On the m-ary partition numbers
We give explicit formulas and recurrences for the m-ary partition numbers and relate them to Toeplitz-Hessenberg determinants. Some of these results are direct analogues of similar statements for the classical (unrestricted) partition numbers.
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irk-123456789-1527882019-06-13T01:25:35Z On the m-ary partition numbers Kachi, Y. Tzermias, P. We give explicit formulas and recurrences for the m-ary partition numbers and relate them to Toeplitz-Hessenberg determinants. Some of these results are direct analogues of similar statements for the classical (unrestricted) partition numbers. 2015 Article On the m-ary partition numbers / Y. Kachi, P. Tzermias // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 67-76. — Бібліогр.: 39 назв. — англ. 1726-3255 2010 MSC:05A17, 11P81 http://dspace.nbuv.gov.ua/handle/123456789/152788 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We give explicit formulas and recurrences for the m-ary partition numbers and relate them to Toeplitz-Hessenberg determinants. Some of these results are direct analogues of similar statements for the classical (unrestricted) partition numbers. |
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On the m-ary partition numbers |
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On the m-ary partition numbers |
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On the m-ary partition numbers |
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On the m-ary partition numbers |
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On the m-ary partition numbers |
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on the m-ary partition numbers |
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On the m-ary partition numbers / Y. Kachi, P. Tzermias // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 67-76. — Бібліогр.: 39 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-14T04:16:58Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 1, pp. 67–76
© Journal “Algebra and Discrete Mathematics”
On the m-ary partition numbers
Yasuyuki Kachi and Pavlos Tzermias
Communicated by R. I. Grigorchuk
Abstract. We give explicit formulas and recurrences for the
m-ary partition numbers and relate them to Toeplitz-Hessenberg
determinants. Some of these results are direct analogues of similar
statements for the classical (unrestricted) partition numbers.
1. Introduction
Let m be a fixed integer such that m > 2. For a positive integer n,
let bm(n) denote the number of m-ary partitions of n, i.e. the number of
ways of writing n as a sum of powers of m using non-negative exponents
with repetitions allowed and the order of the summands not being taken
into account. We also set bm(0) = 1.
The special case of binary partitions, i.e. the case m = 2, was appar-
ently first studied by Euler in 1750. The recurrence formulas
b2(2n + 1) = b2(2n), b2(2n) = b2(2n − 1) + b2(n),
for n > 0 are well-known. The sequence {b2(n) : n ∈ N} is labelled
A018819 in the On-Line Encyclopedia of Integer Sequences (available at
http://oeis.org/AO18819).
Various number-theoretic properties as well as the asymptotic behavior
of the m-ary partition sequence {bm(n) : n ∈ N} and some of its variants
2010 MSC: 05A17, 11P81.
Key words and phrases: m-ary partition numbers, generating functions, Toeplitz-
Hessenberg determinants.
68 m-ary partition numbers
(for the binary and for the general case) have been extensively studied in
the literature. We refer the reader to the work of Alkauskas [1,2], Andrews
[4], Calkin and Wilf [8], Churchhouse [10], de Bruijn [12], Flajolet and
Sedgewick [15], Gupta [17–19], Hirschhorn and Loxton [21], Hirschhorn and
Sellers [22], Mahler [23–25], Meinardus [27], Murty and Murty [29, Ch. 7],
Pennington [30], Richmond [33], Rödseth [34], Rödseth and Sellers [35,36]
for more information, emphasizing the fact that the latter list is by no
means exhaustive.
The generating function for the sequence {bm(n) : n ∈ N} is
fm(x) =
∞
∑
n=0
bm(n)xn =
∞
∏
n=0
∞
∑
i=0
ximn
=
∞
∏
n=0
(1 − xmn
)−1,
which implies the functional equation fm(xm) = (1 − x)fm(x). Besides
its combinatorial importance, the generating function fm(x) is of special
interest in transcendental number theory. For example, as Mahler has
shown in [25] (see also its corrigendum [24]), the values of f2(x) are always
transcendental for every algebraic number x such that 0 < |x| < 1.
In this paper we give explicit formulas and recurrence relations for
the sequence {bm(n) : n ∈ N}. We also relate these formulas to Toeplitz-
Hessenberg determinants (first introduced in [37] and [20]). Some of
these results are direct analogues of similar statements for the classical
(unrestricted) partition numbers.
For a positive integer n, let ξm(n) denote the sum of all divisors of n
that are powers of m with non-negative exponent. Note that ξm(1) = 1
and ξm(n) > 1, for all n > 1. We also define ξm(0) = 1. The sequence
{ξm(n) : n ∈ N} is the m-ary analogue of the classical sum-of-divisors
sequence.
Our first result in this paper is an elementary yet explicit formula for
bm(n) given below. We are not aware of any other formula in the existing
literature that concretely expresses the integer bm(n) as a finite sum of
rational numbers. For more profound analogous results for the classical
partition function we refer the reader to the formulas of Rademacher in
[31] and [32] and of Bruinier and Ono in [7]. It would be interesting to
look for m-ary analogues of the latter formulas, if such analogues exist.
Theorem 1. For every positive integer n, the m-ary partition number
bm(n) is expressed as the finite sum
bm(n) =
∑
n1+2n2+3n3+···=n
ξm(1)n1ξm(2)n2ξm(3)n3 · · ·
n1! n2! n3! · · · 1n12n23n3 · · ·
.
Y. Kachi, P. Tzermias 69
Remark 1. It is a well-known and easy to prove fact that the number of
summands on the right-hand side of the latter formula equals p(n), i.e.
the number of partitions of the integer n.
Now, for a positive integer n, let dm(n) ∈ N be defined as follows:
If n is not the sum of distinct powers of m, let dm(n) = 0. If n is
the sum of an even number of distinct powers of m, let dm(n) = 1. If n
is the sum of an odd number of distinct powers of m, let dm(n) = −1.
We also set dm(0) = 1. Note that
∞
∑
n=0
dm(n) xn =
∞
∏
n=0
(1 − xmn
) =
1
fm(x)
.
Note also that, in the binary case, we have d2(n) = (−1)t(n), where t(n)
is the n-th term of the Prouhet-Thue-Morse sequence, which is discussed
extensively in [3]. We now express the m-ary partition numbers bm(n)
as determinants of certain Toeplitz-Hessenberg matrices with entries in
the set {−1, 0, 1}, specifically with entries given by the numbers dm(n).
This is the m-ary analogue of a formula obtained by Malenfant [26]
for the classical partition function. Conversely, we can also express the
numbers dm(n) as determinants of certain Toeplitz-Hessenberg matrices
with entries given by the numbers bm(n):
Theorem 2. For every positive integer n, we have
bm(n) = (−1)ndet
dm(1) 1 0 · · · 0
dm(2) dm(1) 1 · · · 0
...
...
...
. . .
...
dm(n − 1) dm(n − 2) dm(n − 3) · · · 1
dm(n) dm(n − 1) dm(n − 2) · · · dm(1)
.
Similarly,
dm(n) = (−1)ndet
bm(1) 1 0 · · · 0
bm(2) bm(1) 1 · · · 0
...
...
...
. . .
...
bm(n − 1) bm(n − 2) bm(n − 3) · · · 1
bm(n) bm(n − 1) bm(n − 2) · · · bm(1)
.
Remark 2. The proof of Theorem 1.2 is very simple and based on
Cramer’s rule for solving linear systems. To be more specific, the fact
70 m-ary partition numbers
that the coefficients of the reciprocal of a formal power series can be
expressed as determinants of certain Toeplitz-Hessenberg matrices whose
entries are functions of the coefficients of the original formal power series
is well-known and simple to prove (e.g. page 157 in Comtet’s book [11]).
The same fact has recently been applied in a number of similar contexts
and settings by Chen [9], Malenfant [26] and Vella [38]. On the other hand,
what is far from simple is an ingenious and quite general method employed
by Brioschi [6] relating Toeplitz-Hessenberg determinants to complete
homogeneous forms in the roots of a given polynomial and leading to a
general formula expressing these determinants in terms of multinomial
coefficients. The same formula was apparently discovered earlier by Fergola
[14] using a different method. We refer the reader to Muir’s book [28] for
an extensive and substantiated survey of the historical development of the
theory of determinants. An excellent (albeit significantly shorter) account
of the Brioschi-Fergola general formula is also described in the preprint
[13] by Fera and Talamini, where two different proofs of the formula are
presented. In light of what appears to be a resurgence of recent interest
around this circle of ideas, we believe that the Brioschi-Fergola formula
deseves to be more widely known. We would also like to point out that, to
the best of our knowledge and ability, the Brioschi-Fergola formula applied
to the particular setting of Theorem 1.2 does not immediately reduce
to the formula given in Theorem 1.1 in any obvious way; for one thing,
the summands in the formula given in Theorem 1.1 are not necessarily
integers, while those given by the Brioschi-Fergola formula are.
We also give recurrence formulas for the numbers bm(n) and dm(n).
The recurrence relation for bm(n) is the m-ary analogue of a rather well
known recurrence relation for the classical partition function (see [5],
Entry 52, page 108):
Theorem 3. For n > 1, we have
bm(n) =
1
n
n−1
∑
j=0
ξm(n−j) bm(j) and dm(n) =
−1
n
n−1
∑
j=0
ξm(n−j) dm(j).
2. Proofs
Proof of Theorem 1. As usual, let log(·) denote the natural logarithm.
As formal power series, we have
log(fm(x)) = log
(
∞
∏
n=0
(1 − xmn
)−1
)
= −
∞
∑
n=0
log(1 − xmn
)
Y. Kachi, P. Tzermias 71
=
∞
∑
n=0
∞
∑
i=1
ximn
i
=
∞
∑
n=0
∞
∑
i=1
ximn
imn
mn.
Define integers ak, for k ∈ {1, 2, . . .}, by setting ak = k, if k is a power of
m, and ak = 0, if k is not a power of m. Then
log(fm(x)) =
∞
∑
k=1
xk
k
∑
d|k
ad =
∞
∑
k=1
xk
k
ξm(k).
Therefore,
fm(x) = elog(fm(x)) =
∞
∑
n=0
1
n!
(
∞
∑
k=1
xk
k
ξm(k)
)n
.
For a sequence of natural numbers n1, n2, . . ., with only finitely many
non-zero terms, the multinomial coefficient
(
n1 + n2 + · · ·
n1, n2, · · ·
)
=
(n1 + n2 + · · · )!
n1! n2! · · ·
makes sense and one can use the “infinite version" of the multinomial
theorem
(
∞
∑
k=1
yk
)n
=
∑
n1+n2+···=n
(
n
n1, n2, · · ·
)
∞
∏
k=1
yk
nk ,
to get
fm(x) =
∞
∑
n=0
1
n!
∑
n1+n2+···=n
(
n
n1, n2, · · ·
)
∞
∏
k=1
(
xk
k
ξm(k)
)nk
=
∞
∑
n=0
1
n!
∑
n1+n2+···=n
(
n
n1, n2, · · ·
)
∞
∏
k=1
xknk
knk
ξm(k)nk
=
∞
∑
n=0
1
n!
∑
n1+n2+···=n
(
n
n1, n2, · · ·
)
xn1+2n2+···
1n12n2 · · ·
ξm(1)n1ξm(2)n2 · · ·
=
∞
∑
r=0
∑
n1+2n2+···=r
(
n1 + n2 + · · ·
n1, n2, · · ·
)
ξm(1)n1ξm(2)n2 · · ·
(n1 + n2 + · · · )!1n12n2 · · ·
xr
=
∞
∑
r=0
∑
n1+2n2+3n3+···=r
ξm(1)n1ξm(2)n2ξm(3)n3 · · ·
n1! n2! n3! · · · 1n12n23n3 · · ·
xr.
72 m-ary partition numbers
For all r > 0, the coefficient of xr in the latter series must equal bm(r)
and the theorem follows.
Proof of Theorem 2. Comparing coefficients of the same powers of x
on both sides of the equality
(
∞
∑
i=0
dm(i) xi
) (
∞
∑
i=0
bm(i) xi
)
= 1,
gives the linear system
r
∑
i=0
dm(r − i) bm(i) = 0, for every r > 1.
Combining the latter equations for r = 1, . . . , n, with bm(0) = dm(0) = 1,
results in the n × n linear system AX = B, where
A =
dm(n − 1) dm(n − 2) · · · 1
dm(n − 2) dm(n − 3) · · · 0
...
...
. . .
...
1 0 · · · 0
,
X =
bm(1)
bm(2)
...
bm(n)
, B =
−dm(n)
−dm(n − 1)
...
−dm(1)
.
The determinant of A equals (−1)
n(n+3)
2 . Also, the determinant of the
matrix obtained from A by replacing its last column by B equals
−det
dm(n − 1) dm(n − 2) · · · dm(1) dm(n)
dm(n − 2) dm(n − 3) · · · 1 dm(n − 1)
...
...
. . .
...
...
dm(1) 1 · · · 0 dm(2)
1 0 · · · 0 dm(1)
= (−1)ndet
dm(n) dm(n − 1) dm(n − 2) · · · dm(1)
dm(n − 1) dm(n − 2) dm(n − 3) · · · 1
...
...
...
. . .
...
dm(2) dm(1) 1 · · · 0
dm(1) 1 0 · · · 0
Y. Kachi, P. Tzermias 73
(−1)
n(n+1)
2 det
dm(1) 1 0 · · · 0
dm(2) dm(1) 1 · · · 0
...
...
...
. . .
...
dm(n − 1) dm(n − 2) dm(n − 3) · · · 1
dm(n) dm(n − 1) dm(n − 2) · · · dm(1)
.
Hence, by Cramer’s rule, the formula for bm(n) in Theorem 2 follows. A
similar argument (interchanging the roles of bm(n) and dm(n)) establishes
the formula for dm(n) in Theorem 2.
Proof of Theorem 3. The idea of the proof is classically known and
due to Euler (as explained by Fuchs and Tabachnikov in Section 3.5 of
[16]). To prove the first recurrence formula, we argue as follows:
As formal power series, we have
−x
d
dx
log
(
1 − xmk
)
= x
d
dx
(
∞
∑
i=1
ximk
i
)
= mk
∞
∑
i=1
ximk
.
Therefore,
x
d
dx
log (fm(x)) = −
∞
∑
k=0
x
d
dx
log
(
1 − xmk
)
=
∞
∑
k=0
∞
∑
i=1
mkximk
=
∞
∑
n=1
ξm(n)xn.
Hence,
x
d
dx
fm(x) = fm(x)
∞
∑
n=1
ξm(n)xn,
which implies that
x
d
dx
∞
∑
n=0
bm(n)xn = fm(x)
∞
∑
n=1
ξm(n)xn.
Hence,
∞
∑
n=1
nbm(n)xn =
(
∞
∑
n=0
bm(n)xn
)(
∞
∑
n=1
ξm(n)xn
)
=
∞
∑
n=1
n−1
∑
j=0
bm(j) ξm(n − j)
xn,
74 m-ary partition numbers
and the recurrence formula follows by comparing the coefficients of xn on
both sides of the latter equality.
The second recurrence formula can be proven by a similar argument:
The relation (proven above)
x
d
dx
fm(x) = fm(x)
∞
∑
n=1
ξm(n)xn,
also implies that
x
d
dx
(
1
fm(x)
)
=
−1
fm(x)
∞
∑
n=1
ξm(n)xn.
Therefore,
x
d
dx
∞
∑
n=0
dm(n)xn =
−1
fm(x)
∞
∑
n=1
ξm(n)xn,
which implies that
∞
∑
n=1
ndm(n)xn = −
(
∞
∑
n=0
dm(n)xn
)(
∞
∑
n=1
ξm(n)xn
)
= −
∞
∑
n=1
n−1
∑
j=0
dm(j) ξm(n − j)
xn,
and the recurrence formula follows by comparing of the coefficients of xn
on both sides of the latter equality.
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Contact information
Y. Kachi Department of Mathematics, University of
Kansas, Lawrence, KS 66045-7523, USA
E-Mail(s): kachi@ku.edu
Web-page: www.math.ku.edu/∼kachi/
P. Tzermias Department of Mathematics, University of Pa-
tras, 26500 Rion (Patras), Greece
E-Mail(s): tzermias@math.upatras.gr
Web-page: www.math.upatras.gr/∼tzermias/
Received by the editors: 02.09.2014
and in final form 24.12.2014.
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