ARITHMETIC PROPERTIES OF 3-REGULAR PARTITIONS IN THREE COLOURS

2021 ◽  
pp. 1-9
Author(s):  
ROBSON DA SILVA ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Infinite Family ◽  
Arithmetic Properties ◽  
Regular Partitions ◽  
Appl Math

Abstract Gireesh and Mahadeva Naika [‘On 3-regular partitions in 3-colors’, Indian J. Pure Appl. Math.50 (2019), 137–148] proved an infinite family of congruences modulo powers of 3 for the function $p_{\{3,3\}}(n)$ , the number of 3-regular partitions in three colours. In this paper, using elementary generating function manipulations and classical techniques, we significantly extend the list of proven arithmetic properties satisfied by $p_{\{3,3\}}(n).$

scholarly journals CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS

2013 ◽  
Vol 09 (08) ◽  
pp. 1885-1894 ◽  
Author(s):  
SU-PING CUI ◽  
NANCY SHAN SHAN GU
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Partition Functions ◽  
Arithmetic Properties ◽  
Partition Analysis ◽  
Congruence Relations ◽  
Modulo P ◽  
Regular Partitions

Andrews and Paule introduced broken k-diamond partitions by using MacMahon's partition analysis. Recently, Fu found a generalization which he called k dots bracelet partitions and investigated some congruences for this kind of partitions. In this paper, by finding congruence relations between the generating function for 5 dots bracelet partitions and that for 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime p, we study arithmetic properties modulo p of k dots bracelet partitions.

scholarly journals Arithmetic properties of Andrews' singular overpartitions

2015 ◽  
Vol 11 (05) ◽  
pp. 1463-1476 ◽  
Author(s):  
Shi-Chao Chen ◽  
Michael D. Hirschhorn ◽  
James A. Sellers
Keyword(s):  
Recent Work ◽  
Generating Function ◽  
General Theorem ◽  
Infinite Family ◽  
Combinatorial Objects ◽  
Arithmetic Properties

In a very recent work, G. E. Andrews defined the combinatorial objects which he called singular overpartitions with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers–Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we show that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions.

NEW INFINITE FAMILIES OF CONGRUENCES MODULO 4 AND 8 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 90 (1) ◽  
pp. 37-46 ◽  
Author(s):  
OLIVIA X. M. YAO
Keyword(s):  
Special Class ◽  
Main Diagonal ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Image Position ◽  
Regular Partitions ◽  
Appl Math

AbstractIn 2012, Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’,Util. Math. 88(2012), 223–235] introduced a special class of totally symmetric plane partitions, called$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$-shell totally symmetric plane partitions. Let$f(n)$denote the number of$1$-shell totally symmetric plane partitions of weight$n$. More recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’,Bull. Aust. Math. Soc.to appear. Published online 27 September 2013] discovered a number of arithmetic properties satisfied by$f(n)$. In this paper, employing some results due to Cui and Gu [‘Arithmetic properties of$l$-regular partitions’,Adv. Appl. Math. 51(2013), 507–523], and Hirschhorn and Sellers, we prove several new infinite families of congruences modulo 4 and 8 for$1$-shell totally symmetric plane partitions. For example, we find that, for$n\geq 0$and$\alpha \geq 1$,$$\begin{equation*} f(8 \times 5^{2\alpha } n+39\times 5^{2\alpha -1})\equiv 0 \pmod 8. \end{equation*}$$

scholarly journals Arithmetic properties of partition triples with odd parts distinct

2015 ◽  
Vol 11 (06) ◽  
pp. 1791-1805 ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Infinite Family ◽  
Arithmetic Properties ◽  
Partition Triples

Let pod -3(n) denote the number of partition triples of n where the odd parts in each partition are distinct. We find many arithmetic properties of pod -3(n) involving the following infinite family of congruences: for any integers α ≥ 1 and n ≥ 0, [Formula: see text] We also establish some arithmetic relations between pod (n) and pod -3(n), as well as some congruences for pod -3(n) modulo 7 and 11.

AN ANALOGUE OF EULER’S IDENTITY AND SPLIT PERFECT PARTITIONS

2018 ◽  
Vol 99 (03) ◽  
pp. 353-361
Author(s):  
MEGHA GOYAL
Keyword(s):  
Generating Function ◽  
New Class ◽  
Appl Math ◽  
Euler’S Identity

We give the generating function of split$(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split$n$-colour partitions. We derive a combinatorial relation between the number of restricted split$n$-colour partitions and the function$\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with$d(a)$copies of each part$a$and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’,Indian J. Pure Appl. Math 22(9) (1991), 737–743].

Ramanujan-type congruences for ℓ-regular partitions modulo 3, 5, 11 and 13

2017 ◽  
Vol 13 (08) ◽  
pp. 1995-2006 ◽  
Author(s):  
Hai-Tao Jin ◽  
Li Zhang
Keyword(s):  
Infinite Family ◽  
Hecke Eigenforms ◽  
Regular Partitions

Let [Formula: see text] be the number of [Formula: see text]-regular partitions of [Formula: see text]. Recently, Hou et al. established several infinite families of congruences for [Formula: see text] modulo [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for [Formula: see text] modulo [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text], we obtain three infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text] respectively using the theory of Hecke eigenforms.

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 41-46 ◽  
Author(s):  
ERNEST X. W. XIA
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Congruence Modulo

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

Arithmetic properties of l-regular overpartitions

2016 ◽  
Vol 12 (03) ◽  
pp. 841-852 ◽  
Author(s):  
Erin Y. Y. Shen
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Combinatorial Interpretation ◽  
Arithmetic Properties ◽  
Vector Partitions

Recently, Andrews introduced the partition function [Formula: see text] as the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we call the overpartitions enumerated by the function [Formula: see text] [Formula: see text]-regular overpartitions. For [Formula: see text] and [Formula: see text], we obtain some explicit results on the generating function dissections. We also derive some congruences for [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] which imply the congruences for [Formula: see text] proved by Andrews. By introducing a rank of vector partitions, we give a combinatorial interpretation of the congruences of Andrews for [Formula: see text] and [Formula: see text].

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