 Pα2
2 · · · Pαss . (1.7)
We denote by βi the order of Pi , i.e., the smallest integer such that Pi divides 1 + zβi inIF2[z], and for all k ≥ 0, we set
Ik = { j : 1 ≤ j ≤ s, α j ≡ 2k(mod 2k+1)},Jk = I0 ∪ I1 ∪ · · · ∪ Ik = { j, 1 ≤ j ≤ s, α j �≡ 0(mod 2k+1)},
and Tk = lcm j∈Jk β j (with Tk = 1 if Jk = ∅).In [3], it is proved that the period qk of the sequence (σ (A, 2kn))n≥1 mod 2k+1 is odd
and
qk = Tk . (1.8)
In this paper, we shall prove (Proposition 2) that given any fixed polynomial P ∈ IF2[z], ofpositive degree N , and such that P(0) = 1, there is a unique set B ⊂ {1, . . . , N } such thatP = PB,N = charA0(B, N ). Next, we will give (Theorem 1) a method which determinesBwhen P = PB,N is of the form (1.7) and when the sets Bi satisfying Pi = charA0(Bi , Ni ),with Ni = degree(Pi ), are known. As an application of the last result, a set A = A0(B, N )is constructed with the property that its period, i.e. the period of (σ (A, n))n≥1 mod 2, is 217,and for which the counting function is asymptotically equal to that of A0({1, 2, 3, 4, 5}, 5)which is a set of period 31. In Section 3, we will characterize the sets A = A0(B, N ) whengiven any positive odd integer q , the period qk of (σ (A, 2kn))n≥1 mod 2k+1 is q for allk ≥ 0; the sets A = A0(B, N ) for which q0 = q are also characterized.
All polynomials and series as well as sums and products that will be considered in thepaper are to be taken with coefficients in {0, 1}.
2. Sets A = A0(B, N) with fixed characteristic polynomial
Let us consider the ring of formal power series IF2[[z]]. For an element of this ring
f (z) = a0 + a1z + a2z2 + · · · .
66 BEN SAID
and for n ≥ 0, we define the polynomial f Mod zn+1 by
( f Mod zn+1)(z) =∑
0≤i≤n
ai zi . (2.1)
Note that if f and g are in IF2[[z]], then
( f g) Mod zn+1 = (( f Mod zn+1)(g Mod zn+1)) Mod zn+1. (2.2)
Let B = {b1, . . . , bh} ⊂ IN , B �= ∅ , b1 < · · · < bh , N ≥ bh , and consider the setA = A0(B, N ) defined by (1.2). Denote by PB,N the characteristic polynomial of A givenby (1.5). The following result defines PB,N in another way.
Proposition 1. Under the above notations, we have
PB,N = char A0(B, N ) = fB(z) Mod zN+1. (2.3)
Proof: Recall that if A = A0(B, N ) and fA(z) = ∑∞n=0 p(A, n)zn is its generating
function considered as an element of IF2[[z]], then from (1.1), we have
fA(z) =∏
a∈A
1
1 − za=
∏
a∈B
1
1 − za
∏
a∈Aa>N
1
1 − za, (2.4)
since by (1.2), A ∩ {1, . . . , N } = B.But, from the definition given to PB,N = charA0(B, N ) in (1.5), we deduce immediately
that
PB,N = fA(z) Mod zN+1,
which together with (2.4) and (2.2) prove (2.3).Now, let
E = {(B, N ) : N ∈ IN ,B ⊂ {1, . . . , N },B �= ∅}. (2.5)
By (1.3), it turns out that the relation ∼ defined on E by
(B, N ) ∼ (B′, N ′) ⇔ A0(B, N ) = A0(B′, N ′), (2.6)
is an equivalence relation. Moreover, if (B, N ) is the class of (B, N ) ∈ E , it is obvious by(1.3) that
(B, N ) = {(B′, N ′) ∈ E : N ′ ≥ N and B′ = A0(B, N ) ∩ {1, . . . , N ′}}∪ {(B′, N ′) ∈ E : N ′ ≤ N and B = A0(B′, N ′) ∩ {1, . . . , N }}
Denote by E/∼ the quotient set, and let
F = {P : P ∈ IF2[z], P(0) = 1 and degree (P) ≥ 1}. (2.7)
ON SOME SETS WITH EVEN VALUED PARTITION FUNCTION 67
From the definition of the characteristic polynomial ofA = A0(B, N ), we can prove withoutdifficulty that the mapping
E/∼ −→ F
(B, N ) �−→ PB,N = char A0(B, N )}
is injective. In fact, we will prove in the following result that it is a one to one correspondence.
Proposition 2. Let N be a positive integer, P ∈ IF2[z], P(0) = 1 and degree (P) = N.There exists a unique set B ⊂ {1, . . . , N }, B �= ∅, such that
P = PB,N = char A0(B, N ).
Note that if (B, N ) ∈ E and P is the characteristic polynomial of A = A0(B, N ) definedby (1.5), then degree(P) = M ≤ N . Thus Proposition 2 implies that there is a uniqueset B′ such that the couple (B′, M) represents the class (B, N ) according to the equivalencerelation given by (2.6). Moreover, with the convention A0(∅, 0) = ∅, the conditions B �= ∅and degree(P) ≥ 1 in (2.5) and (2.7) can be removed.
Proof of Proposition 2: We shall proceed by induction on N . For if N = 1, then P = 1+z,so that by setting B = {1}, we get from (2.3),
PB,1 = charA0(B, 1) = 1
1 − zMod z2 = 1 + z = P,
which proves that the result is true for N = 1. Suppose that the result holds up to anyN ≥ 1, and take P ∈ IF2[z] such that P(0) = 1 and degree (P) = N + 1. If we set P1 = PMod zN+1, then 0 ≤ degree (P1) ≤ N .
• When degree (P1) = 0, we have P = 1 + zN+1, so that by setting B = {N + 1}, we getas above
PB,N+1 = charA0(B, N + 1) = 1
1 − zN+1Mod zN+2 = 1 + zN+1 = P.
• If 1 ≤ degree(P1) = M ≤ N , then from induction hypothesis, there is some B1 ⊂{1, . . . , M} satisfying
P1 = PB1,M = charA0(B1, M).
But if B2 = A0(B, M) ∩ {1, . . . , N }, we have from (1.3), A0(B1, M) = A0(B2, N ), andso
P1 = PB1,M = PB2,N .
68 BEN SAID
Suppose for instance that
fB2 (z) Mod zN+1 �= fB2 (z) Mod zN+2,
and set B = B2. We have from (2.3)
charA0(B, N + 1) = fB2 (z) Mod zN+2 = fB2 (z) Mod zN+1 + zN+1
= PB2,N + zN+1 = P1 + zN+1 = P.
Now, if
fB2 (z) Mod zN+1 = fB2 (z) Mod zN+2,
then by setting B = B2 ∪ {N + 1}, we obtain as above
charA0(B, N + 1) =(
fB2 (z).1
1 − zN+1
)Mod zN+2
= ( fB2 (z)(1 + zN+1)) Mod zN+2 = fB2 (z) Mod zN+1 + zN+1,
since fB2 (0) = 1. Hence
charA0(B, N + 1) = P1 + zN+1 = P.
This completes the proof of the existence of B. The uniqueness of such a B follows fromthe fact that if B′ ⊂ {1 . . . , N } is another set satisfying
fB(z) Mod zN+1 = fB′ (z) Mod zN+1,
then for all i , 1 ≤ i ≤ N , we have
fB(z) Mod zi+1 = fB′ (z) Mod zi+1.
Now, what can one say about the set B defined by the relation P = charA0(B, N ), whenP is of degree N and is the product of two polynomials, P1, P2 ∈ F? In particular, can Bbe expressed in terms of the sets B1,B2 obtained from the relations
P1 = charA0(B1, N1), P2 = charA0(B2, N2),
where N1 and N2 are respectively the degrees of P1 and P2?Let us define the following application
g : P(IN ) × P(IN ) −→ P(IN ) × P(IN )(2.8)
(A, B) �−→ g(A, B) = (g1(A, B), g2(A, B)),
ON SOME SETS WITH EVEN VALUED PARTITION FUNCTION 69
where g1(A, B) = A�B = (A ∪ B)\(A ∩ B) and g2(A, B) = 2(A ∩ B) = 2A ∩ 2B. Weshall write gk for the k composite of g , i.e.,
gk = ((gk)1, (gk)2). (2.9)
Let f A be the generating function of A defined as in (1.1). The following result gives arecursive congruence mod 2 for f A. fB .
Proposition 3. Under the above notations. For all k ≥ 1 and all A, B ∈ P(IN ), we have
f A. fB ≡ f(gk )1(A,B). f(gk )2(A,B) mod 2. (2.10)
Proof: We shall proceed by induction. For if k = 1, then
f A. fB = f A�B f 2A∩B ≡ f A�B . f2(A∩B) mod 2,
since for any C ∈ P(IN ), f 2C ≡ f2C mod 2. Hence
f A. fB ≡ fg1(A,B). fg2(A,B) mod 2,
and the result is true for k = 1. Assuming that the result holds up to any k ≥ 1 then we have
f A. fB ≡ f(gk )1(A,B). f(gk )2(A,B) mod 2, (2.11)
and by induction hypothesis
f(gk )1(A,B). f(gk )2(A,B) ≡ f(gk )1(A,B)�(gk )2(A,B). f2((gk )1(A,B)∩(gk )2(A,B))
= f(gk+1)1(A,B). f(gk+1)2(A,B),
which with (2.11) complete the proof.
Note that if A ∩ B = ∅, then f A. fB = f A�B . fφ = f A∪B , and thus
(gk)2(A, B) = ∅, (gk)1(A, B) = A ∪ B, for all k ∈ IN . (2.12)
We also have another interesting property of the application g.
70 BEN SAID
Proposition 4. Let A and B be in P(IN ), and for k ∈ IN , gk = ((gk)1, (gk)2) is theapplication defined by (2.8) and (2.9). Then
(gk)2(A, B) ⊂ 2k(A ∩ B), (2.13)
(A�B)
∖ ∞⋃
i=1
2i (A ∩ B) ⊂ (gk)1(A, B) ⊂ (A�B) ∪∞⋃
i=1
2i (A ∩ B). (2.14)
Proof: The relation (2.13) is immediate by induction since
(g1)2(A, B) = g2(A, B) = 2(A ∩ B),
(gk+1)2(A, B) = 2((gk)1(A, B) ∩ (gk)2(A, B)).
Let us prove (2.14). Since
(g1)1(A, B) = g1(A, B) = A�B,
the case k = 1 is true. Suppose that the result holds up to any k ≥ 1, then
(gk+1)1(A, B) = (gk)1(A, B)�(gk)2(A, B)
⊂ (gk)1(A, B) ∪ (gk)2(A, B)
⊂(
(A�B) ∪∞⋃
i=1
2i (A ∩ B)
)∪ 2k(A ∩ B)
= (A�B) ∪∞⋃
i=1
2i (A ∩ B).
On the other hand,
(gk+1)1(A, B) = (gk)1(A, B)�(gk)2(A, B)
⊃ (gk)1(A, B) \ (gk)2(A, B)
⊃ (A�B)
∖ ∞⋃
i=1
2i (A ∩ B).
This completes the proof of Proposition 4.
Now, we can assert the main result of this paper.
Theorem 1. Let P1, P2 ∈ IF2[z] ; Pi (0) = 1, degree (Pi ) = Ni ≥ 1. Denote by Bi the setsatisfying (cf. Proposition 2),
Bi ⊂ {1, . . . , Ni } and Pi = charA0(Bi , Ni ).
ON SOME SETS WITH EVEN VALUED PARTITION FUNCTION 71
Moreover, let N = N1 + N2, and set
B′1 = A0(B1, N1) ∩ {1, . . . , N }, (2.15)
B′2 = A0(B2, N2) ∩ {1, . . . , N }. (2.16)
Then, the unique set B ⊂ {1, . . . , N } such that
P = P1 P2 = charA0(B, N ),
is given by
B = limk−→∞
(gk)1(B′1,B′
2) ∩ {1, . . . , N }, (2.17)
where (gk)1 is the first coordinate of the application gk defined previously by (2.8) and(2.9). (Note that by (2.12), the above sequence in (2.17) is finite).
Moreover, if A = A0(B1, N1), A′ = A0(B2, N2) and A” = A0(B, N ), then
A” = limk−→∞
(gk)1(A,A′), (2.18)
where the sequence here is infinite.
Proof: We have from (2.2),(
fB′1. fB′
2
)Mod zN+1 = (
fB′1
Mod zN+1)(
fB′2
Mod zN+1)
Mod zN+1,
so that by (2.15), (2.16),(2.3) and (1.3), we get(
fB′1. fB′
2
)Mod zN+1 = (
fB1 Mod zN1+1)(
fB2 Mod zN2+1))
Mod zN+1
= (P1 P2) Mod zN+1 = P1 P2.
Thus, by applying Proposition 3 with k such that
(gk)2(B′1,B′
2) ∩ {1, . . . , N } = ∅,
(2.17) follows then from (2.12) and Proposition 2.(2.18) follows from (2.17) and the uniqueness of the sets of the form A0(B, N ), as
mentioned in (1.2).
Let as apply Theorem 1 on a concrete example. Consider the polynomials P1 and P2 ofdegrees N1 = 3 , N2 = 5 respectively, and given by
P1(z) = 1 + z + z3,
P2(z) = 1 + z + z3 + z4 + z5.
By the use of Propositions 1 and 2, an easy computation gives B1 = {1, 2, 3} and B2 ={1, 2, 3, 4, 5}. Besides, with a slightly further calculation, we obtain (as already
72 BEN SAID
given in [6])
B′1 = A0(B1, 3) ∩ {1, . . . , 8} = {1, 2, 3, 5, 8},
B′2 = A0(B2, 5) ∩ {1, . . . , 8} = {1, 2, 3, 4, 5, 7, 8}.
Hence the unique set B ⊂ {1, . . . , 8} such that
P = P1 P2 = charA0(B, 8),
is obtained by the following procedure mod 2:
fB Mod z9 ≡ fB′1. fB′
2Mod z9
= f{1,2,3,5,8}. f{1,2,3,4,5,7,8} Mod z9
≡ f{4,7}. f{2,4,6} Mod z9 ≡ f{2,6,7}. f{8} Mod z9
≡ f{2,6,7,8} Mod z9,
which gives B = {2, 6, 7, 8}.Let us consider the set A = A0({2, 6, 7, 8}, 8). Since the period of A′ = A0({1, 2, 3}, 3)
is 7 (cf. [6]), and that of A′′ = A0({1, 2, 3, 4, 5}, 5) is 31 (cf. [4]), (these results can alsobe deduced from (1.8) ), then by the use of (1.8), we can prove without difficulty that theperiod of A is 217. Moreover, for real x ≥ 1, let A(x), A′(x) and A′′(x) be respectively thecounting functions of A, A′ and A′′ then, when x tends to infinity, we have
A(x) ∼ A′′(x). (2.19)
Proof of (2.19): From (2.18) and (1.3), we obtain
A(x) = limk−→∞
(gk)1(A′,A′′) ∩ {1, . . . , �x�},
where A(x) (and so for A′(x), A′′(x)) is defined here by
A(x) = A ∩ {1, 2, . . . , �x�}.But the last sequence is finite, since
(gk+1)1(A′,A′′) = (gk)1(A′,A′′)�(gk)2(A′,A′′),
and by Proposition 4, (gk)2(A′,A′′) ⊂ 2kA′, so that (2kA′)(x) = (2kA′)∩{1, . . . , �x�} = ∅for k >
log xlog 2 . Hence if k ≥ � log x
log 2 � + 1 then
A(x) = (gk)1(A′,A′′) ∩ {1, . . . , �x�},so that by (2.14), we get
((A′�A′′)
∖ ∞⋃
i=1
2iA′)
(x) ⊂ A(x) ⊂(
(A′�A′′) ∪∞⋃
i=1
2iA′)
(x),
ON SOME SETS WITH EVEN VALUED PARTITION FUNCTION 73
which yields
|A(x) − A′′(x)| ≤∑
0≤i≤ log xlog 2
A′(
x
2i
), (2.20)
since for i >log xlog 2 , A′( x
2i ) = 0.In [1], an asymptotic of the counting function A′(x) of A′ = A0({1, 2, 3}, 3) is given, weparticularly have
A′(x) � x
(log x)3/4. (2.21)
Furthermore, it is shown in [4], that the counting function A′′(x) of A′′ = A0({1, 2, 3, 4, 5},5) satisfies
A′′(x) � x log log x
(log x)1/3. (2.22)
Let us cut the sum in (2.20) into two parts:
S1 =∑
0≤i≤ log x2 log 2
A′(
x
2i
),
S2 =∑
log x2 log 2 <i≤ log x
log 2
A′(
x
2i
).
It is obvious that the sum S2 is � √x log x . On the other hand, by (2.21), S1 satisfies
S1 �∑
0≤i≤ log x2 log 2
x
2i (log(x/2i ))3/4� x
(log x)3/4.
Hence (2.20) implies
A(x) − A′′(x) � x
(log x)3/4,
which with (2.22) end the proof of (2.19).
In general, we have an upper bound for the counting function A′′(x) of the set A′′ =A0(B, N ), whenB is given by (2.17). In fact, if A(x) and A′(x) are respectively, the countingfunctions of A = A0(B1, N1) and A′ = A0(B2, N2) then, when x ≥ N ,
A′′(x) ≤ A(x) + A′(x). (2.23)
For the proof, we just have to note that from (2.8) and (2.9), we obtain
(gh+1)1(A,A′) = (gh)1(A,A′)�(gh)2(A,A′),
74 BEN SAID
by which, we can prove by induction the following relation for counting functions
((gh)1(A,A′))(x) + ((gh)2(A,A′))(x) ≤ A(x) + A′(x), for all h ∈ IN . (2.24)
Therefore, by taking h > h0 = � log xlog 2 �, (2.13) implies ((gh)2(A,A′))(x) = 0, and (2.23)
results immediately from (2.24) and (2.18) once written for intersections with {1, . . . , �x�}.Note also that the odd elements of A′′ are exactly those of A�A′ = g1(A,A′), and thus
a lower bound for A′′(x) can be easily deduced.
3. Sets B such that σ(A, n) mod 2, where A = A0(B, N), is of fixed period q.
The characterization of the sets A = A0(B, N ) for which (σ (A, 2kn))n≥1 mod 2k+1 is ofperiod qk = q for all k ≥ 0 (where q is any fixed odd positive integer), and of thosesatisfying q0 = q , is not difficult with the use of (1.8). In fact, if P is the characteristicpolynomial of the set A = A0(B, N ), it follows from (1.8) that the sequence σ (A, n) mod 2is of period q0 = q if and only if P is of the form
P = Qq .R2, (3.1)
where Qq is square free, of order q. In addition to that, the sequences (σ (A, 2kn))n≥1
mod 2k+1 will satisfy qk = q for all k ≥ 0, if and only if P is of the form (3.1) and eachirreducible factor of R is of order a divisor of q.
However, we can apply Theorem 1 and get another way to characterize the sets A =A0(B, N ) with the above properties. Indeed, let
Dq = {D ∈ IF2[z] : D divides 1 + zq and order(D) = q}.
Denote by d the degree of the arbitrary polynomial D and set
B(q) = {B : ∃D ∈ Dq , B ⊂ {1, . . . , d}, D = charA0(B, d)}.
Then the sequence σ (A, n) mod 2, where A = A0(B′, N ′), is of period q0 = q, if and onlyif there are two sets C1 ⊂ 2IN ∩ {1, . . . , M}, B1 ∈ B(q) with D = char A0(B1, d), andsuch that the couple (B′, N ′) is in the class (B, N ), where N = M + d and
B = liml−→∞
(gl)1(B′1, C ′
1) ∩ {1, . . . , N }, (3.2)
with
B′1 = A0(B1, d) ∩ {1, . . . , N },
C ′1 = A0(C1, M) ∩ {1, . . . , N }.
For the proof of (3.2), just use (3.1) and apply Theorem 1 with the remark that the charac-teristic polynomial of a set A = A0(C ′, M ′) (with the convention A0(∅, 0) = ∅) is of theform R2 for some R ∈ IF2[z], R(0) = 1, if and only if C ′ ⊂ 2IN , which is easy to prove.
ON SOME SETS WITH EVEN VALUED PARTITION FUNCTION 75
For the sets satisfying qk = q for all k ≥ 0, we just have to factorize 1 + zq in IF2[z]into pairwise relatively prime irreducible factors
1 + zq = Pα11 Pα2
2 · · · Pαss , αi ≥ 1,
say, and take in (3.1), R of the form
R = Pβ11 Pβ2
2 · · · Pβss , βi ≥ 0,
then we search for the set B ⊂ {1, . . . , r}, where r = degree (R), such that R = charA0
(B, r ), and conclude as above.
References
1. F. Ben Saıd and J.-L. Nicolas, “Even partition functions,” Seminaire Lotharingien de Combinatoire 46 (2001),B46i (http://www.mat.univie.ac.at/∼slc/).
2. F. Ben Saıd, “On a conjecture of Nicolas-Sarkozy about partitions,” Journal of Number Theory 95 (2002),209–226.
3. F. Ben Saıd and J.-L. Nicolas, “Sets of parts such that the partition function is even,” Acta Arithmetica 106(2)(2003), 183–196.
4. J.-L. Nicolas, “On the parity of generalized partition function II,” Periodica Mathematica Hungarica 43 (2001),177–189.
5. J.-L. Nicolas, I.Z. Ruzsa, and A. Sarkozy, “On the parity of additive representation functions,” Jounal of NumberTheory 73 (1998), 292–317.
6. J.-L. Nicolas and A. Sarkozy, “On the parity of generalized partition function,” in Number Theory for theMillennium III (M.A. Bennet and al, A.K. Peters eds.), Natick, Massachusetts, (2002), pp. 55–72.
