Characterizing the number of coloured $m$-ary partitions modulo $m$, with and without gaps

0.1 MB·

by  I. P. Goulden

CHARACTERIZING THE NUMBER OF COLOURED m−ARY PARTITIONS
MODULO m, WITH AND WITHOUT GAPS
I.P.GOULDENANDPAVELSHULDINER
7 Abstract. In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization
1 forthenumberofm-arypartitionsmodulom,withandwithoutgaps. Inthispaperweextendtheseresults
0 to the case of coloured m-ary partitions, with and without gaps. Our method of proof is different, giving
2 explicitexpansionsforthegeneratingfunctionsmodulom.

n
a
J

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1. Introduction
2
An m-ary partition is an integer partition in which each part is a nonnegative integer power of a fixed
O] 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.
C
Recently,Andrews,FraenklandSellers[AFS15]foundanexplicitexpressionthatcharacterizesthenumber
.
h of m-ary partitions of a nonnegative integer n modulo m; remarkably,this expression depended only on the
at coefficients inthe basem representationofn. Subsequently Andrews, FraenkelandSellers[AFS16]followed
m this up with a similar result for the number of m-ary partitions without gaps, of a nonnegative integer n
modulom;again,theywereabletoobtaina(morecomplicated)explicitexpression,andagainthisexpression
[
depended only on the coefficients in the base m representation of n. See also Edgar [E16] and Ekhad and
1
Zeilberger [EZ15] for more on these results.
v
The study of congruences for integer partition numbers has a long history, starting with the work of
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7 Ramanujan(see,e.g.,[R19]). Forthespecialcaseofm-arypartitions,anumberofauthorshavestudiedcon-
0 gruenceproperties,includingChurchhouse[C69]form=2,Rødseth[R70]formaprime,andAndrews[A71]
7 for arbitrary positive integers m ≥ 2. The numbers of m-ary partitions without gaps had been previously
0
considered by Bessenrodt, Olsson and Sellers [BOS13] for m=2.
.
1 In this note, we consider m-ary partitions, with and without gaps, in which the parts are coloured. To
0 specify the number of colours for parts of each size, we let k = (k ,k ,...) for positive integers k ,k ,...,
0 1 0 1
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and say that an m-ary partition is k-coloured when there are k colours for the part mj, for j ≥ 0. This
1 j
: means that there are kj different kinds of parts of the same size mj. Let b(mk)(n) denote the number of
v
i k-coloured m-ary partitions of n, and let c(mk)(n) denote the number of k-coloured m-ary partitions of n
X without gaps. For the latter, some part mj of any colour must occur as a part whenever some part mj+1 of
r any colour (not necessarily the same colour) occurs as a part, for every nonnegative integer j.
a
WeextendtheresultsofAndrews,FraenkelandSellersin[AFS15]and[AFS16]tothecaseofk-colouredm-
arypartitions,wheremisrelativelyprime to(k −1)!andtok !forj ≥1. Ourmethodofproofisdifferent,
0 j
givingexplicitexpansionsforthegeneratingfunctions modulom. Theseexpansionsdependonthefollowing
simple result.
Proposition 1.1. For positive integers m,a with m relatively prime to (a−1)!, we have
(1−q)−a ≡(1−qm)−1m−1 a−1+ℓ qℓ (modm).
a−1
ℓ=0 (cid:18) (cid:19)
X
Date:January26,2017.
2010 Mathematics Subject Classification. Primary05A17, 11P83;Secondary05A15.
Keywords and phrases. partition,congruence, generatingfunction.
TheworkofIPGwassupportedbyanNSERCDiscoveryGrant.
1
Proof. From the binomial theorem we have
∞
(1−q)−a = a−1+ℓ qℓ.
a−1
ℓ=0(cid:18) (cid:19)
X
Now using the falling factorial notation (a−1+ℓ)a−1 =(a−1+ℓ)(a−2+ℓ)···(1+ℓ) we have
a−a−1+1 ℓ =((a−1)!)−1(a−1+ℓ)a−1.
(cid:18) (cid:19)
But
(a−1+ℓ+m)a−1 ≡(a−1+ℓ)a−1 (modm),
for any integer ℓ, and ((a−1)!)−1 exists in Z since m is relatively prime to (a−1)!, which gives
m
a−1+ℓ+m a−1+ℓ
(1) ≡ (modm),
a−1 a−1
(cid:18) (cid:19) (cid:18) (cid:19)
and the result follows. (cid:3)
2. Coloured m-ary partitions
In this section we consider the following generating function for the numbers b(k)(n) of k-coloured m-ary
m
partitions:
∞ ∞
B(k)(q)= b(k)(n)qn = 1−qmj −kj.
m m
nX=0 jY=0(cid:16) (cid:17)
The following result gives an explicit expansion for B(k)(q) modulo m.
m
Theorem 2.1. If m is relatively prime to (k −1)! and to k ! for j ≥1, then we have
0 j
m−1 k −1+ℓ ∞ m−1 k +ℓ
B(k)(q)≡ 0 0 qℓ0 j j qℓjmj (modm).
m k −1  k 
ℓX0=0(cid:18) 0 (cid:19) !jY=1 ℓXj=0(cid:18) j (cid:19)
 
Proof. Consider the finite product
P = i 1−qmj −kj, i≥0.
i
jY=0(cid:16) (cid:17)
We prove that
(2) Pi ≡ m−1 k0k−1−+1ℓ0 qℓ0 1−qmi+1 −1 i m−1 kjk+ℓj qℓjmj (modm),
ℓX0=0(cid:18) 0 (cid:19) !(cid:16) (cid:17) jY=1 ℓXj=0(cid:18) j (cid:19)
 
by inductiononi. As a basecase,the resultfor i=0 followsimmediately fromProposition1.1with a=k .
0
Now assume that (2) holds for some choice of i≥0, and we obtain
P = i+1 1−qmj −kj = 1−qmi+1 −ki+1P
i+1 i
jY=0(cid:16) (cid:17) (cid:16) (cid:17)
≡ m−1 k0−1+ℓ0 qℓ0 1−qmi+1 −ki+1−1 i m−1 kj +ℓj qℓjmj (modm)
k −1  k 
ℓX0=0(cid:18) 0 (cid:19) !(cid:16) (cid:17) jY=1 ℓXj=0(cid:18) j (cid:19)
 
≡ m−1 k0−1+ℓ0 qℓ0 1−qmi+2 −1i+1 m−1 kj +ℓj qℓjmj (modm),
k −1  k 
ℓX0=0(cid:18) 0 (cid:19) !(cid:16) (cid:17) jY=1 ℓXj=0(cid:18) j (cid:19)
 
where the second last equivalence follows from the induction hypothesis, and the last equivalence follows
from Proposition 1.1 with a=k +1, q =qmi+1.
i+1
2
This completes the proof of (2) by induction on i, and the result follows immediately since
B(k)(q)= lim P .
m i→∞ i
(cid:3)
Now we give the explicit expression for the coefficients modulo m that follows from the above expansion
of the generating function B(k)(q).
m
Corollary 2.2. For n≥0, suppose that the base m representation of n is given by
n=d +d m+...+d mt, 0≤t.
0 1 t
If m is relatively prime to (k −1)! and to k ! for j ≥1, then we have
0 j
k −1+d t k +d
b(k)(n)≡ 0 0 j j (modm).
m k −1 k
(cid:18) 0 (cid:19)j=1(cid:18) j (cid:19)
Y
Proof. In the expansion of the series B(k)(q) given in Theorem 2.1, the monomial qn arises uniquely with
m
the specializations ℓ = d , j = 0,...,t and ℓ = 0, j ≥ t. But with these specializations, we have
j j j
kj+ℓj = kj =1, and the result follows immediately. (cid:3)
kj kj
(cid:0) (cid:1) (cid:0) (cid:1)
Specializing the expression given in Corollary 2.2 to the case k = 1 for j ≥ 0 provides an alternative
j
proofto Andrews,FraenkelandSellers’characterizationofm−arypartitionsmodulo m, whichwasgivenas
Theorem 1 of [AFS15].
3. Coloured m-ary partitions without gaps
In this section we consider the following generating function for the numbers c(k)(n) of k-coloured m-ary
m
partitions without gaps:
C(k)(q)=1+ ∞ c(k)(n)qn =1+ ∞ i 1−qmj −kj −1 .
m m
nX=0 Xi=0jY=0(cid:18)(cid:16) (cid:17) (cid:19)
The following result gives an explicit expansion for C(k)(q) modulo m.
m
Theorem 3.1. If m is relatively prime to (k −1)! and to k ! for j ≥1, then we have
0 j
C(k)(q)≡1+ m k0−1+ℓ0 qℓ0 ∞ 1−qmi+1 −1 i m−1 kj +ℓj −1 qℓjmj (modm).
m k −1  k 
ℓX0=1(cid:18) 0 (cid:19) !Xi=0(cid:16) (cid:17) jY=1 ℓXj=0(cid:26)(cid:18) j (cid:19) (cid:27)
 
Proof. Consider the finite product
R = i 1−qmj −kj −1 , i≥0.
i
jY=0(cid:18)(cid:16) (cid:17) (cid:19)
We prove that
(3) Ri ≡ m k0k−1−+1ℓ0 qℓ0 1−qmi+1 −1 i m−1 kjk+ℓj −1 qℓjmj (modm),
ℓX0=1(cid:18) 0 (cid:19) !(cid:16) (cid:17) jY=1 ℓXj=0(cid:26)(cid:18) j (cid:19) (cid:27)
3  
by inductiononi. As a basecase,the resultfor i=0 followsimmediately fromProposition1.1with a=k .
0
Now assume that (3) holds for some choice of i≥0, and we obtain
R = i+1 1−qmj −kj −1 = 1−qmi+1 −ki+1 −1 R
i+1 i
jY=0(cid:18)(cid:16) (cid:17) (cid:19) (cid:18)(cid:16) (cid:17) (cid:19)
≡ m k0−1+ℓ0 qℓ0 1−qmi+1 −ki+1−1− 1−qmi+1 −1
k −1
ℓX0=1(cid:18) 0 (cid:19) !(cid:26)(cid:16) (cid:17) (cid:16) (cid:17) (cid:27)
i m−1
k +ℓ
× j j −1 qℓjmj (modm)
 k 
jY=1 ℓXj=0(cid:26)(cid:18) j (cid:19) (cid:27)
 
≡ m k0−1+ℓ0 qℓ0 1−qmi+2 −1i+1 m−1 kj +ℓj −1 qℓjmj (modm),
k −1  k 
ℓX0=1(cid:18) 0 (cid:19) !(cid:16) (cid:17) jY=1 ℓXj=0(cid:26)(cid:18) j (cid:19) (cid:27)
 
where the second last equivalence follows from the induction hypothesis, and the last equivalence follows
from Proposition 1.1 with a=k +1, q =qmi+1 and a=1, q =qmi+1.
i+1
This completes the proof of (3) by induction on i, and the result follows immediately since
∞
C(k)(q)=1+ R .
m i
i=0
X
(cid:3)
Corollary 3.2. For n≥1, suppose that n is divisible by m, with base m representation given by
n=d ms+...+d mt, 1≤s≤t,
s t
where 1 ≤ d ≤ m−1, and 0 ≤ d ,...,d ≤ m−1. If m is relatively prime to (k −1)! and to k ! for
s s+1 t 0 j
j ≥1, then for 0≤d ≤m−1 we have
0
k −1−d k +d −1 t i k +d
c(k)(n−d )≡ 0 0 ε +(−1)s−1 s s −1 j j −1 (modm),
m 0 k −1  s k k 
(cid:18) 0 (cid:19) (cid:26)(cid:18) s (cid:19) (cid:27)i=sj=s+1(cid:26)(cid:18) j (cid:19) (cid:27)
X Y
 
where ε =0 if s is even, and ε =1 if s is odd.
s s
Proof. First note that we have
n−d =m−d +(m−1)m1+...+(m−1)ms−1+(d −1)ms+d ms+1+...+d mt.
0 0 s s+1 t
Now consider the following specializations: ℓ = m−d , ℓ =m−1, j = 1,...,s−1, ℓ = d −1, ℓ = d ,
0 0 j s s j j
j = s+1,...,t, and ℓ = 0, j > t. Then, in the expansion of the series C(k)(q) given in Theorem 3.1, the
j m
monomial qn arises once for each i ≥ 0, in particular with the above specializations truncated to ℓ ,...,ℓ .
0 i
But with these specializations we have
• for j =0:
k −1+ℓ k −1+m−d k −1−d
j j −1= 0 0 = 0 0 , from (1),
k −1 k −1 k −1
(cid:18) j (cid:19) (cid:18) 0 (cid:19) (cid:18) 0 (cid:19)
• for j =1,...,s−1:
k +ℓ k −1
j j −1= j −1=0−1=−1,
k k
(cid:18) j (cid:19) (cid:18) j (cid:19)
and
s−1 i s−1
k +ℓ
j j −1 = (−1)i =ε ,
s
k
i=0j=1(cid:26)(cid:18) j (cid:19) (cid:27) i=0
XY X
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• for j =s:
k +ℓ k +d −1
j j −1= s s −1,
k k
(cid:18) j (cid:19) (cid:18) s (cid:19)
• for j =s+1,...,t:
k +ℓ k +d
j j −1= j j −1,
k k
(cid:18) j (cid:19) (cid:18) j (cid:19)
• for j >t:
k +ℓ k
j j −1= j −1=1−1=0.
k k
(cid:18) j (cid:19) (cid:18) j(cid:19)
The result follows straightforwardly from Theorem 3.1. (cid:3)
Specializing the expression given in Corollary 3.2 to the case k = 1 for j ≥ 0 provides an alternative
j
prooftoAndrews,FraenkelandSellers’characterizationofm−arypartitionsmodulomwithoutgaps,which
was given as Theorem 2.1 of [AFS16].
References
A71. G.E.Andrews,Congruence properties of the m-ary partition function,J.NumberTheory3(1971), 104–110.
AFS15. George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, Characterizing the Number of m-ary Partitions Modulo
m,AmericanMathematicalMonthly 122(2015), 880–885.
AFS16. GeorgeE.Andrews,AviezriS.Fraenkel,JamesA.Sellers,m-arypartitionswithnogaps: Acharacterizationmodulo
m,DiscreteMathematics 339(2016), 283–287.
BOS13. C.Bessenrodt, J.B.Olsson,J.A.Sellers,Unique path partitions: characterization and congruences, AnnalsComb.
17(2013), 591–602.
C69. R.F.Churchhouse,Congruence properties of the binarypartition function,Proc.CambridgePhilos.Soc.66(1969),
371–376.
E16. Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m, Rocky Mountain J. Math. (to
appear),arXiv1603.00085 math.CO
EZ15. ShaloshB.EkhadandDoronZeilberger,ComputerizingtheAndrews-Fraenkel-SellersProofsontheNumberofm-ary
partitions mod m (and doing MUCH more!),arXiv1511.06791 math.CO
R19. S. Ramanujan, Some properties of p(n), the number of partitions of n, Proc. Cambridge Philos. Soc. 19 (1919),
207–210.
R70. Ø.Rødseth,Some arithmetical properties of m-ary partitions,Proc.CambridgePhilos.Soc.68(1970), 447–453.
Dept. of CombinatoricsandOptimization,University ofWaterloo, Canada
E-mail address: [email protected], [email protected]
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