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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]

Abstract: We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree due to H. Schröter, R. Russell and S. Ramanujan. Our approach generalizes that of S. Kim, who has given a bijective proof for two of these five identities, namely the ones modulo 7 (also known as the Farkas–Kra identity) and modulo 3. As a consequence of our method, we determine bijective proofs also for the two highly nontrivial identities modulo 5 and 11, thus leaving open combinatorially only the one modulo 23. [Copyright &y& Elsevier]

Abstract: We show how certain suitably modified N-modular diagrams of integer partitions provide a nice combinatorial interpretation for the general term of Zeilbergerʼs KOH identity. This identity is the reformulation of OʼHaraʼs famous proof of the unimodality of the Gaussian polynomial as a combinatorial identity. In particular, we determine, using different bijections, two main natural classes of modular diagrams of partitions with bounded parts and length, having the KOH terms as their generating functions. One of our results greatly extends recent theorems of J. Quinn et al., which presented striking applications to quantum physics. [Copyright &y& Elsevier]

Abstract: We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes dividing the number, , of plane partitions contained inside an arbitrary box of given sides are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of ) fails to have the WLP in characteristic . We wonder how powerful can be this connection between combinatorial commutative algebra and partition theory. We present a first result in this direction, by deducing, using our algebraic techniques for the WLP, some explicit information on the rational primes dividing . [ABSTRACT FROM AUTHOR]

Abstract: We discuss Green’s paper  from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine a new infinite class of symmetric -vectors that cannot be Gorenstein -vectors, which was left open in the recent work . This includes the smallest example, previously unknown, . As M. Green’s results depend heavily on the characteristic of the base field, so will ours. The Appendix contains a new argument, kindly provided to us by M. Green, for Theorems 3 and 4 of , since we had found a gap in the original proof of those results during the preparation of this manuscript. [Copyright &y& Elsevier]

Abstract: Three basic properties which standard graded artinian -algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP). In this paper we will assume that the base field has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out (see [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for artinian -algebras, J. Algebra 262 (2003) 99–126]) that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). In [J. Migliore, F. Zanello, The Hilbert functions which force the weak Lefschetz property, J. Pure Appl. Algebra 210 (2) (2007) 465–471], J. Migliore and the first author, using Green’s theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP in the above cited reference. Our methods include the Herzog–Popescu theorem on quotients of -algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic. [Copyright &y& Elsevier]

Abstract: We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, and are both level h-vectors, then is also level for each integer . In the Gorenstein case, i.e. when , we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say and , of a Gorenstein h-vector. These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets. We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension. [Copyright &y& Elsevier]

Abstract: The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [A. Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Algebra 32 (12) (2004) 4601–4611]) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If is an artinian algebra enjoying the WLP, do all artinian algebras with the same Hilbert function as and Betti numbers lower than those of have the WLP as well? Also, as a consequence of our result, we have another (simpler) proof of the fact that all codimension 2 algebras enjoy the WLP (this fact was first proven in [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian -algebras, J. Algebra 262 (2003) 99–126], where it was shown that even the Strong Lefschetz Property holds). [Copyright &y& Elsevier]

Abstract: The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen–Macaulay algebra in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of . All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211–226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if has a pure MFR. Therefore, it seems a reasonable–and useful–idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen–Macaulay algebras. In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose -vector is that of a compressed algebra. In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra , which can be expressed exclusively in terms of the -vector of , and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of is not pure. Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras : namely, when is compressed, or when its -vector ends with . Also, we will prove our lower bound when begins with , where , and our upper bound when ends with , where . [Copyright &y& Elsevier]

The purpose of this note is to supply an upper and a lower bound (which are in general sharp) for the h-vector of a level algebra which is relatively compressed with respect to any arbitrary level algebra A. The useful concept of relatively compressed algebra was recently introduced in Migliore et al. (2005) (whose investigations mainly focused on the particular case of A a complete intersection). The key idea of this note is the simple observation that the level algebras which are relatively compressed with respect to A coincide (after an obvious isomorphism) with the generic level quotients of suitable truncations of A. Therefore, we are able to apply to relatively compressed algebras the main result of our recent work, Zanello (2007). [ABSTRACT FROM AUTHOR]

Abstract: is a level h-vector! This example answers negatively the open question as to whether all codimension 3 level h-vectors are unimodal. Moreover, using the same (simple) technique, we are able to construct level algebras of codimension 3 whose h-vectors have exactly N “maxima,” for any positive integer N. These non-unimodal h-vectors, in particular, provide examples of codimension 3 level algebras not enjoying the Weak Lefschetz Property (WLP). Their existence was also an open problem before. In the second part of the paper we further investigate this fundamental property, and show that there even exist codimension 3 level algebras of type 3 without the WLP. [Copyright &y& Elsevier]

First, we construct a bijection between the set of h -vectors and the set of socle-vectors of Artinian algebras. As a corollary, we find the minimum codimension that an Artinian algebra with a given socle-vector can have. Then, we study the main problem in the article: Determining when there is a unique socle-vector for a given h -vector. We solve the problem completely if the codimension is at most 3. [ABSTRACT FROM AUTHOR]

In this article, we study standard graded Artinian level algebras, in particular those whose socle-vector has type 2. Our main results are: the characterization of the level h -vectors of the form (1, r ,..., r , 2) for r = 4; the characterization of the minimal free resolutions associated to each of the h -vectors above when r = 3; a sharp upper-bound (under certain mild hypotheses) for the level h -vectors (1, r ,..., a , 2) of arbitrary codimension r and type 2, which depends on the next to last entry a . [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]

Focuses on the interesting case of the factorial domain of algebraic integers in a number field. Identification of the unique factorial domain; Unification of the algebraic concept that characterizes rings; Measurement of the lengths of all chains of prime ideals.

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]

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators. [ABSTRACT FROM AUTHOR]

Abstract: We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree entry of a Gorenstein h-vector, in terms of its entry in degree i. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given r and i, all Gorenstein h-vectors of codimension r and socle degree (this function being explicitly computed) are unimodal up to degree . This immediately gives a new proof of a theorem of Stanley that all Gorenstein h-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the ith entry of a Gorenstein h-vector may assume, in terms of codimension, r, and socle degree, e. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case and , as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree . [Copyright &y& Elsevier]

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case. [ABSTRACT FROM AUTHOR]

Abstract: We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function , we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from to , and Hesse configurations in . [Copyright &y& Elsevier]