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## Papers by Richard P. Stanley

Complete list of publications

- Some curious sequences constructed with the greedy algorithm (with A. M. Odlyzko) (5 pages) pdf

Unpublished notes dated January, 1978, concerning sequences of integers containing no three terms in arithmetic progression. - Hyperplane arrangements, interval orders and trees (20 pages)
*Proc. Nat. Acad. Sci.***93**(1996), 2620--2625.

Some connections between the three objects of the title are investigated. - Hipparchus, Plutarch, Schröder and Hough (12 pages)
*American Mathematical Monthly***104**(1997), 344-350. pdf

A solution to an ancient combinatorial riddle. Much more detailed and scholarly account by F. Acerbi. A follow-up by Susanne Bozien. - Graph colorings and related symmetric functions: ideas and applications (28 pages)
*Discrete Mathematics***193**(1998), 267-286.

A sequel to "A symmetric function generalization of the chromatic polynomial of a graph,"*Advances in Math.***111**(1995), 166-194. - Hyperplane arrangements, parking functions and tree inversions (14 pages)

in*Mathematical Essays in Honor of Gian-Carlo Rota*(B. Sagan and R. Stanley, eds.), Birkhäuser, Boston/Basel/Berlin, 1998, pp. 359-375.

Connections between the three objects of the title, and a generalization involving*k*-parking functions and rooted*k*-forests. - A
*q*-deformation of a trivial symmetric group action (with Phil Hanlon) (26 pages)*Transactions of the American Mathematical Society***350**(1998), 4445-4459.

Solutions to problems raised by Varchenko and Zagier concerning a*q*-deformation of the element of the group algebra of the symmetric group*S*equal to the sum of all the elements of_{n}*S*._{n} - Polygon dissections and standard Young tableaux (4 pages)
*Journal of Combinatorial Theory, series A***76**(1996), 175-177.

A simple bijection between dissections of a convex*(n+2)*-gon with*d*diagonals not interecting in their interiors and standard Young tableaux of shape*(d+1, d+1, 1*^{n-1-d}). - Lê numbers of arrangements and matroid identities (16 pages)

(with D. B. Massey, R. Simion, D. Vertigan, D. J. A. Welsh, and G. M. Ziegler),*J. Combinatorial Theory, Series B***70**(1997), 118-133.

Some identities involving the Möbius function of a matroid, motivated by the Lê number of a hypersurface singularity. - Parking functions and noncrossing partitions (14 pages)
*Electronic J. Combinatorics***4**, R20 (1997).

Some connections between parking functions, noncrossing partitions, symmetric functions, and a local action of the symmetric group. - Deformations of Coxeter hyperplane arrangements (with Alexander Postnikov) (41 pages, version of 29 March 2000)
*J. Combinatorial Theory (A)***91**(2000), 544-597.

Devoted mainly to the computation of the characteristic polynomial of some hyperplane arrangements related to the braid arrangement and other Coxeter arrangements. Includes the connection between the Linial arrangement, alternating trees, local binary search trees, and other combinatorial objects. Also included is a "Riemann hypothesis" for the characteristic polynomial of the Linial arrangement and related arrangements. - Flag-symmetry of the poset of shuffles and a local action of the symmetric group (with Rodica Simion) (34 pages)
*Discrete Math.***204**(1999), 369-396.

New combinatorial properties of Curtis Greene's poset of shuffles. - A combinatorial miscellany (with Anders Björner) (167 pages, final version of 5 September 2010)
*L'Enseignement Math.*, Monograph No. 42, 2010.

An expository paper of various topics in algebraic and enumerative combinatorics, intended for both mathematicians and nonmathematicians. Includes discussions of integer partitions, plane partitions, the Schensted algorithm, increasing and decreasing subsequences, reduced decompositions, enumeration of tilings, combinatorics and topology, evasiveness, complexity of sorting and distinctness, and face numbers of polytopes. Errata - Spanning trees and a conjecture of Kontsevich (13 pages, publication version)
*Annals of Combinatorics***2**(1999), 351-363.

Kontsevich conjectured that the number of zeros over the field*F*of a certain polynomial connected with the spanning trees of a graph_{q}*G*is a polynomial function of*q*. We have not been able to settle this conjecture, but we show the connection with such topics as the Matrix-Tree Theorem and orthogonal geometry. A sequel to this paper was written by John Stembridge and appears in the same issue of*Annals of Combinatorics*. Update. - Domino tilings with barriers (with Jim Propp) (10 pages)
*J. Combinatorial Theory (A)***87**(1999), 347-356 .

Proves a result about the independence of certain random domino tilings of the Aztec diamond. - Positivity problems and conjectures in algebraic combinatorics (35 pages, version of 24 September 1999) PDF

In*Mathematics: Frontiers and Perspectives*(V. Arnold, M. Atiyah, P. Lax, and B. Mazur, eds.), American Mathematical Society, Providence, RI, 2000, pp. 295-319.

A survey of problems and conjectures in algebraic combinatorics related to showing that certain numbers are nonnegative. The three main areas covered are (1)*f*-vectors, (2) representation theory and symmetric functions, and (3) real zeros and total positivity. Update (14 September 2004). - A polytope related to empirical distributions, plane trees, parking functions, and the associahedron (with Jim Pitman) (40 pages)
*Discrete and Computational Geometry*,**27**(2002), 603-634.

The title says it all, except there are also connections with plane partitions. - A generalized riffle shuffle and quasisymmetric functions (19 pages, version of 31 May 2001)
*Annals of Combinatorics*,**5**(2001), 479-491.

This paper concerns a probability distribution on the symmetric group generalizing the riffle shuffle of Bayer, Diaconis, and others. There are close connections with the theory of quasisymmetric and symmetric functions. - A note on the symmetric powers of the standard representation of
*S*(with David Savitt) (7 pages)_{n}*Electronic J. Combinatorics***7**, R6 (2000).

The main result is that the dimension of the space spanned by the characters of the symmetric powers of the standard*n*-dimensional representation of the symmetric group*S*is asymptotic to_{n}*n*.^{2}/2 - Rodica Simion, January 18, 1955 -- January 7, 2000 (5 pages)
*Pi Mu Epsilon Journal***11**(2000), 83-86.

An appreciation of a truly special person. - Recent progress in algebraic combinatorics (23 pages, version of 19 March 2002)
*Bull. Amer. Math. Soc.***40**(2003), 55-68.

A survey of recent work on (1) the saturation conjecture for Littlewood-Richardson coefficients, (2) the*n!*and*(n+1)*conjectures, and (3) longest increasing subsequences of permutations.^{n-1} - On the enumeration of skew Young tableaux (15 pages)
*Advances in Applied Math.***30**(2003), 283-294.

Exact formulas and asymptotic estimates for the number of skew Young tableaux of shape*a/b*where (1)*b*is fixed and*a*has size*n*, and (2) both*a*and*b*are fixed. - The rank and minimal border strip decompositions of a skew partition (31 pages)
*J. Combinatorial Theory (A)***100**(2002), 349-375.

Some properties of the rank of a skew partition (originally defined by Nazarov and Tarasov) and a related investigation of the minimal border strip decompositions and minimal border strip tableaux of a skew partition. Update. - Irreducible symmetric group characters of rectangular shape (11 pages, version of 17 December 2002)
*Sém. Lotharingien de Combinatoire*(electronic)**50**(2003), B50d.

A new formula for the values of an irreducible symmetric character corresponding to a partition of rectangular shape, and some comments and conjectures on a generalization. - Rodica Simion and shuffle posets (3 pages)
*Advances in Applied Math.***28**(2002), 282-284.

Reminiscences on the only joint paper I wrote with Rodica Simion. - Some remarks on sign-balanced and maj-balanced posets (30 pages, version of 14 January 2004)
*Advances in Applied Math.***34**(2005), 880-902.

An investigation of (labelled) posets for which exactly half the linear extensions have an even number of inversions (i.e., are even permutations) and posets for which exactly half the linear extensions have even major index. - Recent developments in algebraic combinatorics (30 pages) pdf
*Israel J. Math.***143**(2004), 317-340.

A continuation of "Recent progress in algebraic combinatorics" above, with three sections: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric*h*-vectors and intersection cohomology. - The mathematical knight (with N. Elkies) (23 pages) (pdf)
*The Mathematical Intelligencer***25**, no. 1 (Winter 2003), 22--34

A survey of chess problems and puzzles, featuring the knight, that would be of interest to mathematicians. - A map on the space of rational functions (with G. Boros, J. Little, V. Moll, and E. Mosteig) (16 pages)
*Rocky Mountain J. Math.***35**(2005), 1861-1880.

A study of the dynamical properties of a certain map*F*defined on the space of rational functions. The long time behavior of a subclass involves properties of Eulerian polynomials. - A super-class walk on upper-triangular matrices (with E. Arias-Castro and P. Diaconis) (26 pages)
*J. Algebra***278**(2004), 739-765.

An analysis of a random walk on the group of*n*×*n*upper-triangular matrices over a finite field, based on the character theory of Andre, Carter, and Yan. - Properties of some character tables related to the symmetric groups (with C. Bessenrodt and J. Olsson) (16 pages, version of 27 February 2004)
*J. Algebraic Combinatorics***21**(2005), 163-177.

Determination of invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups*S*and their double covers._{n} - An application of set theory to cosmology (one page) (PDF)

Correction of a defective theory of cosmology. - Bottom Schur functions (with P. Clifford) (16 pages)
*Electronic J. Combinatorics***11**(1) (2004), R67.

Properties of the symmetric function consisting of the terms of least degree of a Schur function, when we define the degree of the power sum*p*to be 1. Erratum._{i} - Coefficients and roots of Ehrhart polynomials (with M. Beck, J. De Loera, M. Develin, and J. Pfeifle)
*Contemp. Math.***374**(2005), 15-36.

Restrictions on the coefficients and zeros of various classes of Ehrhart polynomials of integer polytopes, and a conjecture on the Ehrhart polynomial of a cyclic polytope. - Tilings (pdf) (with F. Ardila) (21 pages)

A survey of tilings in the plane for a nonmathematial audience. Based on a Clay Public Lecture at the IAS/Park City Mathematics Institute in July, 2004.*Math. Intelligencer*,**(2010), 32-43.** **Crossings and nestings of matchings and partitions (pdf) (with William Y. C. Chen (陈永川), Eva Y. P. Deng (邓玉平), Rosena R. X. Du (杜若霞), and Catherine H. Yan (颜华菲)) (24 pages) (version of 9 November 2005)***Trans. Amer. Math. Soc.***359**(2007), 1555-1575.

Applications of oscillating tableaux and vacillating tableaux to the enumeration of matchings and set partitions with conditions on crossings and nestings. Selected one of the top 100 Chinese scientific papers of 2007.**Ordering events in Minkowski space (18 pages) (version of 11 June 2005)***Advances in Applied Math.***37**(2006), 514-525.

Given*k*points in (*n*+1)-dimensional Minkowski space, in how many orders can they occur in different reference frames? What sets of orders are possible? These questions are investigated using the theory of hyperplane arrangements. Update.**Chains in the Bruhat order (with Alexander Postnikov) (36 pages)***J. Algebraic Combinatorics***29**(2009), 133-174. .

An investigation of a family of polynomials whose values are degrees of Schubert varieties in the generalized complex flag manifold*G/B*. The polynomials are given by weighted sums over saturated chains in the Bruhat order.**Queue problems revisited (pdf) (12 pages)***Suomen Tehtäväniekat***59**, no. 4 (2005), 193-203.

A paper that will be mainly of interest to mathematical chess problem aficionados. It does include several classes of posets whose number of linear extensions can be computed explicitly.**The descent set and connectivity set of a permutation (12 pages)***J. Integer Sequences***8**(2005), article 05.3.8.

The connectivity set of a permutation is defined and is shown to be a kind of dual to the descent set. For a generalization to any finite Coxeter group, see M. Marietti,*European J. Combinatorics***29**(2008), 1555-1562.**An analogue of Young's lattice for compositions (with Anders Björner) (20 pages, version of 4 November 2005)**

Combinatorial, algebraic, and topological properties of a graded poset whose elements of rank*n*are indexed by compositions of*n*.**Longest alternating subsequences of permutations (19 pages, version of 15 November 2005)***Michigan Math. J.***57**(2008), 675-687.

Combinatorial and statistical properties of the length of the longest alternating subsequence of a permutation of 1,2,...,*n*.**Increasing and decreasing subsequences and their variants (34 pages, pdf file).***Proc. Internat. Cong. Math., Madrid 2006)*, American Mathematical Society, 2007, pp. 545-579.

A survey paper on the subject of the title. Includes such variants as pattern avoidance, alternating subsequences, and matchings. This is the final version as it will appear in the*Proceedings*, except that reference [114] will be updated.**Alternating permutations and symmetric functions (37 pages, version of 18 August 2006)***J. Combinatorial Theory Series A***114**(2007), 436-460..

The theory of symmetric functions is used to enumerate various classes of alternating permutations, such as those with a given cycle type. Update.**A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group (6 pages, version of 26 July 2006)**

A conjectured generalization of the main result of "Irreducible symmetric group characters of rectangular shape". The formula for rectangular shapes is (conjecturally) generalized to arbitrary shapes. Update.**Pairs of noncrossing free Dyck paths and noncrossing partitions (with W. Y. C. Chen (陈永川), S. X. M. Pang (庞兴梅), and Ellen X. Y. Qu (曲晓英)) (8 pages, pdf file)***Discrete Math.***309**(2009), 2834--2838.

A bijecton between pairs of noncrossing free Dyck paths of length 2*n*and noncrossing partitions of [2*n*+1] with*n*+1 blocks, based on the bijection between vacillating tableaux and set partitions.**Promotion and evacuation (24 pages, pdf file, submitted version)***Electronic J. Combinatorics*, volume 15(2) (2008-2009), R9.

A survey of the theory of promotion and evacuation developed originally by Schützenberger, with a discussion of some generalizations.**Some combinatorial properties of hook lengths, contents, and parts of partitions (20 pages, pdf file, version of 25 March 2009)***Ramanujan Journal***23**(2010), 91-105.

Proof of a generalization of a conjecture of Guoniu Han involving the polynomiality of certain sums over partitions.**Some Hecke algebra products and corresponding random walks (with Rosena R. X. Du (杜若霞)) (9 pages, pdf file, version of 5 September 2008)***J. Algebraic Combinatorics***31**(2010), 159-168.

Explicit expansion of some products in the Hecke algebra of the symmetric group in terms of the standard basis {*T*}, and an interpretation of this expansion in terms of a random walk on the symmetric group._{w}**Two enumerative results on cycles of permutations (11 pages, pdf file, version of 15 April 2009)***European J. Combinatorics***32**(2011), 937-943.

Proof that if two*n*-cycles*u*,*v*are chosen uniformly at random from the symmetric group*S*for_{n}*n*odd, then the probability that 1 and 2 appear in the same cycle of the product*uv*is 1/2. Another result concerns a formula for a generating function that counts the number of cycles of the product of the cycle (1,2,...,*n*) with a permutation of fixed cycle type. It is also shown that every zero of this generating function has real part 0.**Polynomial coefficient enumeration (with Tewodros Amdeberhan) (36 pages, pdf file, version of 21 November 2008)**

Various results on the number of nonzero coefficients of a polynomial over the rationals or the number of coefficients with a given value of a polynomial over a finite field.**A survey of alternating permutations (31 pages, pdf file, version of 8 March 2010).***Contemporary Mathematics***531**(2010), 165-196.

This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the*cd*-index of the symmetric group.**Permutations (39 pages, pdf file, version of 1 November 2009)**

A survey of three topics on permutations: increasing and decreasing subsequences, alternating permutations, and reduced decompositions. These are notes for the 2010 AMS Colloquium Lectures.**Formulae for Askey-Wilson moments and enumeration of staircase tableaux (with S. Corteel, D. Stanton, and L. Williams) (28 pages, pdf file, version of 13 August 2010)**

A direct combinatorial formula, arising from the asymmetric exclusion process on a one-dimensional lattice, is given for the moments of Askey-Wilson polynomials. In the process, new combinatorial properties of staircase tableaux are derived.**Refining the Stern diatomic sequence (with H. Wilf) (10 pages, pdf file, version of 6 September 2010)**

A refinement of the celebrated Stern diatomic sequence, in which we consider the number of ways to write*n*as a sum of powers of 2, where each power is used at most twice, and where*k*powers are used exactly twice.**Note.**Herb Wilf (1931-2012) was excited that we finally would write a joint paper, but then he discovered that most of the results had recently appeared on the arXiv. Although our paper therefore contains nothing new, I will keep it posted here in memory of one of the founders of modern enumerative/algebraic combinatorics.**An equivalence relation on the symmetric group and multiplicity-free flag***h*-vectors (21 pages, pdf file, version of 25 January 2012)*J. Combinatorics***3**(2012), 277-298.

Properties of the equivalence relation on*S*generated by the interchange of adjacent consecutive integers. The equivalence class of the identity element can be identified with the set of linear extensions of a certain poset, leading to a classification and enumeration of graded posets whose flag_{n}*h*-vector takes on only the values 0, -1, 1.**Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs (with Beifang Chen) (pdf file, 364.7 KB)***Graphs and Combinatorics***28**(2012), 751-779.

An investigation of modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes.**Two remarks on skew tableaux***Electronic J. Combinatorics*, vol. 18(2) (2011-12), P16.

One result on the number of skew Young tableaux of a fixed shape, and a second result on evaluating a skew Schur function at (1, 1/2^{2k}, 1/3^{2k}, ...) for*k*=1,2,3.**Separation probabilities for products of permutations (with Olivier Bernardi, Rosena R. X. Du (杜若霞), and Alejandro H. Morales)***Combinatorics, Probability and Computing***23**(2014), 201-222.

Results on mixing properties of permutations obtained as a product of two uniformly random permutations of fixed types.**On the rank function of a differential poset (with Fabrizio Zanello),***Electronic J. Combinatorics*, vol. 19(2) (2012), P13.

Various properties of the rank function of an*r*-differential poset, including a super-polynomial lower bound on the number of elements of a given rank*n*.**Some congruence properties of symmetric group character values**

Not for publication.

A simplified proof of a result of Macdonald on the number of partitions λ of*n*for which the number of standard Young tableaux of shape λ is not divisible by some fixed prime. Some generalizations and open problems are briefly mentioned.**Counting conjugacy classes of elements of finite order in Lie groups (with Tamar Friedmann)***Europ. J. Combinatorics***36**(2014), 86-96.

Explicit formulas for the number of conjugacy classes of elements of finite order in unitary, symplectic, and orthogonal Lie groups, as well as the number of such conjugacy classes whose elements have a specified number of distinct eigenvalues.**The string landscape: on formulas for counting vacua (with Tamar Friedmann)***Nuclear Physics B***869**(2013), 74-88.

Formulas for counting certain classes of vacua in the string/M theory landscape, in the context of the moduli space of M-theory compactifications on singular manifolds with*G*_{2}holomony.**Unimodality of partitions with distinct parts inside Ferrers shapes (with Fabrizio Zanello)***Europ. J. Combinatorics***49**(2015), 194-202.

Investigation of the unimodality of the rank-generating function of the poset of partitions contained inside a given shifted Ferrers shape. Includes the asymptotic behavior of the coefficients of the*q*-binomial coefficient [*a*+*k*,*k*] for fixed*k*.**Some asymptotic results on***q*-binomial coefficients (with Fabrizio Zanello)*Ann. Comb.***20**(2016), 623-634.

We look at the asymptotic behavior of the coefficients of the*q*-binomial coefficient (*a*+*k*\choose*k)*_{q}for fixed*k*.**Valid orderings of real hyperplane arrangments***Discrete and Computational Geometry***53**(2015), 951-964.

Addendum

What are the possible orders in which a line through a fixed point*p*in*R*can intersect the hyperplanes of a (finite) real hyperplane arrangement? Some aspects of this question are considered, in particular, a connection with the Dilworth truncation of a matroid.^{d}**The lecture hall parallelopiped (with Fu Liu (刘拂))***Annals of Combinatorics***18**(2014), 473-488.

We obtain some alternate proofs and generalizations of work of Savage et al. on-lecture hall polytopes, which are closely related to the lecture hall partitions of Eriksson and Bousquet-Mélou.*s***Smith normal form of a multivariate matrix associated with partitions (with Christine Bessenrodt).***Journal of Algebraic Combinatorics***41**(2015), 73-82.

Generalization of a result of Carlitz-Roselle-Scoville on certain combinatorially defined matrices whose determinant equals 1. We refine the matrix entries so that they are multivariate polynomials, and we compute not just the determinant, but more strongly the Smith normal form of the matrices.**How the Upper Bound Conjecture was proved***Annals of Combinatorics***18**(2014), 533-539.

Discussion of how I proved the Upper Bound Conjecture for Spheres.**A formula for the specialization of skew Schur functions (with Xiaomei Chen (陈小米))***Annals of Combinatorics***20**(2016), 539-548.

We give a formula for a skew Schur function evaluated at (1,*q*,*q*^{2}, ...), generalizing a result of Okounkov and Olshanskii about the number of standard Young tableaux of a skew shape.**The Catalan case of Armstrong's conjecture on simultaneous core partitions (with Fabrizio Zanello)***SIAM J. Discrete Math.***29**(2015), 658-666.

Drew Armstrong conjectured a formula for the average size of a (*p*,*q*)-core when*p*and*q*are coprime. We prove this conjecture in the case*q*=*p*+1.**A distributive lattice connected with arithmetic progressions of length three (with Fu Liu (刘拂))***Ramanujan J.***(2015), 203-226.**

Proof of two enumerative conjectures of Noam Elkies arising from a problem contributed by Ron Graham to the Numberplay subblog of the*New York Times*Wordplay blog.**The Smith normal form of a matrix associated with Young's lattice (with Tommy Wuxing Cai (蔡 吴 兴))***Proc. Amer. Math. Soc.***143**(2015), 4695-4703.

Proof of a conjecture of Miller and Reiner on the Smith normal form of the operation (*d*/*dp*_{1})*p*_{1}operating on homogeneous symmetric functions of degree*n*.**Supersolvability and freeness for ψ-graphical arrangements (with Lili Mu (牟丽丽))***Discrete and Computational Geometry*,**53**(2015), 965-970.

A sequel to Valid orderings of real hyperplane arrangments. A conjectured characterization of supersolvable ψ-graphical arrangements and some properties of free ψ-graphical arrangements are proved.**The Smith normal form distribution of a random integer matrix (with Yinghui Wang (王颖慧))**

Let*k*be a positive integer. We consider the distribution of the Smith normal form of*m*×*n*integer matrices with entries between -*k*and*k*in the limit as*k*→∞.**A refined enumeration of hex trees and related polynomials (with Hana Kim)***Europ. J. Combinatorics*, to appear.

A refined enumeration of a certain class of trees, and some properties of the zeros of some related polynomials.**The Smith normal form of a specialized Jacobi-Trudi matrix**

We determine the Smith normal form over the ring*Q*[*n*] of the Jacobi-Matrix for the Schur function*s*_{λ}, under the specialization*x*=1 for 1≤_{i}*i*≤*n*and*x*=0 for_{i}*i*>*n*. A*q*-analogue is also given.**Update.**The proof of Theorem 1.1 is not quite correct. For more details see Section 2 of this paper by Gao, Xie, and Yang.**New examples of period collapse (with Dan Cristofaro-Gardiner and Teresa Xueshan Li (李雪珊))**

Some examples are given of convex polytopes whose Ehrhart functions are quasipolynomials with smaller period than predicted by their vertex coordinates. In particular, some*irrational*polytopes have polynomial Ehrhart functions.**Smith normal form in combinatorics***J. Combinatorial Theory (A)***144**(2016), 476-495.

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A survey of some connections between Smith normal form and combinatorics.**Some aspects of (***r,k*)-parking functions (with Yinghui Wang (王颖慧))

(*r,k*)-parking functions are a generalization of ordinary parking functions (the case*r*=*k*=1). We discuss numerous properties of these parking functions and relationships among them, mostly related to a symmetric function associated to an (*r,k*)-parking function.**Some Schubert shenanigans**

A conjectured determinant evaluation related to Schubert polynomials which implies that the weak order on the symmetric group*S*is strongly Sperner._{n}**Update.**Conjecture 4.1 was first proved by Anna Wiegandt and later by Greta Panova. The lower bound 1/4 near the bottom of page 7 was improved to about .251 by Morales, Pak, and Panova on page 25 of this paper.

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