May 2016:
Here is an
article
at the College of Arts & Science
about a Mathmagic talk I gave to the math majors.
See also the two Vimeo
links
one
and
two.
It also
appeared
at UKNow.
Here is the first part of my lecture
"The Mathematics of Juggling"
given April 8, 2009,
during Geek Week 2009 at the University of Kentucky,
together with my fearless
graduate student
Eric Clark.
The
five parts
can be found
here.
On April 7, 2010, Eric Clark and I gave
the juggling lecture again.
See the article
Juggling the Equation
in the
The Kentucky Kernel.
Note however that Katie Hume estimated the size of the audience
to be 120.
On posets and Hopf algebras,
Adv. Math.
119 (1996), 1--25.
pdf
k-Eulerian posets,
Order.
18 (2001), 227--236.
pdf
Homology of Newtonian coalgebras,
European J. Combin.
23 (2002), 919--927,
with M. Readdy.
pdf
The antipode of the noncrossing partition lattice,
Adv. in Appl. Math.
110 (2019), 76--85,
with A. Happ.
See also papers listed under the heading
Flag vectors of polytopes.
Combinatorial topology
Generalizations of Baxter's theorem and cubical homology,
J. Combin. Theory Ser. A
69 (1995), 233--287,
with G. Hetyei.
Non-constructible complexes and the bridge index,
European J. Combin.
22 (2001), 475--491,
with M. Hachimori.
pdf
The topology of the independence complex,
European J. Combin.
27 (2006), 906--923,
with G. Hetyei.
pdf
The complex of non-crossing diagonals of a polygon,
J. Combin. Theory Ser. A
117 (2010), 642--649,
with B. Braun.
pdf
The Frobenius complex,
Ann. Comb.
16 (2012), 215--232,
with E. Clark.
The topology of restricted partition posets,
J. Algebraic Combin.
37 (2013), 643--666,
with J. Jung.
pdf
The cubical matching complex,
Ann. Comb.
18 (2014), 75--81.
The van der Waerden complex,
J. Number Theory,
172 (2017), 287--300,
with L. Govindaiah, P. S. Park and M. Readdy.
pdf
Filters in the partition lattice,
J. Algebraic Combin.
47 (2018), 403--439,
with D. Hedmark.
pdf
Enumerative combinatorics and partially ordered sets
Schröder parenthesizations and chordates,
J. Combin. Theory Ser. A
67 (1994), 127--139,
with M. Méndez.
A bijective proof of infinite variated Good's inversion,
Adv. Math.
103 (1994), 221--259,
with M. Méndez.
pdf
Juggling and applications to q-analogues,
Discrete Math.
157 (1996), 107--125,
with M. Readdy.
pdf
Sheffer posets and r-signed permutations,
Ann. Sci. Math. Québec
19 (1995), 173--196,
with M. Readdy.
pdf
The r-cubical lattice and
a generalization of the cd-index,
European J. Combin.
17 (1996), 709--725,
with M. Readdy.
pdf
The Hankel determinant of exponential polynomials,
Amer. Math. Monthly
107 (2000), 557--560.
pdf
A bijective answer to a question of Zvonkin,
Ann. Comb.
4 (2000), 195--197.
pdf
Determinants involving q-Stirling numbers,
Adv. in Appl. Math.
31 (2003), 630--642.
pdf
Enumerative properties of Ferrers graphs,
Discrete Comput. Geom.
special issue in honor of Louis J. Billera,
32 (2004), 481--492,
with S. van Willigenburg.
pdf
Classification of the factorial functions of
Eulerian binomial and Sheffer posets,
J. Combin. Theory Ser. A,
114 (2007), 339--359.
with M. Readdy.
pdf
The Möbius function of partitions with
restricted block sizes,
Adv. in Appl. Math.
39 (2007), 283--292,
with M. Readdy.
pdf
Exponential Dowling structures,
European J. Combin.
30 (2009), 311--326,
with M. Readdy.
pdf
Explicit expressions for the
extremal excedance set statistics,
European J. Combin.
31 (2010), 270--279,
with E. Clark.
The excedance algebra,
Discrete Math.
313 (2013), 1429--1435,
with E. Clark.
A sign-reversing involution for an extension of
Torelli's Pfaffian identity,
Discrete Math.,
332 (2014), 69--74,
with N. B. Fox.
pdf
A poset view of the major index,
Adv. in Appl. Math.,
62 (2015), 1--14,
with M. Readdy.
pdf
The Gaussian coefficient revisited,
J. Integer Seq.,
19 (2016), Art. 16.7.8,
with M. Readdy.
pdf
Parking cars of different sizes,
Amer. Math. Monthly,
123 (2016), 1045--1048,
with A. Happ.
pdf
A restricted growth word approach to partitions
with odd/even size blocks,
J. Integer Seq.,
20 (2017), Art. 17.5.5,
with D. Hedmark and C. Hettle.
pdf
Parking cars after a trailer,
Australas. J. Combin.,
70 (2018), 402--406,
with A. Happ.
pdf
q-Stirling identities revisited,
Electron J. Combin.,
25 (2018), 18 pp.
with Y. Cai and M. Readdy.
pdf
Box polynomials and the excedance matrix,
with A. Happ, D. Hedmark and C. Hettle.
pdf
Catalan-Spitzer Permutations,
Enumerative Combinatorics and Applications,
4 (2024), S2R15,
with G. Hetyei and M. Readdy.
pdf
On the structure of the d-indivisible noncrossing partition posets,
with G. Hetyei.
pdf
Flag vectors of polytopes
The cd-index of zonotopes and arrangements,
in Mathematical essays in honor of Gian-Carlo Rota
(B. E. Sagan and R. P. Stanley, eds.),
Birkhäuser,
Boston, 1998, pages 23--40
with L. Billera and M. Readdy.
pdf
The c-2d-index of oriented matroids,
J. Combin. Theory Ser. A
80 (1997), 79--105,
with L. Billera and M. Readdy.
pdf
Coproducts and the cd-index,
J. Algebraic Combin.
8 (1998), 273--299,
with M. Readdy.
pdf
On flag vectors, the Dowling lattice and braid arrangements,
Discrete Comput. Geom.
21 (1999), 389--403,
with M. Readdy.
pdf
Cutting polytopes and flag f-vectors,
Discrete Comput. Geom.
23 (2000), 261--271,
with
D. Johnston,
R. Rajagopalan
and
M. Readdy.
pdf
Flags and shellings of Eulerian cubical posets,
Ann. Comb.
4 (2000), 199--226,
with G. Hetyei.
pdf
Monotonicity of the cd-index for polytopes,
Math. Z.
233 (2000), 421--441,
with L. Billera.
pdf
The toric h-vectors of partially ordered sets,
Trans. Amer. Math. Soc.
352 (2000), 4515--4531,
with M. Bayer.
pdf
Inequalities for cd-indices of
joins and products of polytopes,
Combinatorica
23 (2003), 427--452,
with H. Fox.
pdf
Lifting inequalities for polytopes,
Adv. Math.
193 (2005), 205--222.
pdf
Inequalities for zonotopes,
in
Mathematical Sciences Research Institute Publication
on Combinatorial and Computational Geometry,
(J. E. Goodman, J. Pach and E. Welzl, eds.),
Cambridge University Press, Cambridge, England, 2005,
pages 277--286.
pdf
Flag vectors of posets
Decomposition theorem for the cd-index
of Gorenstein* posets,
J. Algebraic Combin.
26 (2007), 225--251,
with K. Karu.
pdf
The Tchebyshev transforms of the first and second kind,
Ann. Comb.
14 (2010), 211--244,
with M. Readdy.
pdf
On the non-existence of an R-labeling,
Order,
28 (2011), 437--442,
with M. Readdy.
pdf
Level Eulerian posets,
Graphs and Combinatorics,
28 (2013), 857--882,
with G. Hetyei and M. Readdy.
pdf
Balanced and Bruhat graphs,
Ann. Comb.,
24 (2020), 587--617,
with M. Readdy.
Generating functions for the cd-indices of simplices and cubes,
Enumerative Combinatorics and Applications,
4 (2024), S2R18.
pdf
Two classes of level Eulerian posets,
Discrete Math.,
347 (2024), 114127.
pdf
Flag enumeration on manifolds
Affine and toric arrangements,
Discrete Comput. Geom.
41 (2009), 481--512,
with M. Readdy and M. Slone,
pdf
Euler flag enumeration of Whitney stratified spaces,
Adv. Math.
268 (2015), 85--128,
with M. Goresky and M. Readdy,
pdf
Manifold arrangements,
J. Combin. Theory Ser. A,
125 (2014), 214--239,
with M. Readdy,
pdf
Prisms and pyramids of shelling components,
Ann. Comb.
22 (2018), 769--779.
Polytopes
The f-vector of the descent polytope,
Discrete Comput. Geom.
45 (2011), 410--424,
with D. Chebikin.
pdf
Simion's type B associahedron is a pulling triangulation of
the Legendre polytope,
Discrete Comput. Geom.
60 (2018), 98--114,
with G. Hetyei and M. Readdy.
pdf
A bijective answer to a question of Simion,
J. Integer Seq.,
22 (2019), Art. 19.1.2,
with G. Hetyei and M. Readdy.
pdf
The boustrophedon transform for descent polytopes,
Ann. Comb.
23 (2019), 67--72,
with Alex Happ.
Classification of uniform flag triangulations of the boundary
of the full root polytope of type A,
Acta Math. Hungar.
163 (2021), 462--511,
with G. Hetyei and M. Readdy.
pdf
Invariant theory
Apolarity and canonical forms for homogeneous polynomials,
European J. Combin.
14 (1993), 157--181,
with G.-C. Rota.
pdf
On apolarity and generic canonical forms,
J. Algebra
213 (1999), 167--194.
pdf
Canonical forms of two by two by two matrices,
J. Algebra
213 (1999), 195--224.
pdf
Hyperplane arrangements
On valuations, the characteristic polynomial
and complex subspace arrangements,
Adv. Math.
134 (1998), 32--42,
with M. Readdy.
pdf
The Dowling transform of subspace arrangements,
J. Combin. Theory Ser. A
91 (2000), 322--333,
with M. Readdy.
pdf
A geometric approach to acyclic orientations,
Order
26 (2009), 283--288,
with M. Slone.
pdf
Counting faces in the extended Shi arrangement,
Adv. in Appl. Math.
109 (2019), 55--64.
See also papers listed under the heading
Flag vectors of polytopes.
Permutation statistics
Mixed volumes and slices of the cube,
J. Combin. Theory Ser. A
81 (1998), 121--126,
with M. Readdy and E. Steingrímsson.
pdf
Maximizing the descent statistic,
Ann. Comb.
2 (1998), 111--129,
with S. Mahajan.
pdf
The excedance set of a permutation,
Adv. in Appl. Math.
24 (2000), 284--299,
with E. Steingrímsson.
pdf
A combinatorial proof of the log-concavity of the numbers of
permutations with k runs,
J. Combin. Theory Ser. A
90 (2000), 293--303,
with M. Bóna.
pdf
Yet another triangle for the Genocchi numbers,
European J. Combin.
21 (2000), 593--600,
with E. Steingrímsson.
pdf
The asymptotics of almost alternating permutations,
Adv. in Appl. Math.
28 (2002), 421--437.
pdf
Cyclotomic factors of the descent set polynomial,
J. Combin. Theory Ser. A
116 (2009), 247--264,
with D. Chebikin, P. Pylyavskyy and M. Readdy.
pdf
Excedances of affine permutations,
Adv. in Appl. Math.
special issue in honor of Dennis Stanton,
46 (2011), 175--191,
with E. Clark.
pdf
The descent set polynomial revisited,
European J. Combin.,
51 (2016), 47--68,
with N. B. Fox.
pdf
The signed descent set polynomial revisited,
Discrete Math.
339 (2016), 2263--2266,
with N. B. Fox.
On the powers of the descent set statistic,
Adv. in Appl. Math.
96 (2018), 1--17,
with A. Happ.
pdf
See also papers listed under the heading
Enumerative combinatorics and Partially ordered sets.
Analytic approaches to permutation enumeration
A probabilistic approach to the descent statistic,
J. Combin. Theory Ser. A
98 (2002), 150--162
with M. Levin and M. Readdy.
pdf
Asymptotics of the Euler number of bipartite graphs,
Adv. in Appl. Math.
44 (2010), 155--167,
with Y. Farjoun.
pdf
A spectral approach to consecutive pattern-avoiding permutations,
J. Comb.
2 (2011), 305--353,
with S. Kitaev and P. Perry.
pdf
Descent pattern avoidance,
Adv. in Appl. Math.
49 (2012), 375--390,
with J. Jung.
pdf
On the generating function for consecutively
weighted permutations,
European J. Combin.
41 (2014), 262--265.
pdf
Cyclically consecutive permutation avoidance,
SIAM J. Discrete Math.
30 (2016), 1385--1390.
pdf
The expectation of the Vandermonde product squared
for uniform random variables,
Adv. in Appl. Math.
118 (2020), 102030.
Combinatorics of Shimura varieties
Some combinatorial identities appearing in the calculation
of the cohomology of Siegel modular varieties,
Algebr. Comb.
2 (2019), 863--878,
with Sophie Morel and Margaret Readdy.
pdf
A generalization of combinatorial identities
for stable discrete series constants,
J. Comb. Algebra
6 (2022), 109--183,
with Sophie Morel and Margaret Readdy.
pdf
Geometry
Heron's formula from a Pythagoras-type theorem,
The Mathematical Gazette
68 (1984), 124--126.
Ehrhart-Macdonald reciprocity extended,
preprint 2005,
with M. Beck.
pdf
Hamiltonian cycles on Archimedean solids are twisting free,
Amer. Math. Monthly
121 (2014), 158--161.
pdf
Coxeter arrangements in three dimensions,
Beitr. Algebra Geom.
57 (2016), 1--7,
with C. Klivans and N. Reading.
pdf
The Wästlund partition in higher dimensions,
Amer. Math. Monthly
127 (2020), 175--178.
Sharing pizza in n dimensions,
Trans. Amer. Math. Soc.,
375 (2022), 5829--5857,
with S. Morel and M. Readdy.
pdf
Pizza and 2-structures,
Discrete Comput. Geom.,
70 (2023), 1221--1244,
with S. Morel and M. Readdy.
pdf
Conjectures for cutting pizza with Coxeter arrangements,
to appear in Exp. Math.
pdf
Graph Theory
The average reliability of a graph,
Discrete Appl. Math.
177 (2014), 19--33,
with J. Brown and D. Cox.
Number of cycles in the graph of 312-avoiding permutations,
J. Combin. Theory Ser. A
129 (2015), 1--18,
with S. Kitaev and E. Steingrímsson.
pdf
An unbiased marriage theorem,
Amer. Math. Monthly
122 (2015), 59.
The number of spanning trees of the Bruhat graph,
Adv. in Appl. Math.
125 (2021), 102150.
Bounding monochromatic triangles using squares,
Math. Magazine
94 (2021), 383--386.
Graph Theory in Higher Dimensions
Bounds on the number of compatible k-simplices
matching the orientation of the (k-1)-skeleton of a simplex,
Combinatorica,
41 (2021), 209--236,
with K. Chandrasekhar.
Material Science
A geometric formulation of the law of Aboav--Weaire
in two and three dimensions,
Journal of Physics A: Mathematical and Theoretical,
45 (2012), 065001, 1--17,
with M. Lazar and J. Mason.
A counting proof for when 2 is a quadratic residue,
Amer. Math. Monthly,
127 (2020) 750--751,
with K. Chandrasekhar and F. Beukers.
pdf
Combinatorial games
The blind bartender's problem,
J. Combin. Theory Ser. A
70 (1995), 249--266,
with C. Skinner.
pdf
Playing nim on a simplicial complex,
Electron J. Combin.
3 (1996), 33 pp.
with E. Steingrímsson.
pdf
Linear Algebra
A conceptual proof of Cramer's rule,
Math. Magazine
77 (2004), 308.
pdf
Magic
Decoding the Hamming code,
Math Horizons, special issue
on Codes, Cryptography and National Security,
13 (2006), April issue, pages 16--17.
pdf
Here is a
pdf file
with the seven cards and the base card.
Either print the file on a double sided
printer, or print the file and run it
through a copy machine to make it double sided.
Then cut out the cards with a pair of scissors.
Fano, Galois, Hamming and a card trick,
The College Mathematics Journal,
52 (2021), 274--280.
Miscellaneous
Advertise!
FOCUS
26 (2006), August/September issue, number 6, pages 28--29.
pdf