Search

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F , over any arithmetic progression. Namely, if g a , b (x) denotes the number of even coefficients of F in degrees n ≡ b (mod a) such that n ≤ x , then we show that g a , b (x) / x is unbounded for x large. Note that our result is very close to the best bound currently known even in the special case of the partition function p (n) (namely, x log log x , proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of p (n) , combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of p (n) first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem. [ABSTRACT FROM AUTHOR]

We continue our study of the density of the odd values of eta-quotients, here focusing on the m -regular partition functions b m for m even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let m = 2 j m 0 with m 0 odd. If 2 j < m 0 , then the odd density of b m is 1/2; moreover, such density is equal to 1/2 on every (nonconstant) subprogression A n + B. If 2 j > m 0 , then b m , which is already known to have density zero, is identically even on infinitely many non-nested subprogressions. This and all other conjectures of this paper are consistent with our "master conjecture" on eta-quotients presented in the previous work. In general, our results on b m for m even determine behaviors considerably different from the case of m odd. Also interesting, it frequently happens that on subprogressions A n + B , b m matches the parity of the multipartition functions p t , for certain values of t. We make a suitable use of Ramanujan-Kolberg identities to deduce a large class of such results; as an example, b 28 (49 n + 12) ≡ p 3 (7 n + 2) (mod 2). Additional consequences are several "almost always congruences" for various b m , as well as new parity results specifically for b 11. We wrap up our work with a much simpler proof of the main result of a recent paper by Cherubini-Mercuri, which fully characterized the parity of b 8. [ABSTRACT FROM AUTHOR]

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of m -regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all m ≤ 28 , we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area. [ABSTRACT FROM AUTHOR]

A famous conjecture of Parkin-Shanks predicts that p (n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any A ≡ ± 1 (mod 6) the multipartition function p A (n) has positive odd density, then so does p (n). Similarly, the positive odd density of any p A (n) with A ≡ 3 (mod 6) would imply that of p 3 (n). Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture. [ABSTRACT FROM AUTHOR]

Recently, Hirschhorn and the first author considered the parity of the function a (n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a (2 m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a (n) for all n ≢ 7 (mod 8). We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case n ≡ 7 (mod 8) , where, interestingly, the behavior of a (n) modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, a (8 m + 7) is odd precisely 5 0 % of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open. [ABSTRACT FROM AUTHOR]

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that p ( n ) is odd exactly 50% of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality. A striking consequence is that, under suitable existence conditions, if any t -multipartition function is odd with positive density and t ≢ 0 ( mod 3 ) , then p ( n ) is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today. Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]

The study of the h-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open questions. In this note, we commence a study of those unimodal Gorenstein h-vectors that do not satisfy the Stanley-Iarrobino property. Our main results, which are characteristic free, show that such h-vectors exist: 1) In socle degree e if and only if e≥6; and 2) in every codimension five or greater. The main case that remains open is that of codimension four, where no Gorenstein h-vector is known without the Stanley-Iarrobino property. We conclude by proposing the following very general conjecture: The existence of any arbitrary level h-vector is independent of the characteristic of the base field. [ABSTRACT FROM AUTHOR]

We classify all possible h-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension ⩽17 and in socle degree 5 and codimension ⩽25. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein h-vector is 13 for socle degree 4 and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein h-vector is (1, 13, 12, 13, 1), which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a longstanding open question in this area. All of our results are characteristic free. [ABSTRACT FROM AUTHOR]

We look at the asymptotic behavior of the coefficients of the q-binomial coefficients (or Gaussian polynomials) $${\left(\begin{array}{c}{a+k}\\{k}\end{array}\right)_{q}}$$ , when k is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations. [ABSTRACT FROM AUTHOR]

We investigate the rank-generating function F λ of the poset of partitions contained inside a given shifted Ferrers shape λ . When λ has four parts, we show that F λ is unimodal when λ = 〈 n , n − 1 , n − 2 , n − 3 〉 , for any n ≥ 4 , and that unimodality fails for the doubly-indexed, infinite family of partitions of the form λ = 〈 n , n − t , n − 2 t , n − 3 t 〉 , for any given t ≥ 2 and n large enough with respect to t . When λ has b ≤ 3 parts, we show that our rank-generating functions F λ are all unimodal. However, the situation remains mostly obscure for b ≥ 5 . In general, the type of results that we obtain present some remarkable similarities with those of the 1990 paper of D. Stanton, who considered the case of partitions inside ordinary (straight) Ferrers shapes. Along the way, we also determine some interesting q -analogs of the binomial coefficients, which in certain instances we conjecture to be unimodal. We state several other conjectures throughout this note, in the hopes to stimulate further work in this area. In particular, one of these will attempt to place into a much broader context the unimodality of the posets M ( n ) of staircase partitions, for which determining a combinatorial proof remains an outstanding open problem. [ABSTRACT FROM AUTHOR]

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the q-binomial coefficients ... . Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of ... can get arbitrarily large, when we assume that b is fixed and a is large enough. [ABSTRACT FROM AUTHOR]

Using entirely algebraic or elementary methods, partly inspired by recent ideas of P. Monsky on quadratic representations, we determine a new asymptotic lower bound for the number of odd values of a fundamental modular function in number theory, the Klein j -function. Namely, the number of n ≤ x such that the Klein j -function — or equivalently, the cubic partition function — is odd has order at least x log ⁡ log ⁡ x log ⁡ x , for x → ∞ . This improves recent results of Berndt, Yee and Zaharescu and Chen and Lin, approaching significantly the best bound currently known for p ( n ) , obtained using modular forms. Then, in the final section, we show how to employ modular forms to slightly refine our bound. In fact, our brief argument, combining a recent result of Nicolas–Serre with a classical theorem of Serre on level 1 modular forms, more generally provides a lower bound for the odd values of any positive power of the generating function of p ( n ) . [ABSTRACT FROM AUTHOR]

The set of f-vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable problem, and its structure is very irregular and complicated. The purpose of this note, where we combine a few different algebraic and combinatorial techniques, is to lend some further evidence to this fact. We first show that pure (in fact, Cohen-Macaulay) f-vectors can be nonunimodal with arbitrarily many peaks, thus improving the corresponding results known for level Hilbert functions and pure O-sequences. We provide both an algebraic and a combinatorial argument for this result. Then, answering negatively a question of the second author and collaborators posed in the recent AMS Memoir on pure O-sequences, we show that the interval property fails for the set of pure f-vectors, even in dimension 2. [ABSTRACT FROM AUTHOR]

A beautiful recent conjecture of Armstrong predicts the average size of a partition that is simultaneously an s-core and a t-core, where s and t are coprime. Our goal is to prove this conjecture when t = s + 1. These simultaneous (s, s + 1)-core partitions, which are enumerated by Catalan numbers, have average size (s+1 3 )/2. [ABSTRACT FROM AUTHOR]

We look at the number L (n) of O -sequences of length n. Recall that an O -sequence can be defined algebraically as the Hilbert function of a standard graded k -algebra, or combinatorially as the f -vector of a multicomplex. The sequence L (n) was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants c 1 and c 2 and all n > 2 , e c 1 n ≤ L (n) ≤ e c 2 n log n. It remains an open problem to determine an exact asymptotic estimate for L (n). [ABSTRACT FROM AUTHOR]

Abstract Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever (1 , h 1 , ... , h i , ... , h e − i , ... , h e − 1 , 1) and (1 , h 1 , ... , h i + α , ... , h e − i + α , ... , h e − 1 , 1) are both Gorenstein Hilbert functions for some α ≥ 2 , then (1 , h 1 , ... , h i + β , ... , h e − i + β , ... , h e − 1 , 1) is also Gorenstein, for all β = 1 , 2 , ... , α − 1. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case e ≤ 5 , in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry. [ABSTRACT FROM AUTHOR]

The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely, the conjecture that the partition function p(n) is equidistributed modulo 2.Our main result will relate the densities, say, δt, of the odd values of the t-multipartition functions pt(n), for several integers t. In particular, we will show that if δt>0 for some t∈{5,7,11,13,17,19,23,25}, then (assuming it exists) δ1>0; that is, p(n) itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that p(n) is odd for x values of n≤x. In general, we conjecture that δt=1/2 for all t odd, i.e., that similarly to the case of p(n), all multipartition functions are in fact equidistributed modulo 2.Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu. [ABSTRACT FROM AUTHOR]

We compare the integration by parts of contact forms – leading to the definition of the interior Euler operator – with the so-called canonical splittings of variational morphisms. In particular, we discuss the possibility of a generalization of the first method to contact forms of lower degree. We define a suitable Residual operator for this case and, working out an original conjecture by Olga Rossi, we recover the Krupka–Betounes equivalent for first order field theories. A generalization to the second order case is discussed. [ABSTRACT FROM AUTHOR]