1. Introduction
We recall that a composition of a positive integer
n is a sequence
of positive integers whose sum is
n, i.e.,
The positive integers in the sequence are called parts [
1]. When the order of integers
does not matter, Representation (
1) is known as an integer partition and can be rewritten as
where each positive integer
i appears
times in the partition. For consistency, a partition of
n is written with the summands in nonincreasing order. As usual, we denote by
the number of integer partitions of
n. Partitions can be graphically visualized with Young diagrams. For example, the five partitions of four can be seen in
Figure 1. It is clear that
. For convenience, we define
.
Euler showed that the generating function of
can be expressed as an elegant infinite product
Here and throughout the paper, we use the following customary
q-series notation:
Because the infinite product
diverges when
and
, whenever
appears in a formula, we assume
. All identities may be understood in the sense of formal power series in
q.
A plane partition
of the positive integer
n is a two-dimensional array
of non-negative integers
such that
which is weakly decreasing in rows and columns:
It can be considered as the filling of a Young diagram with weakly decreasing rows and columns, where the sum of all these numbers is equal to
n. Different configurations are counted as different plane partitions. As usual, we denote by
the number of plane partitions of
n. The plane partitions of four are presented in
Figure 2. We see that
. For convenience, we define
.
An equivalent definition is that a plane partition is a finite subset of
with the property that if
and
, then
must be an element of
. Here,
means
,
and
. There is a nice way to represent a plane partition as a three-dimensional object: this is achieved by replacing each part
k of the plane partition by a stack of
k unit cubes. Thus, we obtain a pile of unit cubes. The piles of cubes corresponding to the plane partitions in
Figure 2 are shown in
Figure 3.
Plane partitions were introduced to mathematics by Major Percy Alexander MacMahon [
2] as generalizations of partitions of integers. MacMahon [
3] (Section 429) offered a surprisingly simple formula for the generating function for all plane partitions
contained in an
box. According to Macdonald [
4] (Equation (
2) on p. 81), this formula can be written as
Letting
, we obtain an elegant product formula for the generating function for all plane partitions, i.e.,
and the expansion starts as
The arrangement of the plane partitions of four in
Figure 2 or
Figure 3 is not random. According to M. K. Azarian [
5] (Theorem 1.1),
can be interpreted as the number of different ways to run up a staircase with
n steps, taking steps of possibly different sizes, where the order is not important and there is no restriction on the number or the size of each step taken. Any partition
can be considered a staircase with
n steps where
(
) is just a label associated with the
kth step. On the other hand, any partitions
can be converted into plane partitions by insertion of line feeds at some or all places of the commas. The plane partition obtained in this way can be interpreted as a way to run up the staircase labeled by
, i.e., the number of parts on the
kth line of the plane partition represents the size of the
kth step. We remark that there are plane partitions that cannot be obtained in this way. For example, the plane partition in
Figure 4 cannot be obtained from the partition
by insertion of line feeds at some or all places of the commas.
Since the length of rows in plane partitions must be nonincreasing, there are only ways to comply with this rule. Thus, we easily deduce the following inequality for :
Theorem 1. For ,with strict inequality if and only if . Rewriting the partitions of four as
the case
of Theorem 1 reads as follows:
In this paper, motivated by Theorem 1, we want to show that can be expressed as a sum over all the partitions of n in terms of binomial coefficients. This new formula considers the multiplicity of the parts greater than one.
Considering (
4), the case
of Theorem 2 reads as follows:
In this context, we remark that the first differences of can be be expressed as a sum over the partitions of n into parts greater than one in terms of binomial coefficients.
According to (
3), we have
The partitions of four into parts greater than pne are
and
. Therefore, case
of Theorem 2 reads as follows:
The following result shows that the partial sums of can be be expressed as a sum over all the partitions of n in terms of binomial coefficients.
According to (
3), we have
Considering (
4), case
of Theorem 4 reads as follows:
Upon reflection, one expects that there might be more general results where our Theorems 2–4 are the first entries. For any positive integer
m, we denote by
the number of
m-tuples of plane partitions of non-negative integers
where
. It is clear that
and
For any positive integer
m,
, the partial sums of
and the first differences of
can be expressed as a sum over all the partitions of
n in terms of binomial coefficients.
Theorem 5. For and ,
- 1.
- 2.
- 3.
The remainder of the paper is organized as follows. In
Section 2, we consider the complete homogeneous symmetric functions and introduce Theorem 6. This general result allows us provision of an analytic proof of Theorem 5 by considering specializations of complete homogeneous symmetric functions. In
Section 3, we provide other applications of Theorem 6 by considering the strict plane partitions and the symmetric plane partitions. In the last section, we consider a sum over all the partitions of
n in order to provide a new expression for the generating function of
. Finding a combinatorial interpretation in terms of plane partitions for this sum over all partitions of
n remains an open problem.
2. Proof of Theorem 5
If
is an integer partition with
, then the monomial symmetric function
is the sum of the monomial
and all distinct monomials obtained from this by a permutation of variables. For instance, with
and
, we have
If every monomial in a symmetric function has total degree
k, then we say that this symmetric function is homogeneous of degree
k. Proofs and more details about monomial symmetric functions can be found in Macdonald’s book [
4].
The
kth complete homogeneous symmetric function
is the sum of all monomials of total degree
k in these variables, i.e.,
where
indicates that
is a partition of
n. It is well known that the complete homogeneous symmetric functions are characterized by the following formal power series identity in
t:
Considering the complete homogeneous symmetric functions, we introduce the following result:
Theorem 6. Let m and n be positive integers. Then,where are independent variables and is a sequence of non-negative integers. Proof. We are to prove the theorem by induction on
n. For
, we have
and the base case of induction is finished. We suppose that relation
is true for any integer
N,
. We can write
where we invoke the well-known Cauchy multiplications of two power series. □
We are now in the position to prove Theorem 5. By Theorem 6, with
replaced by
and
z replaced by
q, we obtain
The limiting case
of this relation reads as
By this identity, with
replaced by
, we obtain
and the first identity of Theorem 5 follows easily by equating the coefficients of
in this relation.
The proof of the second identity of Theorem 5 is quite similar to the proof of the first identity. We take into account the fact that the generating function for the partial sums of
is given by
We let
to be the Kronecker delta function. By (6), with
replaced by
, we obtain
The limiting case
of this relation reads as
and the second identity of Theorem 5 follows easily by equating the coefficients of
in this relation.
In order to prove the last identity of Theorem 5, we consider the fact that the generating function for the first differences of
is given by
By (6), with
replaced by
, we obtain
The limiting case
of this relation reads as
and the last identity of Theorem 5 follows easily by equating the coefficients of
in this relation.