Partition Identities Research Articles

Schmidt-type theorems via weighted partition identities

Schmidt-type theorems via weighted partition identities

Refinements of Beck-type partition identities

Franklin's identity generalizes Euler's identity and states that the number of partitions of n with j different parts divisible by r equals the number of partitions of n with j different parts repeated at least r times. In this article, we give a refinement of Franklin's identity when j=1. We prove Franklin's identity when j=1, r=2 for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter M over the number of parts in all partitions into distinct parts with perimeter M equals the number of partitions with perimeter M whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.

Generalisations of Capparelli's and Primc's identities, I: Coloured Frobenius partitions and combinatorial proofs

The partition identities of Capparelli and Primc were originally discovered via representation theoretic techniques, and have since then been studied and refined combinatorially, but the question of giving a very broad generalisation remained open. In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra An−1(1).In this first paper, we focus on combinatorial aspects. We give a n2-coloured generalisation of Primc's identity by constructing a n2×n2 matrix of difference conditions, Primc's original identities corresponding to n=2 and n=3. While most coloured partition identities in the literature connect partitions with difference conditions to partitions with congruence conditions, in our case, the natural way to generalise these identities is to relate partitions with difference conditions to coloured Frobenius partitions. This gives a very simple expression for the generating function. With a particular specialisation of the colour variables, our generalisation also yields a partition identity with congruence conditions.Then, using a bijection from our new generalisation of Primc's identity, we deduce a large family of identities on (n2−1)-coloured partitions which generalise Capparelli's identity, also in terms of coloured Frobenius partitions. The particular case n=2 is Capparelli's identity and one of the cases where n=3 recovers an identity of Meurman and Primc.In the second paper, we will focus on crystal theoretic aspects. We will show that the difference conditions we defined in our n2-coloured generalisation of Primc's identity are actually energy functions for certain An−1(1) crystals. We will then use this result to retrieve the Kac-Peterson character formula and derive a new character formula as a sum of infinite products for all the irreducible highest weight An−1(1)-modules of level 1.

Multiplication theorems for self-conjugate partitions

In 2011, Han and Ji proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by Berkovich and Garvan. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.Mathematics Subject Classifications: 05A15, 05A17, 05A19, 05E05, 05E10, 11P81Keywords: Hook-length formulas, BGP-ranks, Integers partitions, Littlewood decomposition, core partitions

Repeated integration and explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\)

Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

Beyond Göllnitz' theorem II: Arbitrarily many primary colors

In 2003, Alladi, Andrews and Berkovich proved a four parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively proved a ten-colored partition identity, and then related, by another bijection, our identity to the Alladi-Andrews-Berkovich identity.In this second paper, we state and bijectively prove an n(n+1)2-colored partition identity beyond Göllnitz' theorem for any number n of primary colors, along with the full set of the n(n−1)2 secondary colors as the product of two distinct primary colors, generalizing the identity proved in the first paper. Like the ten-colored partitions, our family of n(n+1)2-colored partitions satisfy some simple minimal difference conditions while avoiding forbidden patterns. Furthermore, the n(n+1)2-colored partitions have some remarkable properties, as they can be uniquely represented by oriented rooted forests which record the steps of the bijection.

Diagonal Hooks and a Schmidt-Type Partition Identity

In a recent paper of Andrews and Paule, several Schmidt-type partition identities are considered within the framework of MacMahon's Partition Analysis. Following their work, we derive a new Schmidt-type identity concerning diagonal hooks of partitions. We provide an analytic proof based on MacMahon's Partition Analysis and a combinatorial proof through an involution on the set of partitions. We also establish connections between Schmidt-type distinct partitions and partitions with nonpositive and negative cranks.

A generalization of multiple zeta values. Part 2: Multiple sums

{\bf Abstract:} Multiple zeta values have become of great interest due to their numerous applications in mathematics and physics. In this article, we present a generalization, which we will refer to as multiple sums, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of partition identities that we use to prove a reduction theorem that expresses multiple sums as a combination of simple sums. We present a variety of applications including applications concerning polynomials and MZVs such as generating functions and expressions for $\zeta(\{2p\}_m)$ and $\zeta^\star(\{2p\}_m)$. Finally, we establish the connection between multiple sums and a type of sums called recurrent sums. By exploiting this connection, we provide additional partition identities for odd and even partitions.

A generalization of multiple zeta values. Part 1: Recurrent sums

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities

Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities but did not establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from which we successfully deduce all the five q-series identities of Ramanujan. In addition to this, we also establish a few interesting weighted partition identities from our generalized identity. In the mid 1980s, Bressoud and Subbarao found an interesting identity connecting the generalized divisor function with a weighted partition function, which they proved by means of a purely combinatorial argument. Quite surprisingly, we found an analytic proof for a generalization of the identity of Bressoud and Subbarao, starting from the fourth identity of the aforementioned five q-series identities of Ramanujan.

Partition Identities for Two-Color Partitions

Three new partition identities are found for two-color partitions. The first relates to ordinary partitions into parts not divisible by 4, the second to basis partitions, and the third to partitions with distinct parts. The surprise of the strangeness of this trio becomes clear in the proof.

Refinement of some partition identities of Merca and Yee

Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.

On certain partition bijections related to Euler's partition problem

We give short elementary expositions of combinatorial proofs of some variants of Euler's partition identity that were first addressed analytically by George Andrews, and later combinatorially by others. The method using certain matrices to concisely explain these bijections, based on ideas first used in a previous manuscript by the author, enables us to also give new generalizations of two of these results.

Even–odd partition identities of Rogers–Ramanujan type

We prove a theorem which adds a new companion to the Rogers–Ramanujan identities. This new companion counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two families of partition identities.

On mex-related partition functions of Andrews and Newman

The minimal excludant, or “mex” function, on a set S of positive integers is the least positive integer not in S. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely \(p_{t,t}(n)\), and provide complete parity characterizations of \(p_{1,1}(n)\) and \(p_{3,3}(n)\). In this article, we study the parity of \(p_{t,t}(n)\) when \(t=2^{\alpha }, 3\cdot 2^{\alpha }\) for all \(\alpha \ge 1\). We prove that \(p_{2^{\alpha },2^{\alpha }}(n)\) and \(p_{3\cdot 2^{\alpha }, 3\cdot 2^{\alpha }}(n)\) are almost always even for all \(\alpha \ge 1\). Using a result of Ono and Taguchi on nilpotency of Hecke operators, we also find infinite families of congruences modulo 2 satisfied by \(p_{2^{\alpha },2^{\alpha }}(n)\) and \(p_{3\cdot 2^{\alpha }, 3\cdot 2^{\alpha }}(n)\) for all \(\alpha \ge 1\).

On sum-of-tails identities

In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two new results - one for Ramanujan's function σ(q) and another for the function recently introduced by Andrews and Ballantine. Later we introduce a new generalization FFWc(n) of a function of Fokkink, Fokkink and Wang and derive an identity for its generating function. This gives, as a special case, a recent representation for the generating function of spt(n) given by Andrews, Garvan and Liang. We also obtain some weighted partition identities along with new representations for two of Ramanujan's third order mock theta functions through combinatorial techniques.

Andrews-Gordon type series for Schur's partition identity

We construct an evidently positive multiple series as a generating function for partitions satisfying the multiplicity condition in Schur's partition theorem. Refinements of the series when parts in the said partitions are classified according to their parities or values mod 3 are also considered. Direct combinatorial interpretations of the series are provided.

A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

We show for a prime power number of parts $m$ that the first differences of partitions into at most $m$ parts can be expressed as a non-negative linear combination of partitions into at most $m-1$ parts. To show this relationship, we combine a quasipolynomial construction of $p(n,m)$ with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of $p(n,m)$ and the new partition identity. We extend these results by establishing conditions for when partitions of $n$ with parts coming from a finite set $A$ can be expressed as a non-negative linear combination of partitions with parts coming from a finite set $B$.

Integral and series representations of q-polynomials and functions: Part III theta, Ramanujan and q-Bessel functions

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [Formula: see text]-Lommel and [Formula: see text]-Bessel polynomials. We introduce and study [Formula: see text]-analogues of the spherical Bessel functions.

New proof to Somos\u2019s Dedekind eta-function identities of level 10

Michael Somos used PARI/GP script to generate several Dedekind eta-function identities by using computer. In the present work, we prove two new Dedekind eta-function identities of level 10 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 10 proved by B. R. Srivatsa Kumar and D. Anu Radha. As an application of this, we establish colored partition identities.