Integer Partitions Research Articles

Some new congruences for simultaneously s-regular and t-distinct partition function

A partition of a positive integer n is said to be simultaneously s-regular and t-distinct partition if none of the parts is divisible by s and parts appear fewer than t times. In this paper, we present some new congruences for simultaneously s-regular and t-distinct partition function denoted by M d (n) with (s, t) (2, 5), (3, 4), (4, 9), (5\alpha, 5\beta ), (7\alpha, 7\beta ), (p, p), where\alpha and \beta are any positive integers and p is any prime.

Proof of a conjecture on congruences of the sum of minimal excludants

Very recently, Baruah et al. refined a result of Andrews and Newman on the sum of minimal excludants over all the partitions of a positive integer [Formula: see text] by establishing the generating functions of two functions [Formula: see text] and [Formula: see text] which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively. They proved some congruences on [Formula: see text] and [Formula: see text] modulo 4 and 8 and posed a conjecture on [Formula: see text] and [Formula: see text] at the end of their paper. In this paper, we confirm this conjecture by using theta function identities.

Growing integer partitions with uniform marginals and the equivalence of partition ensembles

We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level n, it passes through any partition λ of n with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed.Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of n are asymptotically independent as n→∞ and to find their limiting distributions, which are, somewhat surprisingly, different.

Modelling smurfing patterns in cryptocurrencies with integer partitions

Modelling smurfing patterns in cryptocurrencies with integer partitions

Interior vertices and edges in integer partitions

This paper characterises the shape of the Young diagram associated with integer partitions in terms of two newly invented statistics which are defined in the text: namely interior vertices and horizontal edges. Generating functions and associated asymptotic results are developed for both interior vertices and horizontal edges.

Integer partitions detect the primes

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n ≥ 2 is prime if and only if [Formula: see text]where the [Formula: see text] are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions [Formula: see text] we prove that there are infinitely many such prime detecting equations with constant coefficients, such as [Formula: see text].

Distributions of Hook lengths in integer partitions

Distributions of Hook lengths in integer partitions

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

Billiard partitions, Fibonacci sequences, SIP classes, and quivers

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d d -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths.

A Note on Graph Burning of Path Forests

Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected graph of order $m^2$ has burning number at most $m$. Earlier, we showed that the conjecture also holds for a path forest, which is disconnected, provided each of its paths is sufficiently long. However, finding the least sufficient length for this to hold turns out to be nontrivial. In this note, we present our initial findings and conjectures that associate the problem to some naturally impossibly burnable path forests. It is noteworthy that our problem can be reformulated as a topic concerning sumset partition of integers.

Hook lengths in self-conjugate partitions

In 2010, G.-N. Han obtained the generating function for the number of size t t hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even t t . If n t ( λ ) n_t(\lambda ) is the number of size t t hooks in a partition λ \lambda and S C \mathcal {SC} denotes the set of self-conjugate partitions, then for even t t we have ∑ λ ∈ S C x n t ( λ ) q | λ | = ( − q ; q 2 ) ∞ ⋅ ( ( 1 − x 2 ) q 2 t ; q 2 t ) ∞ t 2 . \begin{equation*} \sum _{\lambda \in \mathcal {SC}} x^{n_t(\lambda )} q^{\vert \lambda \vert } = (-q;q^2)_{\infty } \cdot ((1-x^2)q^{2t};q^{2t})_{\infty }^{\frac {t}{2}}. \end{equation*} As a consequence, if a t ⋆ ( n ) a_t^{\star }(n) is the number of such hooks among the self-conjugate partitions of n n , then for even t t we obtain the simple formula a t ⋆ ( n ) = t ∑ j ≥ 1 q ⋆ ( n − 2 t j ) , \begin{equation*} a_t^{\star }(n)=t\sum _{j\geq 1} q^{\star }(n-2tj), \end{equation*} where q ⋆ ( m ) q^{\star }(m) is the number of partitions of m m into distinct odd parts. As a corollary, we find that t ∣ a t ⋆ ( n ) t\mid a_t^{\star }(n) , which confirms a conjecture of Ballantine, Burson, Craig, Folsom and Wen.

Arrow relations in lattices of integer partitions

We give a complete characterisation of the single and double arrow relations of the standard context K(Ln) of the lattice Ln of partitions of any positive integer n under the dominance order, thereby addressing an open question of Ganter, 2020/2022.

ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS

Abstract An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith.199 (2021), 33–53] introduced an analogue $\overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for $\overline {a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline {a}_t(n)$ modulo $p_i^{\,j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq.27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by $\overline {a}_5(n)$ . We study the arithmetic densities of $\overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd (m,6)$ =1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline {a}_3(n)$ modulo arbitrary powers of 2.

On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties

AbstractIn recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function F of at most exponential growth satisfying the condition $$F(\mathbb {N})\subset \mathbb {R}_{+}$$ F ( N ) ⊂ R + , we have $$F(a)F(b)>F(a+b)$$ F ( a ) F ( b ) > F ( a + b ) for sufficiently large positive integers a, b. Moreover, we show that if the sequence $$(F(n))_{n\ge n_{0}}$$ ( F ( n ) ) n ≥ n 0 is log-concave and $$\limsup _{n\rightarrow +\infty }F(n+n_{0})/F(n)<F(n_{0})$$ lim sup n → + ∞ F ( n + n 0 ) / F ( n ) < F ( n 0 ) , then F satisfies the Bessenrodt–Ono type inequality.

Constrained optimal grouping of cloud application components

Cloud applications are built from a set of components often deployed as containers, which can be deployed individually on separate Virtual Machines (VMs) or grouped on a smaller set of VMs. Additionally, the application owner may have inhibition constraints regarding the co-location of components. Finding the best way to deploy an application means finding the best groups of components and the best VMs, and it is not trivial because of the complexity coming from the number of possible options. The problem can be mapped onto may known combinatorial problems as binpacking and knapsack formulations. However, these approaches often assume homogeneus resources and fail to incorporate the inhibition constraints. The main contribution of this paper are firstly a novel formulation of the grouping problem as constrained Coalition Structure Generation (CSG) problem, including the specification of the value function which fulfills the criteria of a Characteristic Function Game (CFG). The CSG problem aims to determine stable and disjoint groups of players collaborating to optimize the joint outcome of the game, and a CFG is a common representation of a CSG, where each group is assigned a value and where the value of the game is the sum of the groups’ contributions. Secondly, the Integer-Partition (IP) CSG algorithm has been modified and extended to handle constraints. The proposed approach is evaluated with the extended IP algorithm, and a novel exhaustive search algorithm establishing the optimum grouping for comparison. The evaluation shows that our approach with the modified algorithm evaluates on average significantly less combinations than the CSG state-of-the-art algorithm. The proposed approach is promising for optimized constrained Cloud application management as the modified IP algorithm can optimally solve constrained grouping problems of attainable sizes.

Cycles in Austrian Solitaire

Austrian Solitaire is a variation of Bulgarian Solitaire. It may be described as a card game, a method of asset inventory management, or a discrete dynamical system on integer partitions. We prove that the limit cycles in Austrian Solitaire do not depend on the initial configuration; in other words, each state space is connected. We show that a full Farey sequence completely characterizes these unique (and balanced) cycles.

Modular relations involving generalized digamma functions

Generalized digamma functions ψk(x), studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the Hurwitz zeta function. In this paper, a modular relation of the form Fk(α)=Fk(1/α) containing infinite series of ψk(x), or, equivalently, between the generalized Stieltjes constants γk(x), is obtained for any k∈N. When k=0, it reduces to a famous transformation given on page 220 of Ramanujan's Lost Notebook. For k=1, an integral containing Riemann's Ξ-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as α→0 and α→∞. Carlitz-type and Guinand-type finite modular relations involving ψj(m)(x),0≤j≤k,m∈N∪{0}, are also derived, thereby extending previous results on the digamma function ψ(x). The extension of Guinand's result for ψj(m)(x),m≥2, involves an interesting combinatorial sum h(r) over integer partitions of 2r into exactly r parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for ψj′(x) is subtle and requires delicate analysis.

Integer partitions with restricted odd and even parts

Integer partitions with restricted odd and even parts

Some separable integer partition classes

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus [Formula: see text]. We also extend separable integer partition classes with modulus [Formula: see text] to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.

Eventual log-concavity of k-rank statistics for integer partitions

Let Nk(m,n) denote the number of partitions of n with Garvan k-rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k-rank for k=1 and k=2, respectively. In this paper, we prove that the sequences (Nk(m,n))|m|≤n−k−71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently.