Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem. Under the conditions that even parts or odd parts appear an even number of times, Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems and derived corresponding generating functions. In the Rogers-Ramanujan-Gordon theorem, there are two parameters $k$ and $a$, where $k-1$ is the maximum number of consecutive parts $l$ and $l+1$, and $a-1$ is the maximum number of parts equal to $1$. Andrews' first theorem deals with the case $k\equiv a \;(\rm{mod}\;2)$, while the second theorem concerns the case where $k$ is even and $a$ is odd. These two partition identities have different infinite product forms on the right-hand side. In this paper, we consider the case $k ot\equiv a \;(\rm{mod}\;2)$ and use Bailey's lemma to obtain an Andrews-Gordon type identity whose right-hand side coincides with that of Andrews' identity for the case $k\equiv a \;(\rm{mod}\;2)$. By fixing the number of peaks of the corresponding lattice paths, we also derive a recurrence system whose solution agrees with the product-side generating function. We were unable to find a suitable combinatorial interpretation of the in
For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $AC_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $λ\colon FAC_k(G) \to AC_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FAC_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $λ\colon FAC_k(G) \to AC_k(G)$ is an isomorphism.
In 1986, Andrews studied the function $σ(q)$ from Ramanujan's ``Lost" Notebook, and made several conjectures on its Fourier coefficients $S(n)$, which count certain partition ranks. In 1988, Andrews-Dyson-Hickerson famously resolved these conjectures, relating the coefficients $S(n)$ to the arithmetic of $\mathbb Q(\sqrt{6})$; this relationship was further expounded upon by Cohen in his work on Maass waveforms, and was more recently extended by Zwegers and by Li and Roehrig. A closer inspection of Andrews' original work on $σ(q)$ reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function $v_1(q)$, also from Ramanujan's ``Lost" Notebook, a $q$-hypergeometric series with partition-theoretic Fourier coefficients $V_1(n)$, and prove two of Andrews' conjectures on $V_1(n)$ which are parallel to his original conjectures on $S(n)$. Our methods differ from those used by Andrews-Dyson-Hickerson, and require a blend of novel techniques inspired by Garoufalidis' and Zagier's recent work on asymptotics of Nahm sums, with classical techniques including the Circle Method in Analytic Number Theory; our methods may also be applied to de
For a positive integer $r$, George Andrews proved that the set of partitions of $n$ in which odd multiplicities are at least $2r + 1$ is equinumerous with the set of partitions of $n$ in which odd parts are congruent to $2r + 1$ modulo $4r + 2$. This was given as an extension of MacMahon's theorem ($r = 1$). Andrews, Ericksson, Petrov and Romik gave a bijective proof of MacMahon's theorem. Despite several bijections being given, until recently, none of them was in the spirit of Andrews-Ericksson-Petrov-Romik bijection. Andrews' theorem has also been extended recently. Our goal is to give a generalized bijective mapping of this further extension in the spirit of Andrews-Ericksson-Petrov-Romik bijection.
We find bivariate generating functions for the $k=1$ cases of recently conjectured colored partition identities of Capparelli, Meurman, A. Primc, and M. Primc that are slight variants of the generating functions for the sum sides of the Andrews-Gordon and Andrews-Bressoud identities, relating them to recent work of Warnaar. This $k=1$ cases turn out to be equivalent to identities of Jing, Misra, and Savage. Finally, we provide bijections for these identities involving two-rowed cylindric partitions, in the spirit of Corteel.
The paper discusses the Andrews-Curtis graph of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one them can be obtained from another by certain elementary transformations. This object appears naturally in the theory of black box finite groups and in the Andrews-Curtis conjecture in algebraic topology. We suggest an approach to the Andrews-Curtis conjecture based on the study of Andrews-Curtis graphs of finite groups, discuss properties of Andrews-Curtis graphs of some classes of finite groups and results of computer experiments with generation of random elements of finite groups by random walks on their Andrews-Curtis graphs.
In 1967, Andrews found a combinatorial generalization of the Göllnitz-Gordon theorem, which can be called the Andrews-Göllnitz-Gordon theorem. In 1980, Bressoud derived a multisum Rogers-Ramanujan-type identity, which can be considered as the generating function counterpart of the Andrews-Göllnitz-Gordon theorem. Lovejoy gave an overpartition analogue of the Andrews-Göllnitz-Gordon theorem for $i=k$. In this paper, we give an overpartition analogue of this theorem in the general case. By using Bailey's lemma and a change of base formula due to Bressoud, Ismail and Stanton, we obtain an overpartition analogue of Bressoud's identity. We then give a combinatorial interpretation of this identity by introducing the Göllnitz-Gordon marking of an overpartition, which yields an overpartition analogue of the Andrews-Göllnitz-Gordon theorem.
The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the previously proposed examples are actually not counterexamples. We hope that the tricks we used in constructing relevant chains of elementary transformations will be useful to those who attempt to establish the Andrews-Curtis equivalence in other situations. On the other hand, we give two rather general and simple methods for constructing balanced presentations of the trivial group; some of these presentations can be considered potential counterexamples to the Andrews-Curtis conjecture. One of the methods is based on a simple combinatorial idea of composition of group presentations, whereas the other one uses "exotic" knot diagrams of the unknot. We also consider the Andrews-Curtis equivalence in metabelian groups and reveal some interesting connections of relevant problems to well-known proble
The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be transformed into the trivial presentation by a finite sequence of "elementary transformations" which are Nielsen transformations together with an arbitrary conjugation of a relator. It is believed that the Andrews-Curtis conjecture is false; however, not so many possible counterexamples are known. It is not a trivial matter to verify whether the conjecture holds for a given balanced presentation or not. The purpose of this paper is to describe some non-deterministic methods, called Genetic Algorithms, designed to test the validity of the Andrews-Curtis conjecture. Using such algorithm we have been able to prove that all known (to us) balanced presentations of the trivial group where the total length of the relators is at most 12 satisfy the conjecture. In particular, the Andrews-Curtis conjecture holds for the presentation <x,y|x y x = y x y, x^2 = y^3> which was one of the well known potential counterexamples.
In a very recent work, G. E. Andrews defined the combinatorial objects which he called {\it singular overpartitions} with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we prove that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions.
The Andrews-Bressoud identities are one of many families of $q$-series identities relating an infinite sum to an infinite product. While the original motivation for studying these series relates to partitions, they can also be viewed in relation to irreducible characters of minimal models in the theory of vertex operator algebras. Furthermore, considering certain Wronskians of the Andrews-Bressoud series for a given modulus produces additional $q$-series, which themselves exhibit interesting and predictable modularity properties. In this paper, we connect the Andrews-Bressoud series to modular forms and prove results about the modularity of their associated Wronskians.
We present what we call a "motivated proof" of the Göllnitz-Gordon-Andrews identities. A similar motivated proof of the Rogers-Ramanujan identities was previously given by G. E. Andrews and R. J. Baxter, and was subsequently generalized to Gordon's identities by J. Lepowsky and M. Zhu. We anticipate that the present proof of the Göllnitz-Gordon-Andrews identities will illuminate certain twisted vertex-algebraic constructions.
It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis made in 1966. We also consider a more restrictive "cancellative" version of the cyclic Andrews--Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews--Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.
We present what we call a "motivated proof" of the Andrews-Bressoud partition identities for even moduli. A "motivated proof" of the Rogers-Ramanujan identities was given by G. E. Andrews and R. J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a "motivated proof" of the somewhat analogous Göllnitz-Gordon-Andrews identities has been found. In the present work, we introduce "shelves" of formal series incorporating what we call "ghost series," which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that these new series will provide insight into the ongoing program of vertex-algebraic categorification of the various "motivated proofs."
In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In a recent paper, the author generalised the first of these theorems to overpartitions, using a new technique which consists in going back and forth between $q$-difference equations on generating functions and recurrence equations on their coefficients. Here, using a similar method, we generalise the second theorem of Andrews to overpartitions.
Recently, Andrews and Yee studied two-variable generalizations of two identities involving partition functions $p_ω(n)$ and $p_ν(n)$ introduced by Andrews, Dixit and Yee. In this paper, we present a combinatorial proof of an interesting identity in their work.
The Andrews-Curtis conjecture claims that every normally generating $n$-tuple of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing $F_n$ by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews-Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag-Solitar groups do not satisfy the generalized Andrews-Curtis conjecture in the sense of Burns and Macedońska.
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked in [5] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [5].
George Andrews [\emph{Bull. Amer. Math. Soc.}, 2007, 561--573] introduced the idea of a \emph{signed partiton} of an integer; similar to an ordinary integer partitions, but where some of the parts could be negative. Further, Andrews reinterpreted the classical Göllnitz--Gordon partition identities in terms of signed partitions. In the present work, we provide interpretations of the sum sides of Rogers--Ramanujan type identities, including a new signed partition interpretation of the Göllnitz--Gordon identities, different from that of Andrews. Both analytic and bijective proofs are presented.
Describe an animal without using the verb look. Can you effectively provide an alternative method for interpreting complex microscopy images while preserving the length scale? The world is filled with features too small for our eyes to see: the setae on a gecko's feet, the cuticles covering a rat's whisker, or the fuzziness of a bat's wing. Furthermore, these structures are non-homogeneous, often shifting from stiff to soft. We provide a workflow for producing low-data, low-cost, and open-source lithograph files, allowing tactile accessibility in microscopy images. The lithographs made with this workflow can be printed on a 350 USD 3D printer using 3D files under 100 Mb, for a total cost per print of 0.75 USD. This work seeks to leverage advanced 3D printing to create tactile graphics and art that make science more accessible and enable tactile exploration of biological structures. This framework in this text is aligned with a GitHub repository that will be constantly updated, allowing tactile media to be created as 3D printing and lithography become more streamlined in the years to come.