Heim, Neuhauser, and Tröger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture. Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$ satisfies the degree $d$ Turán inequality for sufficiently large $n$. The case where $d=2$ is the case of log-concavity.
We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for $L\geq 1$, the number of partitions with $l-s \leq L$ and $s=1$ is greater than the number of partitions with $l-s\leq L$ and $s>1$. Here $l$ and $s$ are the largest part and the smallest part of the partition, respectively.
In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis' results on the rank and crank counts for ordinary partitions.
We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-reg…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.
Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities
University of Florida（佛罗里达大学）；Seoul National University of Science and Technology（首尔科学技术大学）；Ewha Womans University（梨花女子大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
Let $j,n$ be even positive integers, and let $\overline{p}_j(n)$ denote the number of partitions with BG-rank $j$, and $\overline{p}_j(a,b;n)$ to be the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. We give asymptotics for both statistics, and show that $\overline{p}_j(a,b;n)$ is asymptotically equidistributed over the congruence classes modulo $b$. We also show that each of $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy all higher-order Turán inequalities.
We obtain an asymptotic formula for Andrews and Paule's broken $k$-diamond partition function $Δ_k(n)$ where $k=1$ or $2$. Based on this asymptotic formula, we derive that $Δ_k(n)$ satisfies the order $d$ Turán inequalities for $d\geq 1$ and for sufficiently large $n$ when $k=1$ and $ 2$ by using a general result of Griffin, Ono, Rolen and Zagier. We also show that Andrews and Paule's broken $k$-diamond partition function $Δ_k(n)$ is log-concave for $n\geq 1$ when $k=1$ and $2$. This leads to $Δ_k(a)Δ_k(b)\geΔ_k(a+b)$ for $a,b\ge 1$ when $k=1$ and $ 2$.
In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They …
Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov-Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern-Fu-Tang conjecture and to show the Heim-Neuhauser …
We consider differences of one- and two-variable finite products and provide combinatorial proofs of the nonnegativity of certain coefficients. Since the products may be interpreted as generating functions for certain integer partitions, this amounts to showing a partition inequality. This extends results due to Berkovich-Garvan and McLaughlin. We then apply the first inequality and Andrews' Anti-telescoping Method to give a solution to an 'Ehrenpreis Problem' for recently conjectured sum-product identities of Kanade- Russell. That is, we provide some evidence for Kanade-Russell's conjectures …
Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods invo…
Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \geq 94$. More recently, Griffin, Ono, Rolen and Zagier proved more generally that for all $d$, the degree $d$ Jensen polynomials associated to $p(n)$ are hyperbolic for sufficiently large $n$. In this paper, we prove that the same result holds for the $k$-regular partition function $p_k(n)$ for $k \geq 2$. In particular, for any positive i…
In 2007, Andrews and Paule introduced the broken $k$-diamond partition function $Δ_k(n)$, which has received a lot of researches on the arithmetic propertises. In this paper, we prove that $D^3\log Δ_1(n-1)>0$ for $n\geq 5$ and $D^3 \log Δ_2(n-1)>0$ for $n\geq 7$, where $D$ is the difference operator with respect to $n$. We also conjecture that for any $k\geq 1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that for $n\geq n_{k}(r)$, $(-1)^r D^r \log Δ_k(n)>0$. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been pr…
Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo $10$, and additional results modulo $6$ and $10$ for the $M_2$ rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao's rank inequality conjectures for both the rank and the $M_2$ rank modulo $10$ using elementary methods.
The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence $\{a_{n}\}$ is said to satisfy the Turán inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Turán inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequ…
In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $ n $ large enough, the two quantities are different and which of the two is bigger depends on the parity of $ n. $ This solves a recent conjecture formulated by Bringmann and Mahlburg (2012).
In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dyson's rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequality, we show that such non-trivial identities are also rare for the full rank, but on the other hand we obtain an infinite family of non-trivial identities, contrasting the partition theoretic case.
A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition-determined functions imply entro…
Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.
If $ p_k(a,m,n) $ denotes the number of partitions of $n$ into $k$th powers with a number of parts that is congruent to $ a $ modulo $m,$ then $p_2(0,2,n)\sim p_2(1,2,n)$ and the sign of the difference $p_2(0,2,n)- p_k(1,2,n)$ alternates with the parity of $n,$ as proven by recent work of the author (2020). In this paper, we place the problem in a broader framework. By analytic arguments using the circle method and Gauss sums estimates, we show that the same results hold for any $ k\ge2. $ By combinatorial arguments, we show that the sign of the difference $p_k(0,2,n)- p_k(1,2,n)$ depends on t…
Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \cite{atkin}, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \cite{mao10} and for $M_2$ rank differences for partitions with no …
A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coe…
The partition function $pod(n)$ enumerates the partitions of $n$ wherein odd parts are distinct and even parts are unrestricted. Recently, a number of properties for $pod(n)$ have been established. In this paper, for $k\in\{0,2\}$ we consider the partitions of $n$ into distinct parts not congruent to $k$ modulo $4$ and the $4$-regular partitions of $n$ in order to obtain new properties for $pod(n)$. In this context, we derive two new infinite families of linear inequalities involving the function $pod(n)$ and obtain new identities of Watson type.
Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the g…
We consider three different families of Vafa-Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa-Witten invariants is given by combinations of twisted Dedekind $η$-functions. By utilising known properties of these $η$-functions, we obtain exact formulae for each of the invariants and prove that they asymptotically satisfy all higher-order Turán inequalities.
Here we study the roots of the doubly infinite family of Jensen polynomials $J_{\mathrm{PL}}^{d,n}(x)$ associated to MacMahon's plane partition function $\mathrm{PL}(n)$. Recently, Ono, Pujahari, and Rolen proved that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12$, which is equivalent to the polynomials $J_{\mathrm{PL}}^{2,n}(x)$ having real roots. Moreover, they proved, for each $d\geq 2$, that the $J_{\mathrm{PL}}^{d,n}(x)$ have all real roots for sufficiently large $n$. Here we make their result effective. Namely, if $N_{\mathrm{PL}}(d)$ is the minimal integer such that $J_{\mathrm{PL}}…
In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if $M\geq 5$ is an integer and the integers $a$ and $b$ are relatively prime to $M$ and satisfy $1\leq a<b<M/2$, and the $c(m,n)$ are defined by \[ \frac{1}{(sq^a,sq^{M-a};q^M)_{\infty}}-\frac{1}{(sq^b,sq^{M-b};q^M)_{\infty}}:=\sum_{m,n\geq 0} c(m,n)s^m q^n, \] then $c(m, Mn)\geq 0$ for all integers $m\geq 0, n\geq 0$. %If, in addition, $M$ is even, then $c(m, Mn+M/2)\geq 0$ for all integers $m\geq 0, n\geq 0$. A similar result is proved for the integers $d(m,n)$ defined by \[ (-…
Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases, and our main result states that while the two families of moment functions are asymptotically equal, the crank moments are always asymptotically larger than the rank moments. Andrews, Chan, and Kim primarily focused on one case, and proved the stronger result that the first crank moment is strictly larger than the first rank moment for all partitions by…
We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our formulas for the $M_2$-rank at roots of unity lead to asymptotics for certain combinations of $N2(r,m,n)$ (the number of partitions without repeated odd parts of $n$ with $M_2$-rank congruent to $r$ modulo $m$). This allows us to deduce inequalities among certain combinations of $N2(r,m,n)$. In particular, we resolve a few conjectured inequalities of Mao.
In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.
The partition function $p(n)$ has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the "circle method" to estimate the size of $p(n)$, which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined by $\sum_{n = 0}^\infty p_α(n)x^n := \prod_{k=1}^\infty (1-x^k)^{-α}$. In this paper we use the Rademacher circle method to find an exact formula for $p_α(n)$ and study its implications, including log-concavity and the higher-order gener…
We generalize a result of Garvan on inequalities and interpretations of the moments of the partition rank and crank functions. In particular for nearly 30 Bailey pairs, we introduce a rank-like function, establish inequalities with the moments of the rank-like function and an associated crank-like function, and give an associated so called higher order smallest parts function. In some cases we are able to deduce inequalities among the rank-like functions. We also conjecture additional inequalities and a large number of congruences for the higher order smallest parts functions.
Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \begin{equation*} P_n^{g,h}(x):= \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x), \end{equation*} with initial value $P_{0}^{g,h}(x)=1$ encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov--Okounkov polynomials. In this paper we prove that for $g(n)=n$ and fixed $h$ we obtain orthogonal polynomial seq…
We consider the symmetrized moments of three ranks and cranks, similar to the work of Garvan for the rank and crank of a partition. By using Bailey pairs and elementary rearrangements, we are able to find useful expressions for these moments. We then deduce inequalities between the corresponding ordinary moments. In particular we prove that the crank moment for overpartitions is always larger than the rank moment for overpartitions; with recent asymptotics this was known to hold for sufficiently large values of n for each fixed k. Lastly we provide higher order spt functions for overpartitions…
