scholarly journals Partition Statistics for Cubic Partition Pairs

10.37236/615 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Byungchan Kim
Keyword(s):  
Partition Congruences ◽  
Partition Statistics

In this brief note, we give two partition statistics which explain the following partition congruences: \begin{align*} b(5n+4) &\equiv 0 \pmod{5}, \\ b(7n+a) &\equiv 0 \pmod{7}, \text{if $a=2$, $3$, $4$, or $6$}. \end{align*} Here, $b(n)$ is the number of $4$-color partitions of $n$ with colors $r$, $y$, $o$, and $b$ subject to the restriction that the colors $o$ and $b$ appear only in even parts.

scholarly journals Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

10.37236/2550 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Adam M Goyt ◽  
Brady L Keller ◽  
Jonathan E Rue
Keyword(s):  
Fibonacci Numbers ◽  
Natural Extension ◽  
Statistical Distributions ◽  
Set Partition ◽  
Set Partitions ◽  
Partition Statistics

We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k.  The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics.  We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers.  This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations.  In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles.  We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

Distribution and Potential Risks of Frequent HABs Off Zhejiang

2012 ◽  
Vol 610-613 ◽  
pp. 980-986
Author(s):  
Jianxin Lu ◽  
Yan Zhou ◽  
Jun Yu ◽  
Jia Xing Wu ◽  
Hua Long ◽  
...  
Keyword(s):  
Risk Assessment ◽  
Skeletonema Costatum ◽  
Historical Analysis ◽  
Coastal Areas ◽  
Prorocentrum Donghaiense ◽  
Intensity Scale ◽  
Noctiluca Scintillans ◽  
Potential Risks ◽  
Partition Statistics ◽  
Hab Species

Based on the historical analysis and statistics of HAB events from 1933 to 2008, Skeletonema costatum, Prorocentrum donghaiense, Karenia mikimotoi, Noctiluca scintillans were identified as the most frequent HAB species reported in Zhejiang Coastal areas. Risk assessment considering the intensity, scale, duration, species, in accordance with the 0.25°×0.25°partition statistics, was adopted. Conclusion could be made that the extreme high HABs risk located at area A (30.5°N-30.75°N, 122.5°E-123°E), B (29.75°N-30.25°N, 122.25°E-122.75°E), C (28.25°N-28.5°N, 121.75°E-122°E), and D (27.25°N-27.5°N, 121.0°E-121.25°E), in which Prorocentrum donghaiense has the biggest causative risk.

scholarly journals Set partition statistics and q-Fibonacci numbers

2009 ◽  
Vol 30 (1) ◽  
pp. 230-245 ◽  
Author(s):  
Adam M. Goyt ◽  
Bruce E. Sagan
Keyword(s):  
Fibonacci Numbers ◽  
Set Partition ◽  
Partition Statistics

scholarly journals PARTITION CONGRUENCES AND DYSON’S RANK

2014 ◽  
Vol 2 (2) ◽  
pp. 49-60
Author(s):  
Sabuj Das
Keyword(s):  
Generating Functions ◽  
Modular Functions ◽  
Partition Congruences ◽  
Freeman Dyson

In this article the rank of a partition of an integer is a certain integer associated with the partition. The term has first introduced by freeman Dyson in a paper published in Eureka in 1944. In 1944, F.S. Dyson discussed his conjectures related to the partitions empirically some Ramanujan’s famous partition congruences. In 1921, S. Ramanujan proved his famous partition congruences: The number of partitions of numbers 5n+4, 7n+5 and 11n +6 are divisible by 5, 7 and 11 respectively in another way. In 1944, Dyson defined the relations related to the rank of partitions. These are later proved by Atkin and Swinnerton-Dyer in 1954. The proofs are analytic relying heavily on the properties of modular functions. This paper shows how to generate the generating functions for In this paper, we show how to prove the Dyson’s conjectures with rank of partitions.

SUMS OF SQUARES AND PARTITION CONGRUENCES

2020 ◽  
pp. 1-19
Author(s):  
SU-PING CUI ◽  
NANCY S. S. GU
Keyword(s):  
Positive Integer ◽  
Sums Of Squares ◽  
Binomial Coefficients ◽  
Partition Congruences ◽  
Arithmetic Properties ◽  
Positive Integers ◽  
Pair Function

For positive integers $n$ and $k$ , let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$ , by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$ . Furthermore, in view of these arithmetic properties of $r_{k}(n)$ , we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

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