arXiv subjects

Frank G. Garvan

Publications and source records attributed to Frank G. Garvan.

13 recordsLinked to original sources

arXiv2024

Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.

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arXiv2024

Old Meets New: Connecting Two Infinite Families of Congruences Modulo Powers of 5 for Generalized Frobenius Partition Functions

In 2012 Paule and Radu proved a difficult family of congruences modulo powers of 5 for Andrews' 2-colored generalized Frobenius partition function. The family is associated with the classical modular curve of level 20. We demonstrate the existence of a congruence family for a related generalized Frobenius partition function associated with the same curve. We construct an isomorphism between this new family and the original family of congruences via a mapping on the associated rings of modular functions. The pairing of the congruence families provides a new strategy for future work on congruences associated with modular curves of composite level. We show how a similar approach can be made to multiple other recent examples in the literature. We also give some important insights into the behavior of these congruence families with respect to the Atkin--Lehner involution which proved very important in Paule and Radu's original proof.

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arXiv2013

Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $cϕ_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $cϕ_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. Work on Ramanujan--like congruence properties satisfied by the functions $cϕ_k(n)$ continues, with recent works completed by Baruah and Sarmah as well as the author. Unfortunately, in all cases, the authors restrict their attention to small values of $k.$ This is often due to the difficulty in finding a "nice" representation of the generating function for $cϕ_k(n)$ for large $k.$ Because of this, no Ramanujan--like congruences are known where $k$ is large. In this note, we rectify this situation by proving several infinite families of congruences for $cϕ_k(n)$ where $k$ is allowed to grow arbitrarily large. The proof is truly elementary, relying on a generating function representation which appears in Andrews' Memoir but has gone relatively unnoticed.

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arXiv2012

Rank-Crank type PDEs and generalized Lambert series identities

We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson respectively. The first author's Lambert series identities are common generalizations. We also show how Atkin and Swinnerton-Dyer's proof using elliptic functions can be extended to prove these generalized Lambert series identities.

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arXiv2008

The GBG-Rank and t-Cores I. Counting and 4-Cores

Let r_j(π,s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of πmod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*Π*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all eta-products.

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arXiv2007

The BG-rank of a partition and its applications

Let πbe a partition. In [2] we defined BG-rank(π) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a number of partitions (t-cores) of n with BG-rank=j. Here, we provide an elegant combinatorial proof that 5|p_j(5n+4) by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by p_j(5n+4) into five equal classes. This proof uses the orbit construction in [2] and new identity for BG-rank. In addition, we find eta-quotient representation for the generating functions for coefficients a_{t,floor((t+1)/4)}(n), a_{t,-floor((t-1)/4)}(n) when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a_{5,j}(n) with j=0,1,-1.

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arXiv2007

K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [17] and Garvan, Kim and Stanton [10]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.

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arXiv2006

K. Saito's Conjecture for Nonnegative Eta Products

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arXiv2006

Shifted and Shiftless Partition Identities II

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form p(S,n)=p(T,n-a), for all n greater or equal to a. Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M=32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form p(S,n)=p(T,n), for all n not equal to a. In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four parameter theta function identity due to Jacobi. New shifted and shiftless partition identities are proved.

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arXiv2005

Dissecting the Stanley Partition Function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p[i](n) denote the number of partitions pi of n such that O(pi) - O(pi') = i mod 4. Here O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. R. Stanley [13], [14] derived an infinite product representation for the generating function of p[0](n)-p[2](n). Recently, Holly Swisher[15] employed the circle method to show that limit[n->oo] p[0](n)/p(n) = 1/2 (i) and that for sufficiently large n 2 p[0](n) > p(n), if n=0,1 mod 4, 2 p[0](n) < p(n), otherwise. (ii) In this paper we study even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result |p[0](2n) - p[2](2n)| > |p[0](2n+1) - p[2](2n+1)|, n>0. Two proofs of this surprising inequality are given. The first one uses the Gollnitz-Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.

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arXiv2004

On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations

In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0; 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enumerated by p[i](5n + 4) (i = 0, 2) into five equinumerous classes. In this paper we discuss three such statistics: the St-crank, the 2-quotient-rank and the 5-core-crank. The first one, while new, is intimately related to the Andrews-Garvan crank. The second one is in terms of the 2-quotient of a partition. The third one was introduced by Garvan, Kim and Stanton. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod 5. This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5. Finally, we discuss some new formulas for partitions that are 5-cores and discuss an intriguing relation between 3-cores and the Andrews-Garvan crank.

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arXiv2004

On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5

In a recent study of sign-balanced, labelled posets Stanley [13], introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0, 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enumerated by p[i](5n + 4) (i = 0, 2) into five equinumerous classes. In this paper we discuss two such statistics. The first one, while new, is intimately related to the Andrews-Garvan [2] crank. The second one is in terms of the 5-core crank, introduced by Garvan, Kim and Stanton [9]. Finally, we discuss some new formulas for partitions that are 5-cores.

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arXiv2002

Some Observations on Dyson's New Symmetries of Partitions

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank <= k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's ``modular'' partitions with modulus 2. This way we find a new combinatorial proof of Gauss's famous identity. We give a direct combinatorial proof that for n>1 the partitions of n with crank k are equinumerous with partitions of n with crank -k.

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