Lecture Hall Partitions

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

In 2013, Andrews, Garvan and Liang defined Self-conjugate S-partitions. In 2011, Andrews stated the definition of spt(n). This paper shows how to find the Self-conjugate S-partitions of 4 and 5 respectively, and proves the Corollary-1 that is ‘The number of Self-conjugate S-partitions counted according to the weight w is congruent to the total number of smallest parts in all the partitions of n modulo 2’. This paper shows how to generate the generating functions for Msc(n) in different ways. This paper shows how to find the number of partitions of n with an odd number of smallest parts and a total number of even (respectively odd) parts, and shows how to find the number of partitions of n in odd parts without gaps, and also shows how to generate the generating functions for and respectively. This paper shows how to prove the further three Corollaries with the help of examples, and shows how to prove the three Theorems by easy algebraic method.

In this paper, we are going to present sums over partitions. We define a function uₙ(g; q), which can be considered as one generalization of the q-Pochhammer symbol. Using this function uₙ(g; q), many results have been proved for sums of partitions into even distinct parts, odd distinct parts, 1 as the only part, and v = v as a part.

On the largest sizes of certain simultaneous core partitions with distinct parts

Let $$pod_3(n)$$ denote the number of 3-regular partitions of n with distinct odd parts (and even parts are unrestricted). In this article, we prove an infinite family of congruences for $$pod_3(n)$$ using the theory of Hecke eigenforms. We also study the divisibility properties of $$pod_3(n)$$ using arithmetic properties of modular forms.

We introduce group actions on the integer partitions and their variances. Using generating functions and Burnside's lemma, we study arithmetic properties of the counting functions arising from group actions. In particular, we find a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.

Let $Q(n)$ denote the number of partitions of n into distinct parts. Merca [‘Ramanujan-type congruences modulo 4 for partitions into distinct parts’, An. Şt. Univ. Ovidius Constanţa30(3) (2022), 185–199] derived some congruences modulo $4$ and $8$ for $Q(n)$ and posed a conjecture on congruences modulo powers of $2$ enjoyed by $Q(n)$ . We present an approach which can be used to prove a family of internal congruence relations modulo powers of $2$ concerning $Q(n)$ . As an immediate consequence, we not only prove Merca’s conjecture, but also derive many internal congruences modulo powers of $2$ satisfied by $Q(n)$ . Moreover, we establish an infinite family of congruence relations modulo $4$ for $Q(n)$ .

The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

On non-squashing partitions

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections, and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely, we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on ? n is bounded above by 2n, and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.

Let P(n), Q(n) denote, respectively, the set of all unrestricted partitions of n and the set of all partitions of n into distinct parts. For 1 < j < n, we derive formulas that permit the computation of the number of partitions in P(n), Q(n) respectively whose least part is j.

Introduction.Let Pd(n) denote the number of partitions of into positive integers of the form = A1+ +bs with -b + 1 ^ d where strict inequality holds if d\b.Px(n) appears in the first Rogers-Ramanujan identity [5, p. 291].P2(n) appears in the first of the Gllnitz-Gordon partition theorems [2, p. 945], and P3() appears in a partition theorem of I. J. Schur [6, p. 489].For d>3, we have the following result due to H. L. Alder [1, p. 713].(1.1) Let S be any fixed set of positive integers.Then Pd(n) is not identically equal to the number of partitions of into parts taken from S if d> 3.In a different approach, B. Gordon [4, p. 394] has made the following generalization of the Rogers-Ramanujan identities.(1.2) Let a and k be integers with 0<a^k.Let Akia(n) denote the number of partitions of into parts ^0, a (mod 2k +1).Let Bka(n) denote the number of partitions of of the form = >1+ +bs with ^>i + 1, , -i + fc_i^2, and with 1 appearing as a summand at most a-1 times.Then Aka(n) = Bk(1(n).The Gllnitz-Gordon identities have also been generalized in this manner [2, p. 946].(1.3) Let a and k be integers with 0<a/c.Let A^a(n) denote the number of partitions of into parts which are neither =2 (mod 4) nor =0, (2a-1) (mod 4k).Let B_a(n) denote the number of partitions of of the form n = by+ +bs, with b(bi+1, no odd parts are repeated, -i+fc_1^2 with strict inequality if 2[> and at most a-1 parts not exceeding 2 appear.Then Ak*,a(n) = Bffa(n).This leads us to define the following partition function.Let BAik<a(n) denote the number of partitions of of the form n = bx+ +bs, with btbt+i, no parts ^0 (modA+1) are repeated, bi-bi+k_x = A+l with strict inequality if X+l\bt and at most a-I parts not exceeding A + l appear.We note that B0ka(n) appears in (1.2); Bx,k,a(n) appears in (1.3), and 77d_1>2>2()=Pd().Question.If k is sufficiently large with respect to A, does B^ka(n) appear in a partition theorem of the Rogers-Ramanujan typelWe answer this question affirmatively with the following result.Theorem 1.Let O^X^a^k be integers, and 2X-l^k.If X is even, define

In this work, we consider the function pod(n), the number of partitions of an integer n wherein the odd parts are distinct (and the even parts are unrestricted), a function which has arisen in recent work of Alladi. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). We prove a number of results for pod(n) including the following infinite family of congruences: for all α≥0 and n≥0, $$\mathrm{pod}\biggl(3^{2\alpha+3}n+\frac{23\times3^{2\alpha+2}+1}{8}\biggr)\equiv 0\ (\mathrm{mod}\ 3).$$

Profile cutting of thin sheet bright metals can be achieved in a cost effective manner through the use of a pulsed fiber laser in conjunction with a galvanometric scanhead without co-axial assist gas. This is attractive where many batches of distinct small parts are manufactured in volume, for instance within the jewellery and the micro-electronics industries.In this paper the capabilities of a pulsed fiber laser are considered in detail when processing bright metals such as brass, copper and silver. The effect of key laser parameters such as peak power, temporal pulse shape and laser pulse frequency will be considered along with several process parameters such as pulse overlap and beam wobbling techniques. This is with the aim of producing parts at high speed and/or at high quality and to determine the maximum thickness that can be cut at effective speeds. The results will be compared and contrasted with other metals such as stainless steel and aluminium to determine the range of processing strategies needed to cut these materials.Profile cutting of thin sheet bright metals can be achieved in a cost effective manner through the use of a pulsed fiber laser in conjunction with a galvanometric scanhead without co-axial assist gas. This is attractive where many batches of distinct small parts are manufactured in volume, for instance within the jewellery and the micro-electronics industries.In this paper the capabilities of a pulsed fiber laser are considered in detail when processing bright metals such as brass, copper and silver. The effect of key laser parameters such as peak power, temporal pulse shape and laser pulse frequency will be considered along with several process parameters such as pulse overlap and beam wobbling techniques. This is with the aim of producing parts at high speed and/or at high quality and to determine the maximum thickness that can be cut at effective speeds. The results will be compared and contrasted with other metals such as stainless steel and aluminium to determine the range of processing strategies ne...