Generating Functions for Certain Weighted Cranks

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these cumulative and point crank functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for $k$-colored partitions, which was first introduced by Fu and Tang, and investigate the corresponding generating function for this crank weighted by the number of parts in the first subpartition of a $k$-colored partition. We show that the cumulative generating functions are the same as the generating functions for certain unimodal sequences.

Proof of the Bessenrodt–Ono Inequality by Induction

In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern–Fu–Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.

On a problem on restricted k-colored partitions

Let [Formula: see text] be the number of [Formula: see text]-colored partitions of [Formula: see text]. Recently, Keith proved that for [Formula: see text], if [Formula: see text] for all [Formula: see text], then [Formula: see text] is large. We prove that such [Formula: see text] do not exist. Furthermore, for any positive integers [Formula: see text] with [Formula: see text], there exist infinitely many positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text].

Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

Non-hyperoctahedral Categories of Two-Colored Partitions Part II: All Possible Parameter Values

This article is part of a series with the aim of classifying all non-hyperoctahedral categories of two-colored partitions. Those constitute by some Tannaka-Krein type result the representation categories of a specific class of quantum groups. In Part I we introduced a class of parameters which gave rise to many new non-hyperoctahedral categories of partitions. In the present article we show that this class actually contains all possible parameter values of all non-hyperoctahedral categories of partitions. This is an important step towards the classification of all non-hyperoctahedral categories.

Non-hyperoctahedral categories of two-colored partitions part I: new categories

Compact quantum groups can be studied by investigating their representation categories in analogy to the Schur–Weyl/Tannaka–Krein approach. For the special class of (unitary) “easy” quantum groups, these categories arise from a combinatorial structure: rows of two-colored points form the objects, partitions of two such rows the morphisms. Vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes $${\mathcal {O}}$$ O , $${\mathcal {B}}$$ B , $${\mathcal {S}}$$ S and $${\mathcal {H}}$$ H of such categories (inspired, respectively, by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups), we treat the first three—the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. It is purely combinatorial in nature. The quantum group aspects are left out.

Restricted k-color partitions, II

We consider [Formula: see text]-colored partitions, partitions in which [Formula: see text] colors exist but at most [Formula: see text] colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where [Formula: see text] and [Formula: see text] are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the number of partitions in the [Formula: see text] box with a given number of part sizes, and extend to multiple colors a conjecture of Dousse and Kim on unimodality in overpartitions.

Formulas for the number of $k$-colored partitions and the number of plane partitions of $n$ in terms of the Bell Polynomials

We derive closed formulas for the number of $k$-colored partitions and the number of plane partitions of $n$ in terms of the Bell polynomials.

Polynomization of the Bessenrodt–Ono Inequality

In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials $$P_n(x)$$ P n ( x ) . We prove for all real numbers $$x >2 $$ x > 2 and $$a,b \in \mathbb {N}$$ a , b ∈ N with $$a+b >2$$ a + b > 2 the inequality: $$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$ P a ( x ) · P b ( x ) > P a + b ( x ) . We show that $$P_n(x) < P_{n+1}(x)$$ P n ( x ) < P n + 1 ( x ) for $$x \ge 1$$ x ≥ 1 , which generalizes $$p(n) < p(n+1)$$ p ( n ) < p ( n + 1 ) , where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: $$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$ P 2 ( - 3 + 10 ) = P 3 ( - 3 + 10 ) .