Download Characterizing the number of coloured $ m $

CHARACTERIZING THE NUMBER OF COLOURED m−ARY PARTITIONS
MODULO m, WITH AND WITHOUT GAPS
arXiv:1701.07077v1 [math.CO] 24 Jan 2017
I. P. GOULDEN AND PAVEL SHULDINER
Abstract. In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization
for the number of m-ary partitions modulo m, with and without gaps. In this paper we extend these results
to the case of coloured m-ary partitions, with and without gaps. Our method of proof is different, giving
explicit expansions for the generating functions modulo m.
1. Introduction
An m-ary partition is an integer partition in which each part is a nonnegative integer power of a fixed
integer m ≥ 2. An m-ary partition without gaps is an m-ary partition in which mj must occur as a part
whenever mj+1 occurs as a part, for every nonnegative integer j.
Recently, Andrews, Fraenkl and Sellers [AFS15] found an explicit expression that characterizes the number
of m-ary partitions of a nonnegative integer n modulo m; remarkably, this expression depended only on the
coefficients in the base m representation of n. Subsequently Andrews, Fraenkel and Sellers [AFS16] followed
this up with a similar result for the number of m-ary partitions without gaps, of a nonnegative integer n
modulo m; again, they were able to obtain a (more complicated) explicit expression, and again this expression
depended only on the coefficients in the base m representation of n. See also Edgar [E16] and Ekhad and
Zeilberger [EZ15] for more on these results.
The study of congruences for integer partition numbers has a long history, starting with the work of
Ramanujan (see, e.g., [R19]). For the special case of m-ary partitions, a number of authors have studied congruence properties, including Churchhouse [C69] for m = 2, Rødseth [R70] for m a prime, and Andrews [A71]
for arbitrary positive integers m ≥ 2. The numbers of m-ary partitions without gaps had been previously
considered by Bessenrodt, Olsson and Sellers [BOS13] for m = 2.
In this note, we consider m-ary partitions, with and without gaps, in which the parts are coloured. To
specify the number of colours for parts of each size, we let k = (k0 , k1 , . . .) for positive integers k0 , k1 , . . .,
and say that an m-ary partition is k-coloured when there are kj colours for the part mj , for j ≥ 0. This
(k)
means that there are kj different kinds of parts of the same size mj . Let bm (n) denote the number of
(k)
k-coloured m-ary partitions of n, and let cm (n) denote the number of k-coloured m-ary partitions of n
j
without gaps. For the latter, some part m of any colour must occur as a part whenever some part mj+1 of
any colour (not necessarily the same colour) occurs as a part, for every nonnegative integer j.
We extend the results of Andrews, Fraenkel and Sellers in [AFS15] and [AFS16] to the case of k-coloured mary partitions, where m is relatively prime to (k0 − 1)! and to kj ! for j ≥ 1. Our method of proof is different,
giving explicit expansions for the generating functions modulo m. These expansions depend on the following
simple result.
Proposition 1.1. For positive integers m, a with m relatively prime to (a − 1)!, we have
m−1
X a − 1 + ℓ
−a
−1
(1 − q) ≡ (1 − q m )
q ℓ (mod m).
a−1
ℓ=0
Date: January 26, 2017.
2010 Mathematics Subject Classification. Primary 05A17, 11P83; Secondary 05A15.
Key words and phrases. partition, congruence, generating function.
The work of IPG was supported by an NSERC Discovery Grant.
1
Proof. From the binomial theorem we have
−a
(1 − q)
=
∞ X
a−1+ℓ ℓ
q .
a−1
ℓ=0
Now using the falling factorial notation (a − 1 + ℓ)a−1 = (a − 1 + ℓ)(a − 2 + ℓ) · · · (1 + ℓ) we have
a−1+ℓ
−1
= ((a − 1)!) (a − 1 + ℓ)a−1 .
a−1
But
(a − 1 + ℓ + m)a−1 ≡ (a − 1 + ℓ)a−1 (mod m),
−1
for any integer ℓ, and ((a − 1)!)
(1)
and the result follows.
exists in Zm since m is relatively prime to (a − 1)!, which gives
a−1+ℓ+m
a−1+ℓ
≡
(mod m),
a−1
a−1
2. Coloured m-ary partitions
(k)
In this section we consider the following generating function for the numbers bm (n) of k-coloured m-ary
partitions:
∞
∞ −kj
X
Y
mj
(k)
n
.
1
−
q
(q) =
(n)q
=
Bm
b(k)
m
n=0
The following result gives an explicit expansion for
j=0
(k)
Bm (q)
modulo m.
Theorem 2.1. If m is relatively prime to (k0 − 1)! and to kj ! for j ≥ 1, then we have

! ∞ m−1 m−1
X k0 − 1 + ℓ0 Y X kj + ℓj j
(k)

Bm
(q) ≡
q ℓ0
q ℓj m  (mod m).
k0 − 1
k
j
j=1
ℓ0 =0
ℓj =0
Proof. Consider the finite product
Pi =
i Y
j −kj
,
1 − qm
i ≥ 0.
j=0
We prove that
(2)
Pi ≡


!
i
m−1
X kj + ℓj j
i+1 −1 Y
k0 − 1 + ℓ0 ℓ0 
q ℓj m  (mod m),
1 − qm
q
kj
k0 − 1
j=1
m−1
X
ℓ0 =0
ℓj =0
by induction on i. As a base case, the result for i = 0 follows immediately from Proposition 1.1 with a = k0 .
Now assume that (2) holds for some choice of i ≥ 0, and we obtain
Pi+1 =
i+1
Y
j=0
≡
≡
1 − qm
j
−kj
i+1 −ki+1
Pi
= 1 − qm


!
i
m−1
X kj + ℓj i+1 −ki+1 −1 Y
j
k0 − 1 + ℓ0 ℓ0 
1 − qm
q ℓj m  (mod m)
q
k
k0 − 1
j
j=1
ℓj =0
ℓ0 =0


!
m−1
i+1 m−1
X kj + ℓj X k0 − 1 + ℓ0 j
i+2 −1 Y

q ℓj m  (mod m),
1 − qm
q ℓ0
k
k0 − 1
j
j=1
m−1
X
ℓj =0
ℓ0 =0
where the second last equivalence follows from the induction hypothesis, and the last equivalence follows
i+1
from Proposition 1.1 with a = ki+1 + 1, q = q m .
2
This completes the proof of (2) by induction on i, and the result follows immediately since
(k)
Bm
(q) = lim Pi .
i→∞
Now we give the explicit expression for the coefficients modulo m that follows from the above expansion
(k)
of the generating function Bm (q).
Corollary 2.2. For n ≥ 0, suppose that the base m representation of n is given by
n = d0 + d1 m + . . . + dt mt ,
0 ≤ t.
If m is relatively prime to (k0 − 1)! and to kj ! for j ≥ 1, then we have
b(k)
m (n) ≡
t k0 − 1 + d0 Y kj + dj
(mod m).
kj
k0 − 1
j=1
(k)
Proof. In the expansion of the series Bm (q) given in Theorem 2.1, the monomial q n arises uniquely with
the specializations ℓj = dj , j = 0, . . . , t and ℓj = 0, j ≥ t. But with these specializations, we have
kj +ℓj
= kkjj = 1, and the result follows immediately.
kj
Specializing the expression given in Corollary 2.2 to the case kj = 1 for j ≥ 0 provides an alternative
proof to Andrews, Fraenkel and Sellers’ characterization of m−ary partitions modulo m, which was given as
Theorem 1 of [AFS15].
3. Coloured m-ary partitions without gaps
(k)
In this section we consider the following generating function for the numbers cm (n) of k-coloured m-ary
partitions without gaps:
(k)
Cm
(q)
=1+
∞
X
n
c(k)
m (n)q
n=0
∞ Y
i −kj
X
mj
−1 .
1−q
=1+
i=0 j=0
(k)
The following result gives an explicit expansion for Cm (q) modulo m.
Theorem 3.1. If m is relatively prime to (k0 − 1)! and to kj ! for j ≥ 1, then we have


!X
i
m−1
∞ m Y
X
X
−1
j
i+1
kj + ℓj
k0 − 1 + ℓ0 ℓ0
(k)

− 1 q ℓj m  (mod m).
1 − qm
Cm
(q) ≡ 1 +
q
k
k0 − 1
j
j=1
i=0
ℓj =0
ℓ0 =1
Proof. Consider the finite product
Ri =
i Y
j=0
1 − qm
j
−kj
−1 ,
i ≥ 0.
We prove that
(3)
Ri ≡


!
i
m−1
m X kj + ℓj X
j
i+1 −1 Y
k0 − 1 + ℓ0 ℓ0 
− 1 q ℓj m  (mod m),
1 − qm
q
k
k0 − 1
j
j=1
ℓj =0
ℓ0 =1
3
by induction on i. As a base case, the result for i = 0 follows immediately from Proposition 1.1 with a = k0 .
Now assume that (3) holds for some choice of i ≥ 0, and we obtain
i+1
−ki+1
−kj
Y mi+1
mj
− 1 Ri
−1 =
1−q
1−q
Ri+1 =
j=0
≡
≡
!
m −1 −ki+1 −1 X
k0 − 1 + ℓ0 ℓ0
mi+1
mi+1
− 1−q
1−q
q
k0 − 1
ℓ0 =1


i
m−1
Y
X kj + ℓj j

×
− 1 q ℓj m  (mod m)
k
j
j=1
ℓj =0


!
i+1
m−1
m Y
X
X
−1
j
i+2
kj + ℓj
k0 − 1 + ℓ0 ℓ0

− 1 q ℓj m  (mod m),
1 − qm
q
k
k0 − 1
j
j=1
ℓj =0
ℓ0 =1
where the second last equivalence follows from the induction hypothesis, and the last equivalence follows
i+1
i+1
from Proposition 1.1 with a = ki+1 + 1, q = q m
and a = 1, q = q m .
This completes the proof of (3) by induction on i, and the result follows immediately since
(k)
Cm
(q) = 1 +
∞
X
Ri .
i=0
Corollary 3.2. For n ≥ 1, suppose that n is divisible by m, with base m representation given by
n = ds ms + . . . + dt mt ,
1 ≤ s ≤ t,
where 1 ≤ ds ≤ m − 1, and 0 ≤ ds+1 , . . . , dt ≤ m − 1. If m is relatively prime to (k0 − 1)! and to kj ! for
j ≥ 1, then for 0 ≤ d0 ≤ m − 1 we have


X
t
i
Y
k
+
d
−
1
k
+
d
k
−
1
−
d
s
s
j
j
0
0
εs + (−1)s−1
− 1  (mod m),
c(k)
−1
m (n − d0 ) ≡
k
k0 − 1
ks
j
i=s j=s+1
where εs = 0 if s is even, and εs = 1 if s is odd.
Proof. First note that we have
n − d0 = m − d0 + (m − 1)m1 + . . . + (m − 1)ms−1 + (ds − 1)ms + ds+1 ms+1 + . . . + dt mt .
Now consider the following specializations: ℓ0 = m − d0 , ℓj = m − 1, j = 1, . . . , s − 1, ℓs = ds − 1, ℓj = dj ,
(k)
j = s + 1, . . . , t, and ℓj = 0, j > t. Then, in the expansion of the series Cm (q) given in Theorem 3.1, the
n
monomial q arises once for each i ≥ 0, in particular with the above specializations truncated to ℓ0 , . . . , ℓi .
But with these specializations we have
• for j = 0:
k0 − 1 − d0
k0 − 1 + m − d0
kj − 1 + ℓj
,
=
−1=
k0 − 1
k0 − 1
kj − 1
• for j = 1, . . . , s − 1:
kj + ℓj
kj − 1
−1=
− 1 = 0 − 1 = −1,
kj
kj
and
X
s−1
s−1 Y
i X
kj + ℓj
(−1)i = εs ,
−1 =
k
j
i=0
i=0 j=1
4
from (1),
• for j = s:
ks + ds − 1
kj + ℓj
−1=
− 1,
kj
ks
• for j = s + 1, . . . , t:
kj + dj
kj + ℓj
− 1,
−1=
kj
kj
• for j > t:
kj + ℓj
kj
kj
−1=
− 1 = 1 − 1 = 0.
kj
The result follows straightforwardly from Theorem 3.1.
Specializing the expression given in Corollary 3.2 to the case kj = 1 for j ≥ 0 provides an alternative
proof to Andrews, Fraenkel and Sellers’ characterization of m−ary partitions modulo m without gaps, which
was given as Theorem 2.1 of [AFS16].
References
A71.
AFS15.
AFS16.
BOS13.
C69.
E16.
EZ15.
R19.
R70.
G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104–110.
George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, Characterizing the Number of m-ary Partitions Modulo
m, American Mathematical Monthly 122 (2015), 880–885.
George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, m-ary partitions with no gaps: A characterization modulo
m, Discrete Mathematics 339 (2016), 283–287.
C. Bessenrodt, J. B. Olsson, J. A. Sellers, Unique path partitions: characterization and congruences, Annals Comb.
17 (2013), 591–602.
R. F. Churchhouse, Congruence properties of the binary partition function, Proc. Cambridge Philos. Soc. 66 (1969),
371–376.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m, Rocky Mountain J. Math. (to
appear), arXiv 1603.00085 math.CO
Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary
partitions mod m (and doing MUCH more!), arXiv 1511.06791 math.CO
S. Ramanujan, Some properties of p(n), the number of partitions of n, Proc. Cambridge Philos. Soc. 19 (1919),
207–210.
Ø. Rødseth, Some arithmetical properties of m-ary partitions, Proc. Cambridge Philos. Soc. 68 (1970), 447–453.
Dept. of Combinatorics and Optimization, University of Waterloo, Canada
E-mail address: [email protected], [email protected]
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