Related papers: A Recursive Relation for Bipartition Numbers

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such…

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals…

One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was…

I propose two simple ways of generating the partitions of (n+1) from the partitions of n. A recurrence relation for P(n+1), the number of partitions of (n+1), in terms of P(n) and Q(n), where Q(n) denotes the number of partitions of n…

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: $\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_x(d)$, where the sum is over the proper divisors of $n$. We give a geometrical…

By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…

In this article we will derive a combinatorial formula for the partition function p(n). In the second part of the paper we will establish connection between partitions and q-binomial coefficients and give new interpretation for q-binomial…

In this paper, we give a purely bijective proof that two different partition classes that are both combinatorial interpretations of the partition function $p_\nu(n)$, a partition function related to the third order mock theta function…

In a recent paper, Andrews and Merca investigated the number of even parts in all partitions of $n$ into distinct parts, which arise naturally from the Euler-Glaisher bijective proof. They obtained new combinatorial interpretations for this…

In 2003, Maroti showed that one could use the machinery of l-cores and l-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case l=2, using them to give a…

The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$…

Let $p\leq 23$ be a prime and $a_p(n)$ counts the number of partitions of $n$ where parts that are multiple of $p$ come up with $2$ colors. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula…

In the quantum theory, using the notion of partial supersymmetry, in which some, but not all, operators have superpartners we derive the Euler theorem in partition theory. The paraferminic partition function gives another identity in…

The least $r$-gap, $g_r(\lambda)$, of a partition $\lambda$ is the smallest part of $\lambda$ appearing less than $r$ times. In this article we introduce two new partition functions involving least $r$-gaps. We consider a bisection of a…

We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…

The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…

Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $\sigma(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a…

In this paper, for every $n \in \mathbb{N}$, the following relationships between the functions $K_{b}(n)$ and $K_{e}(n)$ and the Bernoulli and Euler numbers are proved: \[ B_{2n} = -\,\frac{(2n)!}{2^{2n}-2}\, K_{b}(n), \qquad E_{2n} =…

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one…

We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge…