Abstract
In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function F of at most exponential growth satisfying the condition \(F(\mathbb {N})\subset \mathbb {R}_{+}\), we have \(F(a)F(b)>F(a+b)\) for sufficiently large positive integers a, b. Moreover, we show that if the sequence \((F(n))_{n\ge n_{0}}\) is log-concave and \(\limsup _{n\rightarrow +\infty }F(n+n_{0})/F(n)<F(n_{0})\), then F satisfies the Bessenrodt–Ono type inequality.
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1 Introduction
Let \(\mathbb {N}\) be the set of non-negative integers, \(\mathbb {N}_{+}\) the set of positive integers and \(\mathbb {R}_{+}=(0,+\infty )\). Moreover, for an integer \(n_{0}\ge 2\) we put \(\mathbb {N}_{\ge n_{0}}=\{n\in \mathbb {N}:\;n\ge n_{0}\}\).
For given \(A\subset \mathbb {N}_{+}\) and \(n\in \mathbb {N}\) by \(p_{A}(n)\) we denote the number of partitions of n with parts in the set A. As usual, we put \(p_{A}(0)=1\). It is well known that the ordinary generating function for the sequence \((p_{A}(n))_{n\in \mathbb {N}}\) takes the form
If \(A=\mathbb {N}_{+}\) then we simply write p(n) instead of \(p_{\mathbb {N}_{+}}(n)\). In this case the partition function is the famous Euler partition function thoroughly studied by Ramanujan and many others. The number of papers devoted to various properties of p(n), or more generally, for \(p_{A}(n)\) for various choice of the set A is enormous. The standard reference is the book of Andrews [2] (see also [3] for less advanced approach).
A few years ago there emerged a broad research devoted to log-behavior of partition statistics. Let us recall that a sequence \((a_n)_{n\in \mathbb {N}}\) of real numbers is said to be log-concave if the inequality
is valid for all sufficiently large values of n. On the other hand, it is called log-convex if the inverse inequality holds. The first who proved the log-concavity of classical partition function p(n) was Nicolas in [27].
Recently, using analytic methods, DeSalvo and Pak [13] reproved the Nicolas’ theorem.
Now, there is a wealth of literature devoted to the log-concavity property for other variations of the partition function. For instance, Bringmann, Kane, Rolen and Tripp [8] investigated the case of the k-colored partition function and partly proved the conjecture formulated by Heim and Nauhauser in [18]. This conjecture is a polynomial generalization of a one stated by Chern, Fu, and Tang in [10]. On the other hand, Dawsey and Masri [11] examined the Andrews spt-function in that direction. Further, Engel [14] proved that the overpartition function \(\overline{p}(n)\) is log-concave for every \(n\ge 2\). Gajdzica [15] discovered a similar phenomenon for the A-partition function when A is finite. O’Sullivan [28] investigated the number of partitions into powers and proved a conjecture of Ulas [30]. Ono, Pujahari and Rolen [29] showed that the plane partition function satisfies the log-concavity property as well.
However, the research devoted to log-behavior of partition functions is not only bounded by the log-concavity or log-convexity properties. Another interesting phenomenon is the so-called Bessenrodt–Ono inequality. More precisely, in 2016, Bessenrodt and Ono [7] showed that for \(a, b\ge 2\) and \(a+b>9\) we have
Their proof is based on the asymptotic estimates due to Lehmer [23]. It is worth noting that there are two alternative approaches to derive the result. Alanazi, Gagola and Munagi [1] showed the Bessenrodt–Ono inequality in combinatorial manner by constructing appropriate injections between some sets of partitions. Heim and Neuhauser [19], on the other hand, presented a proof which is based on the induction on \(a+b\).
Moreover, the inequality (1.1) can be effectively used in practice. For instance, one can apply it to determine the value of
where \(p(\varvec{\lambda })\) denotes the extended partition function defined as \(p(\varvec{\lambda }):=\prod _{i=1}^jp\)\((\lambda _i)\) for \(\mathbf {\lambda }=(\lambda _1,\lambda _2,\ldots ,\lambda _j)\). For additional information, we refer the reader to [7].
There are a lot of publications regarding Bessenrodt–Ono type inequalities. For example, Beckwith and Bessenrodt [5] found out similar properties for the k-regular partition function. Chern, Fu and Tang [10] considered the k-colored partition function. Dawsey and Masri [11] discovered the analogous phenomenon for the Andrews spt-function. Heim, Neuhauser and Tröger [21] examined the issue for the plane partition function. Moreover, Heim and Neuhauser widely generalized the property and investigated the so-called polynomization of the Bessenrodt–Ono inequality for a couple of partition functions [16, 17, 20, 21]. Hou and Jagadeesan [22], and Males [26] discovered the analogues of the Bessenrodt–Ono inequality for the so-called partition rank function.
Actually, there are many more properties related to the log-behavior of partition statistics. We do not discuss them here, but focus on some general criteria for both Bessenrodt–Ono type inequalities and log-concavity problems. It turns out thata plenty of the aforementioned results are proved using some asymptotic estimates. Therefore, it would be convenient to possess several conditions which assert that a function with an appropriate growth is log-concave or fulfills the Bessenrodt–Ono inequality. Essentially, these goals were the main motivations for our investigation.
Let us describe the content of the paper in some details. In Sect. 2 we get a general criterion on the sequence \((F(n))_{n\in \mathbb {N}}\), which guarantees that it satisfies the Bessenrodt–Ono type inequality. In other words, for all sufficiently large \(a, b\in \mathbb {N}, a\ge b\) we have \(F(a)F(b)>F(a+b)\). In particular, as an application we reprove recent result of Heim and Neuhauser which says that the plane partition function satisfies the Bessenrodt–Ono type inequality. In Sect. 3 we present some conditions guaranteeing asymptotic log-concavity of the sequence \((F(n))_{n\in \mathbb {N}}\). As an application we reprove recent result of DeSalvo and Pak [13, Theorem 1.1]. Finally, in the last section, under some mild condition on the sequence \((F(n))_{n\in \mathbb {N}}\) we show that log-concavity implies the Bessenrodt–Ono type inequality. We also prove that the opposite implication is not true. More precisely, we show that for each \(m\in \mathbb {N}_{\ge 2}\) the m-ary partition function \(b_{m}(n)=p_{A}(n)\), where \(A=\{m^{i}:\;i\in \mathbb {N}\}\), satisfies the Bessenrodt–Ono type inequality but is not log-concave.
2 Bessenrodt–Ono Type Inequality Holds for a Class of Sub-exponential Functions
In this section we offer a general approach to Bessenrodt–Ono type inequalities. More precisely, we prove the following general result.
Theorem 2.1
Let \(c_1,c_2,f,F:\mathbb {N}_+\rightarrow \mathbb {R}_+\) be given and such that the inequalities
are valid for all positive integers \(n\ge N_{0}\) for some \(N_0\in \mathbb {N}_+\). Suppose further that the following conditions hold:
-
(1)
\(\exists g:\mathbb {N}_+\rightarrow \mathbb {R}_+, N_1\in \mathbb {N}_+\text { }\forall a\ge b\ge N_1:f(a)+f(b)-f(a+b)\ge g(b)\);
-
(2)
\(\exists h:\mathbb {N}_+\rightarrow \mathbb {R}_+,N_2\in \mathbb {N}_+\text { }\forall a\ge b\ge N_2: c_2(a+b)/c_1(a)\le h(b)\);
-
(3)
\(\exists N_3\in \mathbb {N}_+\text { }\forall n\ge N_3: g(n)\ge \log h(n)-\log c_1(n)\).
Then, the inequality
is satisfied for all \(a,b\ge \max \{N_0, N_1, N_2, N_{3}\}\).
Proof
Let us fix functions \(c_1,c_2,f,F\) and g as in the statement. Since both of the inequalities
and
are true for any positive integers \(a,b\ge N_{0}\), it is enough to prove that the following inequality
is satisfied for all sufficiently large values of a and b. From (1), it follows that the inequality
holds for every \(a\ge b\ge N_1\). On the other hand, (2) implies that the inequality
is valid for any \(a\ge b\ge N_2\). Hence, it is enough to show that the following
holds for all large values of b—but that is a direct consequence of (3), as required. \(\square \)
Although the above theorem is very easy we show that it is strong enough to be applied to classical examples of partition functions.
Example 2.2
Let us consider the partition function p(n). It follows from the Bessenrodt and Ono’s paper [7] that the inequalities
hold for every positive integer n. Therefore, we will use Theorem 2.1 with \(F(n):=p(n)\),
Now, if we assume that \(a\ge b\ge 1\), then we have that
On the other hand, if we assume that \(a\ge 9\) and \(1\le b\le a\), then one can derive that
Hence, it is enough to observe that
is true for all sufficiently large values of n. In fact, one can show that the above is valid for every \(n\ge 22\). Thus, Theorem 2.1 implies that the Bessenrodt–Ono inequality
holds for all \(a\ge b\ge 22\).
Example 2.3
Let us recall that the number pp(n) of plane partitions of n can be computed via the generating function (obtained by MacMahon [24])
From Wright’s formula [32, Formula (2.21)] one can deduce the existence of (ineffective constants) \(\alpha ,\beta ,\gamma ,N>0\) such that for all \(n>N\) the following inequalities
hold. We set \(F(n):=pp(n)\),
and apply Theorem 2.1. For every \(a\ge b\ge N\), we have that
where the first inequality is a consequence of Bernoulli’s inequality. On the other hand, it is not difficult to see that
whenever \(a\ge b\ge \max \{N,(\beta +1)^2\}\). Now, it is straightforward to deduce that
is satisfied for all but finitely many values of n. In conclusion, we get that pp(n) fulfills the Bessenrodt–Ono type inequality for all large parameters a and b.
At this point, it is worth saying that Heim et al. [21, Theorem 1.1] completely solved the Bessenrodt–Ono type inequality for the plane partition function. More precisely, they showed that the inequality
is satisfied for every \(a,b\in \mathbb {N}_{\ge 2}\) with \(a+b\ge 12.\)
3 Log-Concavity Property Holds for a Class of Sub-exponential Functions
Let us recall that a sequence \((F(n))_{n\in \mathbb {N}}\) (or just a function \(F:\;\mathbb {N}\rightarrow \mathbb {R}_{+}\)) is log-concave if
for all \(n\ge n_0\), where \(n_{0}\) is some positive integer.
In this section we get a general result which under mild conditions on the growth of the function \(F:\mathbb {N}\rightarrow \mathbb {R}_{+}\) guarantees that F is log-concave.
Theorem 3.1
Let \(c_1,c_2,f,F:\mathbb {N}\rightarrow \mathbb {R}_+\) be given and such that the inequalities
are valid for all positive integers \(n\ge N_{0}\) for some \(N_0\in \mathbb {N}\). Suppose further that the following conditions hold:
-
(1)
\(\exists N_1\in \mathbb {N}_+\text { }\exists h:\mathbb {N}_+\rightarrow \mathbb {R}_{+}\text { }\forall n\ge N_1: h(n)\le 2f(n)-f(n-1)-f(n+1)\);
-
(2)
\(\exists N_2\in \mathbb {N}_{+}\text { }\forall n\ge N_2: c_2(n+1)c_2(n-1)/c_1^2(n)\le e^{h(n)}\).
Then, for all the values of \(n\ge \max \{N_{0}, N_{1}, N_{2}\}\) the inequality
is true.
Proof
Let \(c_1,c_2,f,F,h\) be as in the statement of the theorem. It follows that both of the inequalities
and
are valid for every \(n\ge N_{0}\). Therefore, it is enough to show that the inequality
is true for all sufficiently large values of n, but this is exactly (2). Hence, we conclude that the inequality
is satisfied for all \(n\ge \max \{N_{0}, N_1, N_2\}\), as required. \(\square \)
Example 3.2
For every positive integer \(n\ge 37\), we have that
where
The inequalities (3.1) follow directly from Chen, Jia and Wang [9, Lemma 2.2] and some numerical computations carried out in Wolfram Mathematica [31].
Now, we want to apply Theorem 3.1 to deduce the log-concavity for p(n). At first, let us observe that the generalized binomial theorem asserts that for every \(|j|<|n|\) the following inequalities
are true, where
and
Therefore, we have that
On the other hand, one can also calculate that
where the last inequality is a consequence of (3.2) and some elementary but tiresome computations. Hence, it is enough to check under what conditions on n the inequality
is true. One can verify that it holds for all \(n\ge 94\). Therefore, if we examine the positivity of \(p^2(n)-p(n-1)p(n+1)\) for every \(1\le n\le 93\), then we obtain Nicolas’ theorem (see, [13, Theorem 1.1] or [27]).
4 Log-Concavity (Usually) Implies Bessendrodt-Ono Type Inequality
In this section we show a strict connection between log-concavity property and the Bessenrodt–Ono type inequality.
Theorem 4.1
Let \(F:\mathbb {N}_+\rightarrow \mathbb {R}_+\) be fixed. Assume further that there exists \(n_0\in \mathbb {N}_+\) such that \((F(n))_{n\ge n_0}\) is log-concave and
Then, the inequality
holds for all sufficiently large numbers \(a, b\in \mathbb {N}_+\).
Proof
At first, let us observe that the log-concavity property implies that the inequality
is valid for each \(n\ge n_0\). Thus, it is enough to check the validity of the inequality in
for all sufficiently large a and b. However, (4.1) points out that
is true for any \(i=0,1,\ldots ,b-n_0-1\). Therefore, the task boils down to proving that the inequality
is true for all large values of a, but it is a direct consequence of the leftover assumption from the statement. This ends the proof. \(\square \)
One can easily notice that besides Theorem 4.1, the above proof implies the following property.
Proposition 4.2
Let \(F:\mathbb {N}_+\rightarrow \mathbb {R}_+\) be given. Assume further that there exists \(n_0\in \mathbb {N}_+\) such that \((F(n))_{n\ge n_0}\) is log-concave and the inequality
is valid for every \(n\ge n_0\). Then, we have that
is satisfied for all \(a,b\ge n_0\).
Example 4.3
Suppose that we know that the classical partition function satisfies
for every \(a\ge 26\). Then, from [13, Theorem 1.1] or [27] and Proposition 4.2 we get that the inequality
holds for all \(a,b\ge 26\).
Theorem 4.4
Let \(F:\mathbb {N}\rightarrow \mathbb {R}_+\) be fixed and such that \(F(0)\ge 1\). Suppose further that the sequence \(\left( F(n)\right) _{n=0}^\infty \) is log-concave for every positive integer n. Then
is true for all positive integers a and b.
Proof
The log-concavity property and the assumption that \(F(0)\ge 1\) imply that
holds for all positive integers a and b. This completes the proof. \(\square \)
Before we proceed to some applications of Theorem 4.4, it is worth pointing out that its first proof was obtained by Asai et al. [4, Theorem 2.1 (a)]. More recently, Benfield and Roy [6] also investigated similar properties for sequences of positive real numbers.
Example 4.5
Since the sequence \((p(n))_{n\ge 26}\) is log-concave by [13, Theorem 1.1] or [27] and we have \(p(26)=2436>1\), Theorem 4.4 implies that the sequence \((p(n+26))_{n\in \mathbb {N}}\) satisfies the Bessenrodt–Ono type inequality.
Example 4.6
Let us set \(q(n):=F_{2n}\), where \(F_j\) denotes the j-th Fibonacci number, where \(F_{0}=0, F_{1}=1\) and \(F_{n}=F_{n-1}+F_{n-2}\). From Cassini identity we deduce that
Thus, the sequence \((q(n))_{n\in \mathbb {N}}\) is log-concave. However, it does not satisfy the Bessenrodt–Ono type inequality. Indeed, one can easily check that \(\phi ^{2n}/\sqrt{5}-1<F_{2n}<\phi ^{2n}/\sqrt{5}\), where \(\phi =(1+\sqrt{5})/2\) is the golden mean. Thus, for \(a+b\ge 3\) we have
As a consequence, despite the fact that the sequence \((q(n))_{n\in \mathbb {N}}\) is log-concave, it does not satisfy the Bessenrodt–Ono type inequality. In conclusion, we see that the assumption \(F(0)\ge 1\) in Theorem 4.4 is crucial.
On the other hand, let us note that for any fixed \(j\in \mathbb {N}_{+}\) the sequence \((q(n+j))_{n\in \mathbb {N}}\) satisfies the Bessenrodt–Ono type inequality.
Theorem 4.1 asserts that a positive log-concave sequence usually satisfies the Bessenrodt–Ono inequality. Therefore, there appears a natural question whether the inverse statement is true. In general, it is not the case as the following example shows.
Example 4.7
For an arbitrary positive integer \(m\ge 2\), the m-ary partition function \(b_m(n)\) is just \(p_{A}(n)\) for \(A=\{m^{i}:\;i\in \mathbb {N}\}\), i.e., \(b_{m}(n)\) is the number of representations of n as sums of powers of m. Let us recall that \(b_{m}(0)=1, b_{m}(mn+i)=b_{m}(mn)\) for \(i=1, \ldots , m-1\) and \(b_{m}(mn)=b_{m}(mn-1)+b_{m}(n)\).
The sequence \((b_m(n))_{n\in \mathbb {N}}\) is not log-concave. Indeed, if we have that \(n\equiv -1\pmod *{m}\), then \(b_m(n-1)=b_m(n)<b_m(n+1)\).
On the other hand, the following is true.
Theorem 4.8
For each \(m\ge 2\), the inequality
is valid for all \(x,y\ge m^2+m\).
Proof
Let us fix \(m\ge 2\) and \(x,y\ge m^2+m\). In such a setting, there exist numbers \(c\ge d\ge m+1\) and \(i,j\in \{0,1,\ldots ,m-1\}\) such that \(x=cm+i\) and \(y=dm+j\). Hence, we get that
The right hand side from the Bessenrodt–Ono type inequality might be estimated as follows
Therefore, it suffices to show that
The above might be further simplified to
At first, let us deal with (4.2). Since \(d\ge m+1\), it follows that m can be taken l times as a part of dm, where \(l\in \{0,1,\ldots ,d\}\). In such a setting, we may require that all of the remaining parts of dm are equal to 1. Furthermore, the assumption that \(d\ge m+1\) also guarantees that \(m^2\) might occur as a part of dm, and if that is the case, then we can either take m as a part or not, which deliver us at least two additional m-ary partitions of dm. Thus, we obtain that
In the case of (4.3), let us observe that if \(m\ge 3\), then \(b_m(cm)\ge b_m(3c)\ge b_m(2c+1)\). On the other hand, if \(m=2\), then we just have \(b_2(2c)=b_2(2c+1)\).
In conclusion, both inequalities (4.2) and (4.3) are satisfied for all \(c,d\ge m+1\), as required. \(\square \)
At the end, it is worth noting that despite having Mahler’s theorem (see [25]), which states that
we can neither use it nor apply Theorem 2.1 to deduce the Bessenrodt–Ono type inequality for the m-ary partition function. In order to apply Theorem 2.1 we need to know functions \(c_1,c_2,f:\mathbb {N}_+\rightarrow \mathbb {R}_+\) such that
However, asymptotic equality (4.4) ensures only that
where \(\lim _{n\rightarrow \infty }d_1(n)=\lim _{n\rightarrow \infty }d_2(n)=1\). This provides too little information in order to use Theorem 2.1. Even stronger estimation,
given by [12] is not sufficient to conclude the Bessenrodt–Ono type inequality for the m-ary partition function. Here the main obstruction is the function \(\Psi :\mathbb {R}_+\rightarrow \mathbb {R}_+\), which is implicitly given. We only know that \(\Psi \) is periodic with period 1 and bounded.
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Acknowledgements
The first author would like to thank Bernhard Heim for fruitful discussion and valuable suggestions. The authors are grateful to the referees, for their careful readings of and revisions to the manuscript, which significantly improved its final version.
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Gajdzica, K., Miska, P. & Ulas, M. On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties. Ann. Comb. (2024). https://doi.org/10.1007/s00026-024-00700-7
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DOI: https://doi.org/10.1007/s00026-024-00700-7