Partition Statistics Research Articles

A variation of the Andrews–Stanley partition function and two interesting q-series identities

Stanley introduced a partition statistic $$srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$$ , where $$\mathcal {O}(\pi )$$ denote the number of odd parts of the partition $$\pi $$ , and $$\pi '$$ is the conjugate of $$\pi $$ . Let $$p_i(n)$$ denote the number of partitions of n with srank $$\equiv i\pmod 4$$ . Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5: $$\begin{aligned} p_0(5n+4)\equiv p_2(5n+4)\equiv 0\pmod 5. \end{aligned}$$ In this paper, we consider an analogous partition statistic $$\begin{aligned} lrank (\pi )=\mathcal {O}(\pi )+\mathcal {O}(\pi '). \end{aligned}$$ Let $$p_i^+(n)$$ denote the number of partitions of n with lrank $$\equiv i \pmod 4$$ . We will establish the generating functions of $$p_0^+(n)$$ and $$p_2^+(n)$$ and show that they satisfy similar properties to $$p_i(n)$$ . We also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences $$\begin{aligned} p_0^+(5n+4)\equiv p_2^+(5n+4)\equiv 0\pmod 5. \end{aligned}$$

Statistics on Partitions Arising from Seaweed Algebras

Using the index theory of seaweed algebras, we explore various new partition statistics. We find relations to some well-known families of partitions as well as a surprising periodicity result.

Algebraic relations between partition functions and the j-function

We obtain identities and relationships between the modular $j$-function, the generating functions for the classical partition function and the Andrews $spt$-function, and two functions related to unimodal sequences and a new partition statistic we call the "signed triangular weight" of a partition. These results follow from the closed formula we obtain for the Hecke action on a distinguished harmonic Maass form $\mathscr{M}(\tau)$ defined by Bringmann in her work on the Andrews $spt$-function. This formula involves a sequence of polynomials in $j(\tau)$, through which we ultimately arrive at expressions for the coefficients of the $j$-function purely in terms of these combinatorial quantities.

On the largest part size and its multiplicity of a random integer partition

Abstract Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let Ln = L n (λ) and M n = M n (λ) be the largest part size and its multiplicity, respectively. For large n, we focus on a comparison between the partition statistics L n and L n M n . In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of L n M n – L n grows as fast as 1 2 log n {1 \over 2}\log n . We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.

Extracting Flooded Roads by Fusing GPS Trajectories and Road Network

Urban roads are the lifeline of urban transportation and satisfy the commuting and travel needs of citizens. Following the acceleration of urbanization and the frequent extreme weather in recent years, urban waterlogging is occurring more than usual in summer and has negative effects on the urban traffic networks. Extracting flooded roads is a critical procedure for improving the resistance ability of roads after urban waterlogging occurs. This paper proposes a flooded road extraction method to extract the flooding degree and the time at which roads become flooded in large urban areas by using global positioning system (GPS) trajectory points with driving status information and the high position accuracy of vector road data with semantic information. This method uses partition statistics to create density grids (grid layer) and uses map matching to construct a time-series of GPS trajectory point density for each road (vector layer). Finally, the fusion of grids and vector layers obtains a more accurate result. The experiment uses a dataset of GPS trajectory points and vector road data in the Wuchang district, which proves that the extraction result has a high similarity with respect to the flooded roads reported in the news. Additionally, extracted flooded roads that were not reported in the news were also found. Compared with the traditional methods for extracting flooded roads and areas, such as rainfall simulation and SAR image-based classification in urban areas, the proposed method discovers hidden flooding information from geospatial big data, uploaded at no cost by urban taxis and remaining stable for a long period of time.

Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph.

Early warning indexes determination of the crop injuries caused by waterlogging based on DHSVM model

Early warning on crop injuries aroused by waterlogging is the guarantee of crop yield in the middle and lower reaches of the Yangtze River. The waterlogging warning measures based on precipitation index were often inaccurate. In this literature, a new method based on DHSVM (Distributed Hydrology Soil Vegetation Model) was used for practice in summer crops at county-level, real-time and high-precision forecasting, which simulates soil groundwater depth and soil volumetric moisture content in grid unit on time step of 1 day. According to the agricultural standard of waterlogging level, the meteorology characteristic data were calculated, and then spatial distribution of the waterlogging eigenvalues was mapped. Waterlogging distribution from Jianli County was clustered into five types of regions with a 90-m spatial resolution via the unsupervised classification method (k-means) algorithm. Based on the partition statistics of waterlogging eigenvalues, the character values of each kind were extracted. Using these eigenvalues as criteria, we predicted the waterlogging in the years from 1970 to 2015. The experimental results show that the average value of kappa coefficient is between 0.87 and 0.93, which illustrates that our method is effective for warning waterlogging level.

On certain weighted 7-colored partitions

Inspired by Andrews’ 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.

Asymptotic analysis of expectations of plane partition statistics

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where $$Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}$$ is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around $$x=1$$ . The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

OLAPS: Online load-balancing in range-partitioned main memory database with approximate partition statistics

Modern database systems can achieve high throughput main-memory query execution by being aware of the dynamics of highly parallel hardware. In such systems, data is partitioned into smaller pieces to reach a better parallelism. Unfortunately, data skew is one of the main problems faced during parallel processing in a parallel main memory database. In some data-intensive applications, parallel range queries over a dynamic range partitioned system are important. Continuous insertions/deletions can lead to a very high degree of data skew and consequently a poor performance of parallel range queries. In this paper, we propose an approach for maintaining balanced loads over a set of nodes as in a system of communicating vessels, by migrating tuples between neighboring nodes. These frequent (or even continuous) data transfers inevitably involve dynamic changes in the partition statistics. To avoid the performance degradation typically associated with this dynamism, we provide a solution based on an approximate Partition Statistics Table. The basic idea behind this table is that both clients and nodes may have an imperfect knowledge about the effective load distribution. They can nevertheless locate any data with almost the same efficiency as using exact partition statistics. Furthermore, maintaining load distribution statistics do not require exchanging additional messages as opposed to the cost of efficient solutions from the state-of-art (which requires at least O(logn) messages). We show through intensive experiments that our proposal supports efficient range queries, while simultaneously guaranteeing storage balance even in the presence of numerous concurrent insertions/deletions generating a heavy skewed data distribution.

Ordered set partition statistics and the Delta Conjecture

The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund–Remmel–Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations.

An Asymptotic Scheme for Analysis of Expectations of Plane Partition Statistics

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where Q(x)=∏j=1∞(1−xj)−j is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around x = 1. The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part (the height of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner, Knopfmacher and Wagner (2014) related to linear integer partition statistics. We base our study on the Hayman's method for admissible power series.

A Plancherel measure associated to set partitions and its limit

In recent years increasing attention has been paid on the area of supercharacter theories, especially to those of the upper unitriangular group. A particular supercharacter theory, in which supercharacters are indexed by set partitions, has several interesting properties, which make it object of further study. We define a natural generalization of the Plancherel measure, called superplancherel measure, and prove a limit shape result for a random set partition according to this distribution. We also give a description of the asymptotical behavior of two set partition statistics related to the supercharacters. The study of these statistics when the set partitions are uniformly distributed has been done by Chern, Diaconis, Kane and Rhoades.

Weighted Rogers–Ramanujan partitions and Dyson crank

In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is n is equal to the number of partitions of n with non-negative crank.

On 1212-Avoiding Restricted Growth Functions

Restricted growth functions (RGFs) avoiding the pattern $1212$ are in natural bijection with noncrossing partitions. Motivated by recent work of Campbell et al., we study five classical statistics bk, ls, lb, rs and rb on $1212$-avoiding RGFs. We show the equidistribution of (ls, rb, lb, bk) and (rb, ls, lb, bk) on $1212$-avoiding RGFs by constructing a simple involution. To our surprise, this result was already proved by Simion 22 years ago via an involution on noncrossing partitions. Our involution, though turns out essentially the same as Simion's, is defined quite differently and has the advantage that makes the discussion more transparent. Consequently, a multiset-valued extension of Simion's result is discovered. Furthermore, similar approach enables us to prove the equidistribution of (mak, rb, rs, bk) and (rb, mak, rs, bk) on $1212$-avoiding RGFs, where "mak" is a set partition statistic introduced by Steingrímsson.Through two bijections to Motzkin paths, we also prove that the triple of classical permutation statistics (exc+1, den, inv — exc) on $321$-avoiding permutations is equidistributed with the triple (bk, rb, rs) on $1212$-avoiding RGFs, which generalizes another result of Simion. In the course, an interesting $q$-analog of the $\gamma$-positivity of Narayana polynomials is found.

Variation on a theme of Nathan Fine. New weighted partition identities

We utilize false theta function results of Nathan Fine to discover four new partition identities involving weights. These relations connect Göllnitz–Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or equal to a given value and another one is defined as the number of different even parts larger than the first integer that is not a part of the partition.

Narrow Spectral Width FP Lasers for High-Speed Short-Reach Applications

While being cost effective and robust to external optical feedback, conventional Fabry–Perot (FP) semiconductor laser diodes cannot be used for fiber-optic communication systems beyond 1 Gb/s with span requirement over several kilometers due to the mode partition noise (MPN) caused by their multimode operation nature. In order to reduce the MPN-induced power penalty, we have proposed a narrow spectral width FP laser, which incorporates a built-in bandpass filter with its peak wavelength detuned from the material gain peak. Both simulation and experimental results show that this design can effectively narrow the root-mean-square width of the optical spectrum approximately by half. Within a given spectral width, we have also studied the impact of the number of longitudinal modes on the mode partition statistics k factor, aiming at further reducing the MPN-induced power penalty. The proposed laser exhibits noticeable better transmission performances over the conventional FP laser in simulation, and is expected to find its potential applications in high-speed short-reach, yet cost-sensitive fiber transmission systems, such as the passive optical access networks and Fibre channels for optical interconnection.

On the asymptotic distribution of cranks and ranks of cubic partitions

Cubic partitions are a special kind of bi-partitions whose name is inspired by a connection between a cubic continued fraction and an arithmetic property of this bi-partition. There are two partition statistics, namely rank and crank, for cubic partitions, which explain cubic partition congruences combinatorially. We obtain asymptotics for the number of cubic partitions of rank (resp. crank) m which reveal the distribution of the rank (resp. crank) among cubic partitions. As applications, we derive asymptotic inequalities between cubic partition rank and crank functions, and we remark upon the similarities and the differences between these and the corresponding inequalities for ordinary partitions.

Central limit theorems for some set partition statistics

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n]={1,2,…,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

Quantum tomography of an electron.

The complete knowledge of a quantum state allows the prediction of the probability of all possible measurement outcomes, a crucial step in quantum mechanics. It can be provided by tomographic methods which have been applied to atomic, molecular, spin and photonic states. For optical or microwave photons, standard tomography is obtained by mixing the unknown state with a large-amplitude coherent photon field. However, for fermions such as electrons in condensed matter, this approach is not applicable because fermionic fields are limited to small amplitudes (at most one particle per state), and so far no determination of an electron wavefunction has been made. Recent proposals involving quantum conductors suggest that the wavefunction can be obtained by measuring the time-dependent current of electronic wave interferometers or the current noise of electronic Hanbury-Brown/Twiss interferometers. Here we show that such measurements are possible despite the extreme noise sensitivity required, and present the reconstructed wavefunction quasi-probability, or Wigner distribution function, of single electrons injected into a ballistic conductor. Many identical electrons are prepared in well-controlled quantum states called levitons by repeatedly applying Lorentzian voltage pulses to a contact on the conductor. After passing through an electron beam splitter, the levitons are mixed with a weak-amplitude fermionic field formed by a coherent superposition of electron-hole pairs generated by a small alternating current with a frequency that is a multiple of the voltage pulse frequency. Antibunching of the electrons and holes with the levitons at the beam splitter changes the leviton partition statistics, and the noise variations provide the energy density matrix elements of the levitons. This demonstration of quantum tomography makes the developing field of electron quantum optics with ballistic conductors a new test-bed for quantum information with fermions. These results may find direct application in probing the entanglement of electron flying quantum bits, electron decoherence and electron interactions. They could also be applied to cold fermionic (or spin-1/2) atoms.