Durfee Symbols Research Articles

Modular forms and congruences for k-marked odd Durfee symbols

In 2007, Andrews introduced the k-marked odd Durfee symbols and studied their parities. Let Dk0(n) denote the number of k-marked odd Durfee symbols of n. He proved that Dk0(n) is even for k≥1 and n≡k−1(mod2). Inspired by Andrews' work, many authors investigated congruences for Dk0(n) modulo 2 or powers of 2. For example, Wang obtained D20(8n+4)≡D20(8n+6)≡0(mod2) which were first conjectured by Andrews. More recently, Xia proved that D2mk0(2n+1)≡0(mod2m+1) with m,k≥1 and n≥0. In this paper, we first study the modular properties of the generating function of D20(n) which is completed to be a weakly modular form of weight 52. As application, we utilize the Hecke operators to establish families of congruences for D20(n) modulo powers of primes ℓ≥5.

Explicit congruences for k-marked Durfee symbols

In 2007, Andrews introduced k-marked Durfee symbols to give a combinatorial interpretation of 2kth symmetrized moment η2k(n) of ranks of partitions of n. Let Dk(n) denote the number of k-marked Durfee symbols of n. Bringmann, Garvan and Mahlburg proved that there exist infinitely many arithmetic progressions An+B such that Dk(An+B)≡0(modpj), where j is a positive integer and p≥5 is a prime. In addition, Keith proved that D4(3n)≡0(mod3). Motivated by their works, we establish some explicit congruences modulo 2, 3, 4 and 8 of Dk(n) for some small k by employing some identities due to Garvan, and Lewis and Santa-Gadea on M(r,m,n) which counts the number of partitions of n with crank congruent to r modulo m.

Asymptotics of Some Plancherel Averages Via Polynomiality Results

Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set {1,ldots ,n}. Here we are interested in asymptotics, as nrightarrow infty , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.

On k-measures and Durfee squares of partitions

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of [Formula: see text]-measure of an integer partition, and proved a surprising identity that the number of partitions of [Formula: see text] which have [Formula: see text]-measure [Formula: see text] is equal to the number of partitions of [Formula: see text] with a Durfee square of side [Formula: see text]. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of partitions of [Formula: see text] which have [Formula: see text]-measure [Formula: see text] for [Formula: see text]. In this paper, we perform these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we obtain a natural generalization for [Formula: see text]-measures.

Durfee squares, symmetric partitions and bounds on Kronecker coefficients

We resolve two open problems on Kronecker coefficients g(λ,μ,ν) of the symmetric group. First, we prove that for partitions λ,μ,ν with fixed Durfee square size, the Kronecker coefficients grow at most polynomially. Second, we show that the maximal Kronecker coefficients g(λ,λ,λ) for self-conjugate partitions λ grow superexponentially. We also give applications to explicit special cases.

New companions to the Andrews–Gordon identities motivated by commutative algebra

We give a proof of a recent conjectural partition identity due to the first author, which was discovered in the framework of commutative algebra. This result gives rise to new companions to the famous Andrews–Gordon identities. Our tools involve graded quotient rings, Durfee squares and rectangles for integer partitions, and q-series identities.

Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions

In this paper, we explore intricate connections between Ramanujan’s theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.

Identities, inequalities and congruences for odd ranks and k-marked odd Durfee symbols

In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N0(m,n) denote the number of odd Durfee symbols of n with odd rank m, and N0(r,m;n) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N0(r,12;n) by utilizing some identities involving Appell-Lerch sums m(x,q,z) and a universal mock theta function g(x,q). Based on these formulas, we determine the signs of N0(r,12;4n+t)−N0(s,12;4n+t) for all 0≤r,s≤6 and 0≤t≤3. In particular, we prove that N0(2,12;4n+1)=N0(4,12;4n+1). Moreover, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D20(8n+s) and D30(16n+t) are even with s∈{4,6} and t∈{1,9,11,13} which were confirmed by Wang. In this paper, we found new congruences for Dk0(n). In particular, for k=2 or 3, we give characterizations of n such that Dk0(n) is odd and prove that Dk0(n) take even values with probability 1 for n≥0.

Inequalities for odd ranks of odd Durfee symbols

Andrews introduced odd Durfee symbols to give an interesting combinatorial interpretation of $$\omega (q)$$ invoked by MacMahon’s modular partitions, where $$\omega (q)$$ is one of the mock theta functions defined by Watson. In analogy with Dyson’s rank, Andrews defined the odd rank of an odd Durfee symbol as the number of entries in the top row minus the number of entries in the bottom row. Let $$N^0(m,n)$$ be the number of odd Durfee symbols of n with odd rank m. In this paper, we employ Wright’s circle method to give an asymptotic formula for $$N^0(m,n)$$ which implies that the inequalities $$\begin{aligned} N^0(m,n)\ge N^0(m+2,n) \end{aligned}$$ and $$\begin{aligned} N^0(m,n)\le N^0(m,n+2) \end{aligned}$$ hold for sufficient large n. Motivated by the work of Chan and Mao, we proved that the above inequalities hold for all nonnegative integers m and n.

On odd ranks of odd Durfee symbols

An odd Durfee symbol of [Formula: see text] is an array of positive odd integers and a subscript [Formula: see text], [Formula: see text] such that [Formula: see text], [Formula: see text], and [Formula: see text]. Andrews defined the odd rank of an odd Durfee symbol as [Formula: see text]. Let [Formula: see text] be the number of odd Durfee symbols of [Formula: see text] with odd rank congruent to [Formula: see text] modulo [Formula: see text]. We decompose the generating function of [Formula: see text] into modular and mock modular parts. Specifically, we derive some special cases of the generating functions of [Formula: see text] for [Formula: see text]. Some generating functions are related to classical mock theta functions.

Arithmetic properties of odd ranks and k-marked odd Durfee symbols

Let N0(m,n) be the number of odd Durfee symbols of n with odd rank m, and N0(a,M;n) be the number of odd Durfee symbols of n with odd rank congruent to a modulo M. We give explicit formulas for the generating functions of N0(a,M;n) and their ℓ-dissections where 0≤a≤M−1 and M,ℓ∈{2,4,8}. From these formulas, we obtain some interesting arithmetic properties of N0(a,M;n). Furthermore, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews (2007) conjectured that D20(n) is even if n≡4 or 6 (mod 8) and D30(n) is even if n≡1,9,11 or 13 (mod 16). Using our results on odd ranks, we prove Andrews' conjectures.

On the distribution of rank and crank statistics for integer partitions

Let $k$ be a positive integer and $m$ be an integer. Garvan's $k$-rank $N_k(m,n)$ is the number of partitions of $n$ into at least $(k-1)$ successive Durfee squares with $k$-rank equal to $m$. In this paper give some asymptotics for $N_k(m,n)$ with $|m|\ge \sqrt{n}$ as $n\rightarrow \infty$. As a corollary, we give a more complete answer for the Dyson's crank distribution conjecture. We also establish some asymptotic formulas for finite differences of $N_k(m,n)$ with respect to $m$ with $m\gg \sqrt{n}\log n$.

Complexity metrics for DMN decision models

Complexity impairs the maintainability and understandability of conceptual models. Complexity metrics have been used in software engineering and business process management (BPM) to capture the degree of complexity of conceptual models. The recent introduction of the Decision Model and Notation (DMN) standard provides opportunities to shift towards the Separation of Concerns paradigm when it comes to modelling processes and decisions. However, unlike for processes, no studies exist that address the representational complexity of DMN decision models. In this paper, we provide an initial set of complexity metrics for DMN models. We gather insights from the process modelling and software engineering fields to propose complexity metrics for DMN decision models. Additionally, we provide an empirical complexity assessment of DMN decision models. For the decision requirements level of the DMN standard 19 metrics were proposed, while 7 metrics were put forward for the decision logic level. For decision requirements, the model size-based metrics, the Durfee Square Metric (DSM) and the Perfect Square Metric (PSM) prove to be the most suitable. For the decision logic level of DMN the Hit Policy Usage (HPU) and the Total Number of Input Variables (TNIV) were evaluated as suitable for measuring DMN decision table complexity.

Durfee squares in compositions

Abstract We study compositions (ordered partitions) of n. More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size s × s. We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition (i.e., is ‘grounded’). Via generating functions and asymptotic analysis, we consider compositions of n whose Durfee squares are of size less than s × s. This is followed by a section on the total and average number of grounded s × s squares. We then count the number of Durfee squares in compositions of n.

Durfee squares in compositions

Рассматриваются композиции (упорядоченные разбиения) числа $n$. В частности, основное внимание уделяется представлению композиций в виде столбчатых диаграмм, которые содержат или не содержат квадраты размера $s \times s$. Квадрат Дэрфи (изучавшийся в теории разбиений) мы определяем как наибольший «лежачий» квадрат, основание которого лежит на основании диаграммы. С помощью производящих функций и асимптотического анализа анализируются разбиения $n$, для которых размеры квадратов Дэрфи не превосходят $s \times s$. Рассматриваются общие и средние количества лежачих квадратов размера $s\times s$ в диаграммах, соответствующих разбиениям числа $n$.

Partition identities and quiver representations

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector.

REFINEMENTS OF THE FIRST AND SECOND POSITIVE CRANK MOMENTS

We revisit Berkovich and Garvan’s two bijections: the first gives symmetry of cranks and the second relates partitions with crank $\leq k$ to those with $k$ in the rank-set of partitions. Using these, we give a combinatorial proof for the relationship between the first positive crank moment and the sum of sizes of Durfee squares. We also study refinements of the first and second positive crank moments.

A Formula for the Partition Function That “Counts”

We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then study $D(n,k)$ as a quasipolynomial. We consider the natural polynomial approximation $\tilde{D}(n,k)$ to the quasipolynomial representation of $D(n,k)$. Numerically, the sum $\sum_{1\leq k \leq \sqrt{n}} \tilde{D}(n,k)$ appears to be extremely close to the initial term of the Hardy--Ramanujan--Rademacher convergent series for $p(n)$.

The asymptotic distribution of Andrews’ smallest parts function

In this paper, we use methods from the spectral theory of automorphic forms to give an asymptotic formula with a power saving error term for Andrews’ smallest parts function spt(n). We use this formula to deduce an asymptotic formula with a power saving error term for the number of 2-marked Durfee symbols associated to partitions of n. Our method requires that we count the number of Heegner points of discriminant −D < 0 and level N inside an “expanding” rectangle contained in a fundamental domain for \({\Gamma_0(N)}\).

Taylor coefficients of mock-Jacobi forms and moments of partition statistics

AbstractWe develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.