Parafunctions of triangular matrices and m-ary partitions of numbers
Using the machinery of paradeterminants and parapermanents developed in [2] we get new relations for some number-theoretical functions natural argument that were studied in [3].
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| Цитувати: | Parafunctions of triangular matrices and m-ary partitions of numbers / S. Stefluk, R. Zatorsky // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 144-152. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1551972019-06-17T01:26:13Z Parafunctions of triangular matrices and m-ary partitions of numbers Stefluk, S. Zatorsky, R. Using the machinery of paradeterminants and parapermanents developed in [2] we get new relations for some number-theoretical functions natural argument that were studied in [3]. 2016 Article Parafunctions of triangular matrices and m-ary partitions of numbers / S. Stefluk, R. Zatorsky // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 144-152. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC:12E10. http://dspace.nbuv.gov.ua/handle/123456789/155197 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Using the machinery of paradeterminants and parapermanents developed in [2] we get new relations for some number-theoretical functions natural argument that were studied in [3]. |
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Article |
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Stefluk, S. Zatorsky, R. |
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Stefluk, S. Zatorsky, R. Parafunctions of triangular matrices and m-ary partitions of numbers Algebra and Discrete Mathematics |
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Stefluk, S. Zatorsky, R. |
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Stefluk, S. |
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Parafunctions of triangular matrices and m-ary partitions of numbers |
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Parafunctions of triangular matrices and m-ary partitions of numbers |
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Parafunctions of triangular matrices and m-ary partitions of numbers |
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Parafunctions of triangular matrices and m-ary partitions of numbers |
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Parafunctions of triangular matrices and m-ary partitions of numbers |
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parafunctions of triangular matrices and m-ary partitions of numbers |
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Інститут прикладної математики і механіки НАН України |
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2016 |
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Parafunctions of triangular matrices and m-ary partitions of numbers / S. Stefluk, R. Zatorsky // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 144-152. — Бібліогр.: 7 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT stefluks parafunctionsoftriangularmatricesandmarypartitionsofnumbers AT zatorskyr parafunctionsoftriangularmatricesandmarypartitionsofnumbers |
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2025-07-14T07:16:23Z |
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2025-07-14T07:16:23Z |
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1837605725211197440 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 1, pp. 144–152
© Journal “Algebra and Discrete Mathematics”
Parafunctions of triangular matrices
and m-ary partitions of numbers
Svitlana Stefluk and Roman Zatorsky
Communicated by R. I. Grigorchuk
Abstract. Using the machinavy of paradeterminants and
parapermanents developed in [2] we get new relations for some
number-theoretical functions natural argument that were studied
in [3].
Introduction
Partition polynomials together with corresponding linear recurrent
equations appear in different areas of mathematics. Therefore, it is im-
portant to develop the general theory of partition polynomials, which
would unify the results obtained in this area of mathematics. One of these
general approaches to studying partition polynomials and its correspond-
ing linear recurrent equations is their study with the help of triangular
matrices (tables) machinery [1, 2].
The present paper continues the study of properties and interrelations
of three number-theoretical functions of a natural argument, which was
started in [3]. These functions are the functions bm(n), ξm(n), ([3], p.68-
69.) respectively generalizing the number p(n) of unordered partitions of a
positive integer n into positive integer summands and the sum of divisors
of a positive integer σ(n), as well as the function dm(n), which for m = 2
equals (−1)t(n), where t(n) is the n-th term of the Prouhet-Thue-Morse
2010 MSC: 12E10.
Key words and phrases: partition polynomials, triangular matrices, paradeter-
minant, parapermanent, m-ary partition.
S. Stefluk, R. Zatorsky 145
sequence [4]. In [3] the authors apply the methods of combinatorial analysis
(generatrix method) and linear algebra. As for us, in order to study these
functions, we use the general theory of partition polynomials developed
with the help of triangular matrix calculus machinery developed by the
first autor. At that, we received new relations between these functions and
all the proofs are considerably simplified. As the result we get a seweral
new relations between functions bm(n), ξm(n) and dm(n) and express
them via paradeterminants and parapermanents.
1. Preliminaries
This section includes some necessary notions and their properties,
which will be needed in the next section.
1.1. Some notions and results concerning triangular matrices
In this section we provide basic notions and results about parade-
terminants and parapermanents that will be used for the proving of the
main results of the paper.
Let K be some number field.
Definition 1 ([1, 2]). A triangular table
A =
a11
a21 a22
...
...
. . .
an1 an2 · · · ann
n
(1)
of numbers from a number field K is called a triangular matrix, an element
a11 is an upper element of this triangular matrix, and a number n is its
order.
Definition 2 ([1, 2]). If A is the triangular matrix (1), then its parade-
terminant and parapermanent are the following numbers, respectively:
ddet(A) =
n
∑
r=1
∑
p1+...+pr=n
(−1)n−r
r
∏
s=1
{ap1+...+ps,p1+...+ps−1+1}, (2)
pper(A) =
n
∑
r=1
∑
p1+...+pr=n
r
∏
s=1
{ap1+...+ps,p1+...+ps−1+1},
146 Parafunctions and m-ary partitions
where the summation is over the set of natural solutions of the equality
p1 + . . . + pr = n and
bij = {aij} =
i
∏
k=j
aik, 1 6 j 6 i 6 n.
For a parapermanent and paradeterminant of a matrix we will use
notations shown in (15) and (16) respectively.
Definition 3 ([1,2]). To each element aij of the given triangular matrix (1)
we correspond a triangular matrix with this element in the bottom left
corner, which we call a corner of the given triangular matrix and denote
by Rij(A).
It is obvious that the corner Rij(A) is a triangular matrix of order
(i − j + 1). The corner Rij(A) includes only those elements ars of the
triangular matrix (1), the indices of which satisfy the relations j 6 s 6
r 6 i.
Sometimes we extend the range of indeces in (1) from 1,. . . ,n to
0,1,. . . ,n+1 and agree that
ddet(R01(A)) = ddet(Rn,n+1(A)) = pper(R01(A))
= pper(Rn,n+1(A)) = 1. (3)
When finding values of the paradeterminant and the parapermanent
of triangular matrices, it is convenient to use algebraic complements.
Definition 4 ([1, 2]). Algebraic complements Dij , Pij to a factorial prod-
uct {aij} of a key element aij of the matrix (1) are, respectively, numbers
Dij = (−1)i+j · ddet(Rj−1,1) · ddet(Rn,i+1), (4)
Pij = pper(Rj−1,1) · pper(Rn,i+1), (5)
where Rj−1,1 and Rn,i+1 are corners of the triangular matrix (1).
Theorem 1 ([1,2]). If A is the triangular matrix (1), then the parafunc-
tions of this matrix can be decomposed by the elements of the last row.
With that, the following equalities hold:
S. Stefluk, R. Zatorsky 147
ddet(A) =
n
∑
s=1
{ans} Dns =
n
∑
s=1
(−1)n+s {ans} · ddet(Rs−1,1), (6)
pper(A) =
n
∑
s=1
{ans} Pns =
n
∑
s=1
{ans} · pper(Rs−1,1), (7)
where
bij = {aij} =
i
∏
k=j
aik, 1 6 j 6 i 6 n.
Theorem 2 (Relation between parapermanent and paradeterminant[1, 2]).
If A is the triangular matrix (1), then the following relation holds
pper (aij)16j6i6n
= ddet
(
(−1)δij+1
aij
)
16j6i6n
. (8)
Corollary 1. For any triangular matrix (bij)16j6i6n, the following equal-
ity holds
ddet(bij)16j6i6n = pper((−1)δij+1bij)16j6i6n.
Theorem 3 ([5]). The following is true
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
a11 a1 0 . . . 0 0 0
a21 a22 a2 . . . 0 0 0
a31 a32 a33 . . . 0 0 0
... . . . . . . . . . . . . . . .
...
an−2,1 an−2,2 an−2,3 . . . an−2,n−2 an−2 0
an−1,1 an−1,2 an−1,3 . . . an−1,n−2 an−1,n−1 an−1
an1 an2 an3 . . . an,n−2 an,n−1 ann
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
〈
a11
a1
a21
a22
a22
a1
a31
a32
a2
a32
a33
a33
... . . . . . .
. . .
a1
an−2,1
an−2,2
a2
an−2,2
an−2,3
a3
an−2,3
an−2,4
. . . an−2,n−2
a1
an−1,1
an−1,2
a2
an−1,2
an−1,3
a3
an−1,3
an−1,4
. . . an−2
an−1,n−2
an−1,n−1
an−1,n−1
a1
an1
an2
a2
an2
an3
a3
an3
an4
. . . an−2
an,n−2
an,n−1
an−1
an,n−1
ann
ann
〉
(9)
148 Parafunctions and m-ary partitions
1.2. Some data from the general theory
of partition polynomials
We will need also the following three results.
Theorem 4 ([2], Theorem 2.5.3). The following three equalities are
equipotent:
An =
〈 x(1)
x(2)
x(1) x(1)
... . . .
. . .
·x(n−1)
x(n−2)
x(n−2)
x(n−3) . . . x(1)
x(n)
x(n−1)
x(n−1)
x(n−2) . . .
x(2)
x(1) x(1)
〉
,
An = x1An−1 − x2An−2 + . . . + (−1)n−1xnA0, A0 = 1,
An =
∑
λ1+2λ2+...+nλn=n
(−1)n−k k!
λ1! · . . . · λn!
xλ1
1 · . . . xλn
n , k = λ1 + . . . + λn.
Theorem 5 ([7]). Let polynomials yn(x1, x2, . . . , xn), n = 0, 1, . . ., satisfy
the recurrence relation
yn = x1yn−1 − x2yn−2 + . . . + (−1)n−2xn−1y1 + (−1)n−1anxny0, (10)
where y0 = 1. Then the relations
yn = ddet
a1x1
a2
x2
x1
x1
... . . .
. . .
an
xn
xn−1
. . . x2
x1
x1
, (11)
yn =
∑
λ1+2λ2+...+nλn=n
(−1)n−k
(
n
∑
i=1
λiai
) (k − 1)!
λ1!λ2!·. . .·λn!
xλ1
1 xλ2
2 · . . . · xλn
n , (12)
where k = λ1 + λ2 + . . . + λn, hold.
Theorem 6 ([2], Theorem 3.6.1). The following formulae of inversion
of partition polynomials written as parafunctions of triangular matrices
are valid:
S. Stefluk, R. Zatorsky 149
1) bi =
〈
τsr
as−r+1
as−r
〉
16r6s6i
, (13)
ai =
〈
τ−1
s,s−r+1
bs−r+1
bs−r
〉
16r6s6i
, i = 1, 2, . . . ; (14)
2) bi =
[
τsr
as−r+1
as−r
]
16r6s6i
,
ai = (−1)i−1
〈
τ−1
s,s−r+1
bs−r+1
bs−r
〉
16r6s6i
, i = 1, 2, . . . ,
where ai, bi are arbitrary real variables, τrs are rational numbers.
2. Parafunctions of triangular matrices and m-ary parti-
tions of numbers
Our first theorem show how functions bm(n), ξm(n), dm(n), studied
in [3] can be expressed via paradeterminant and parapermanent.
Theorem 7. The following equalities hold:
bm(n) =
ξm(1)
ξm(2)
ξm(1)
1
2ξm(1)
... . . .
. . .
ξm(n−1)
ξm(n−2)
ξm(n−2)
ξm(n−3) . . . 1
n−1ξm(1)
ξm(n)
ξm(n−1)
ξm(n−1)
ξm(n−2) . . .
ξm(2)
ξm(1)
1
n
ξm(1)
, (15)
dm(n) = (−1)n
〈 ξm(1)
ξm(2)
ξm(1)
1
2ξm(1)
... . . .
. . .
ξm(n−1)
ξm(n−2)
ξm(n−2)
ξm(n−3) . . . 1
n−1ξm(1)
ξm(n)
ξm(n−1)
ξm(n−1)
ξm(n−2) . . .
ξm(2)
ξm(1)
1
n
ξm(1)
〉
, (16)
ξm(n) = (−1)n−1
〈 bm(1)
2 · bm(2)
bm(1) bm(1)
... . . .
. . .
(n−1)· bm(n−1)
bm(n−2)
bm(n−2)
bm(n−3) . . . bm(1)
n· bm(n)
bm(n−1)
bm(n−1)
bm(n−2) . . .
bm(2)
bm(1) bm(1)
〉
, (17)
150 Parafunctions and m-ary partitions
ξm(n) = (−1)n
〈 dm(1)
2 · dm(2)
dm(1) dm(1)
... . . .
. . .
(n−1)· dm(n−1)
dm(n−2)
dm(n−2)
dm(n−3) . . . dm(1)
n· dm(n)
dm(n−1)
dm(n−1)
dm(n−2) . . .
dm(2)
dm(1) dm(1)
〉
, (18)
bm(n) = (−1)n
〈 dm(1)
dm(2)
dm(1) dm(1)
... . . .
. . .
dm(n−1)
dm(n−2)
dm(n−2)
dm(n−3) . . . dm(1)
dm(n)
dm(n−1)
dm(n−1)
dm(n−2) . . .
dm(2)
dm(1) dm(1)
〉
, (19)
dm(n) = (−1)n
〈 bm(1)
bm(2)
bm(1) bm(1)
... . . .
. . .
· bm(n−1)
bm(n−2)
bm(n−2)
bm(n−3) . . . bm(1)
bm(n)
bm(n−1)
bm(n−1)
bm(n−2) . . .
bm(2)
bm(1) bm(1)
〉
. (20)
Proof. Relations (15), (16) follows from recurrent relations of Theorem 3
(from [3], p. 70). Indeed, each of these equalities is a result of expansion of
the paradeterminants on the right side of (15) or (16) by elements of the
last raw. Relations (17), (18) can be obtained by inversion of (15), (16)
using Theorem 6; (19), (20) follows directly from Theorem 2 in [3], p. 69,
and the above Theorem 3 on the relation between paradeterminants and
determinants.
The following theorem gives recurrent relations between functions
bm(n), ξm(n), dm(n).
Theorem 8. The following equalities hold:
ξm(n) = −
(
bm(1)ξm(n − 1) + bm(2)ξm(n − 2)
+ . . . + bm(n − 1)ξm(1) − nbm(n)ξm(0)
)
, (21)
ξm(n) = −
(
dm(1)ξm(n − 1) + dm(2)ξm(n − 2)
+ . . . + dm(n − 1)ξm(1) + ndm(n)ξm(0)
)
, (22)
S. Stefluk, R. Zatorsky 151
bm(n) = −
(
dm(1)bm(n − 1) + dm(2)bm(n − 2)
+ . . . + dm(n − 1)bm(1) + dm(n)bm(0)
)
, (23)
dm(n) = −
(
bm(1)dm(n − 1) + bm(2)dm(n − 2)
+ . . . + bm(n − 1)dm(1) + bm(n)dm(0)
)
, (24)
where bm(0) = dm(0) = ξm(0) = 1.
Proof. To prove (21) multiply both sides of (17) by (−1)n−1 and expand
paradeterminant on the right side of the obtained equality by elements of
the last row. As the result, we get
(−1)n−1ξm(n) = bm(1)(−1)n−2ξm(n − 1) − bm(2)(−1)n−3ξm(n − 2)
+ . . . + (−1)n−2bm(n − 1)(−1)0ξm(n − (n − 1))
+ (−1)n−1bm(n)(−1)−1ξm(n − n)
and hence (21). Similarly, one can prove the relation (22) using (18).
Relations (23), (24) can be obtained from (19) and (20) respectively. Let
us prove, for example, (23). Multiply both sides of (19) by (−1)n and
expand paradeterminant on the right side of obtained equality by elements
of the last row. Then
(−1)nbm(n) = dm(1)(−1)n−1bm(n − 1) − dm(2)(−1)n−2bm(n − 2)
+ . . . + (−1)n−2dm(n − 1)(−1)1)bm(n − (n − 1))
+ (−1)n−1dm(n)(−1)0bm(n − n),
and the required relation follow immediately.
In the next theorem, we describe partition polynomials as defined
in [6] presenting m-ary numbers bm(n), ξm(n), dm(n).
Theorem 9. The following equalities hold:
dm(n) =
∑
λ1+2λ2+...+nλn=n
(−1)k ξλ1
m (1) · . . . · ξλn
m
λ1! · . . . · λn!1λ1 · . . . · nλn
, (25)
ξm(n) =
∑
λ1+2λ2+...+nλn=n
(−1)k−1 n(k − 1)!
λ1! · . . . · λn!
· bλ1
m (1) · . . . · bλn
m (n), (26)
ξm(n) =
∑
λ1+2λ2+...+nλn=n
(−1)k n(k − 1)!
λ1! · . . . · λn!
· dλ1
m (1) · . . . · dλn
m (n), (27)
152 Parafunctions and m-ary partitions
bm(n) =
∑
λ1+2λ2+...+nλn=n
(−1)k k!
λ1! · . . . · λn!
· dλ1
m (1) · . . . · dλn
m (n), (28)
dm(n) =
∑
λ1+2λ2+...+nλn=n
(−1)k k!
λ1! · . . . · λn!
· bλ1
m (1) · . . . · bλn
m (n), (29)
where k = λ1 + λ2 + . . . + λn.
Proof. Partition polynomial corresponding to parapermanent (15), were
described by Kachi and Tzermias [3, Theorem 1, p. 68]. Paradeterminant of
the same matrix corresponds to the partition polynomial that differs from
the previous one only by sign (−1)n−k. Thus (25) holds. The relations
for partition polynomials (26), (27) and (28), (29) follow directly from
theorems 5 and 4 respectively.
References
[1] Zatorsky R.A. Theory of paradeterminants and its applications // Algebra and
Discrete Mathematics №1, 2007, pp. 109-138.
[2] Zatorsky R.A. Calculus of Triangular Matrices and Its Applications. Ivano-
Frankivsk, Simyk, 2010, 508 p. (in Ukrainian).
[3] Yasuyuki Kachi and Pavlos Tzermias, On the m-ary partition numbers, Algebra
and Discrete Mathematics 19, 1 (2015), 67–76.
[4] J.-P. Allouche, J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence, In: C. Ding
et al. (eds.), Sequences and their Aplications, Proceedings of SETA ’98, Springer-
Verlag London, 1999, 1–16.
[5] R. A. Zatorsky, Researching of Hessenbergs matrix functions, Carpathian Mathe-
matical Publications 3, 1 (2011), 49–55. (in Ukrainian)
[6] Riordan J. Combinatorial identities. New York: Wiley, 1968, 256 p.
[7] R. Zatorsky, S. Stefluk, On one class of partition polynomials, Algebra and Discrete
Mathematics 16, 1 (2013), 127–133.
Contact information
S. Stefluk,
R. Zatorsky
Department of Mathematics and Computer Science,
Precarpathian Vasyl Stefanyk National University 57
Shevchenka Str. Ivano-Frankivsk, 76025 Ukraine
E-Mail(s): ljanys@mail.ru,
romazatorsky@gmail.com
Web-page(s): www.romaz.pu.if.ua
Received by the editors: 24.09.2015
and in final form 19.03.2016.
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