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Unpublished notes dated January, 1978, concerning sequences of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),
26202625.
Some connections between the three objects of the title are
investigated.

Hipparchus, Plutarch, Schröder and Hough (12 pages)
American Mathematical Monthly 104
(1997), 344350. pdf
A solution to an ancient combinatorial riddle. Much
more detailed and scholarly account by
F. Acerbi. A
followup by Susanne Bozien.

Graph colorings and related symmetric functions: ideas and
applications (28 pages)
Discrete Mathematics 193 (1998),
267286.
A sequel to "A symmetric function generalization of the chromatic
polynomial of a graph," Advances in Math.
111 (1995), 166194.

Hyperplane arrangements, parking functions and tree inversions
(14 pages)
in Mathematical Essays in Honor of GianCarlo Rota
(B. Sagan and
R. Stanley, eds.), Birkhäuser,
Boston/Basel/Berlin, 1998, pp. 359375.
Connections between the three objects of the title, and a
generalization involving kparking functions and rooted
kforests.

A qdeformation of a trivial symmetric group action
(with Phil Hanlon) (26 pages)
Transactions of the American Mathematical Society
350 (1998), 44454459.
Solutions to problems raised by Varchenko and Zagier concerning a
qdeformation of the element of the group algebra of the
symmetric group S_{n} equal to the sum of all
the elements of S_{n}.

Polygon dissections and standard Young tableaux (4 pages)
Journal of Combinatorial Theory, series A 76
(1996), 175177.
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^{n1d}).

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), 118133.
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),
544597.
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.

Flagsymmetry of the poset of shuffles and a local action of the
symmetric group (with Rodica Simion) (34 pages)
Discrete Math. 204 (1999), 369396.
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), 351363.
Kontsevich conjectured that the number of zeros over the field
F_{q} of a certain polynomial connected with the
spanning trees of a graph 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 MatrixTree 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),
347356 .
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. 295319.
A survey of problems and conjectures in algebraic combinatorics
related to showing that certain numbers are nonnegative. The three
main areas covered are (1) fvectors, (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), 603634.
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), 479491.
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_{n} (with David
Savitt) (7 pages)
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
ndimensional representation of
the symmetric group S_{n} is asymptotic to
n^{2}/2.
 Rodica Simion, January 18, 1955
 January 7, 2000 (5 pages)
Pi Mu Epsilon
Journal 11 (2000), 8386.
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),
5568.
A survey of recent work on (1) the saturation conjecture for
LittlewoodRichardson coefficients, (2) the n! and
(n+1)^{n1} conjectures, and (3) longest increasing
subsequences of permutations.
 On the enumeration of skew Young
tableaux (15 pages)
Advances in Applied Math. 30 (2003),
283294.
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),
349375.
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),
282284.
Reminiscences on the only joint paper I wrote with Rodica Simion.
 Some remarks on signbalanced and
majbalanced posets (30 pages, version of 14 January 2004)
Advances in Applied Math. 34 (2005),
880902.
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), 317340.
A continuation of "Recent
progress in algebraic combinatorics" above, with three
sections: (1) the Laurent phenomenon, (2) GromovWitten invariants and
toric Schur functions, and (3) toric hvectors and
intersection cohomology.
 The mathematical knight (with N. Elkies) (23
pages) (pdf)
The Mathematical Intelligencer 25, no. 1
(Winter 2003), 2234
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),
18611880.
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 superclass walk on
uppertriangular matrices (with E. AriasCastro
and P.
Diaconis)
(26 pages)
J. Algebra 278 (2004), 739765.
An analysis of a random walk on the group of n×n
uppertriangular 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),
163177.
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_{n} and
their double covers.
 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_{i} to be 1. Erratum.
 Coefficients and roots of
Ehrhart polynomials (with M. Beck, J. De Loera,
M. Develin, and J. Pfeifle)
Contemp. Math. 374 (2005), 1536.
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), 3243.
 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), 15551575.
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),
514525.
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),
133174. .
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),
193203.
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), 15551562.
 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), 675687.
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. 545579.
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), 436460..
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), 28342838.
A bijecton between pairs of noncrossing free Dyck paths of length
2n and noncrossing partitions of [2n+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) (20082009), 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),
91105.
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),
159168.
Explicit expansion of some products in the Hecke algebra of the
symmetric group in terms of the standard basis
{T_{w}}, and an interpretation of this expansion in
terms of a random walk on the symmetric group.
 Two
enumerative results on cycles of permutations (11 pages, pdf file,
version of 15 April 2009)
European J. Combinatorics 32 (2011),
937943.
Proof that if two ncycles
u, v are chosen uniformly at random from the
symmetric group S_{n} for 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),
165196.
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 cdindex 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 AskeyWilson 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 onedimensional lattice, is given for the
moments of AskeyWilson 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 (19312012) 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 multiplicityfree flag hvectors
(21 pages, pdf file, version of 25 January 2012)
J. Combinatorics 3
(2012), 277298.
Properties of the equivalence relation on S_{n}
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 hvector 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), 751779.
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) (201112), 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), 201222.
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 rdifferential poset, including a superpolynomial 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),
8696.
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), 7488.
Formulas for counting certain classes of vacua in the string/M
theory landscape, in the context of the moduli space of Mtheory
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),
194202.
Investigation of the unimodality of the rankgenerating function
of the poset of partitions contained inside a given shifted Ferrers
shape. Includes the asymptotic behavior of the coefficients of
the qbinomial coefficient [a+k,k]
for fixed k.
 Some asymptotic results on qbinomial
coefficients (with Fabrizio Zanello)
Ann. Comb. 20 (2016), 623634.
We look at the asymptotic behavior of the coefficients of
the qbinomial coefficient (a+k
\choose k)_{q} for fixed k.
 Valid orderings of real hyperplane
arrangments
Discrete and Computational Geometry 53
(2015), 951964.
Addendum
What are the possible orders in which a line through a fixed
point p in R^{d} 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.
 The
lecture hall parallelopiped (with Fu Liu (刘拂))
Annals of Combinatorics 18 (2014), 473488.
We obtain some alternate proofs and generalizations of work of
Savage et al. on slecture hall polytopes,
which are closely related to the lecture hall partitions of Eriksson
and BousquetMélou.

Smith normal form of a multivariate matrix associated with
partitions (with Christine Bessenrodt).
Journal of Algebraic Combinatorics 41
(2015), 7382.
Generalization of a result of CarlitzRoselleScoville 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), 533539.
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), 539548.
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),
658666.
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), 203226.
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),
46954703.
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 ψgraphical arrangements (with Lili Mu
(牟丽丽))
Discrete and Computational
Geometry, 53 (2015), 965970.
A sequel to Valid orderings of real hyperplane
arrangments. A conjectured characterization of supersolvable
ψgraphical arrangements and some properties of free
ψgraphical 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 JacobiTrudi matrix
We determine the Smith normal form over the
ring Q[n] of the JacobiMatrix for the Schur
function s_{λ}, under the
specialization x_{i}=1 for
1≤i≤n and x_{i}=0
for i>n. A qanalogue 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 CristofaroGardiner
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),
476495.
Published
version (accessible only to those with suitable entitlements)
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_{n} is strongly Sperner.
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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