Richard Ehrenborg

Here is my current vita.

Co-organizer of the KOI Combinatorics Lectures:

• Fall 2025 meeting at the University of Kentucky.
• Fall 2024 meeting at Indiana University.
• Fall 2023 meeting at Ohio State.
• Spring 2023 meeting at the University of Kentucky.

November 2021: Nothing Up His Sleeve: Mathematician Creates Card Trick to Illustrate Key Concepts in Information Correction.

February 2017: The Double Möbius Strip and cutting it into two.

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.

Upcoming Invitations, Conferences and Workshops

Seminar. (September 2025)

NORCOM 2025 and Einar fest (2025). (June 2024)

Cosmic Lunch, Department of Physics and Astronomy, University of Kentucky. (April 2025)

Seminar, University of Miami. (March 2025)

Oliver Club at Cornell University (October 2023).

FPSAC at Davis (July 2023).

KOI Combinatorics Lectures (April 2022).

Rutgers Experimental Mathematics Seminar (March 2022).

LaCIM, UQAM (October 2022).

2006, 2007, 2008, 2009. 2010, 2011, 2012, 2013. 2014, 2015, 2016, 2017, 2018, 2019 and 2021.

Publications

• Coalgebras and Hopf algebras
  1. On posets and Hopf algebras, Adv. Math. 119 (1996), 1--25. pdf
  2. k-Eulerian posets, Order. 18 (2001), 227--236. pdf
  3. Homology of Newtonian coalgebras, European J. Combin. 23 (2002), 919--927, with M. Readdy. pdf
  4. 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
  5. Generalizations of Baxter's theorem and cubical homology, J. Combin. Theory Ser. A 69 (1995), 233--287, with G. Hetyei.
  6. Non-constructible complexes and the bridge index, European J. Combin. 22 (2001), 475--491, with M. Hachimori. pdf
  7. The topology of the independence complex, European J. Combin. 27 (2006), 906--923, with G. Hetyei. pdf
  8. The complex of non-crossing diagonals of a polygon, J. Combin. Theory Ser. A 117 (2010), 642--649, with B. Braun. pdf
  9. The Frobenius complex, Ann. Comb. 16 (2012), 215--232, with E. Clark.
  10. The topology of restricted partition posets, J. Algebraic Combin. 37 (2013), 643--666, with J. Jung. pdf
  11. The cubical matching complex, Ann. Comb. 18 (2014), 75--81.
  12. The van der Waerden complex, J. Number Theory, 172 (2017), 287--300, with L. Govindaiah, P. S. Park and M. Readdy. pdf
  13. Filters in the partition lattice, J. Algebraic Combin. 47 (2018), 403--439, with D. Hedmark. pdf

• Enumerative combinatorics and partially ordered sets

  14. Schröder parenthesizations and chordates, J. Combin. Theory Ser. A 67 (1994), 127--139, with M. Méndez.
  15. A bijective proof of infinite variated Good's inversion, Adv. Math. 103 (1994), 221--259, with M. Méndez. pdf
  16. Juggling and applications to q-analogues, Discrete Math. 157 (1996), 107--125, with M. Readdy. pdf
  17. Sheffer posets and r-signed permutations, Ann. Sci. Math. Québec 19 (1995), 173--196, with M. Readdy. pdf
  18. The r-cubical lattice and a generalization of the cd-index, European J. Combin. 17 (1996), 709--725, with M. Readdy. pdf
  19. The Hankel determinant of exponential polynomials, Amer. Math. Monthly 107 (2000), 557--560. pdf
  20. A bijective answer to a question of Zvonkin, Ann. Comb. 4 (2000), 195--197. pdf
  21. Determinants involving q-Stirling numbers, Adv. in Appl. Math. 31 (2003), 630--642. pdf
  22. 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
  23. Classification of the factorial functions of Eulerian binomial and Sheffer posets, J. Combin. Theory Ser. A, 114 (2007), 339--359. with M. Readdy. pdf
  24. The Möbius function of partitions with restricted block sizes, Adv. in Appl. Math. 39 (2007), 283--292, with M. Readdy. pdf
  25. Exponential Dowling structures, European J. Combin. 30 (2009), 311--326, with M. Readdy. pdf
  26. Explicit expressions for the extremal excedance set statistics, European J. Combin. 31 (2010), 270--279, with E. Clark.
  27. The excedance algebra, Discrete Math. 313 (2013), 1429--1435, with E. Clark.
  28. A sign-reversing involution for an extension of Torelli's Pfaffian identity, Discrete Math., 332 (2014), 69--74, with N. B. Fox. pdf
  29. A poset view of the major index, Adv. in Appl. Math., 62 (2015), 1--14, with M. Readdy. pdf
  30. The Gaussian coefficient revisited, J. Integer Seq., 19 (2016), Art. 16.7.8, with M. Readdy. pdf
  31. Parking cars of different sizes, Amer. Math. Monthly, 123 (2016), 1045--1048, with A. Happ. pdf
  32. 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
  33. Parking cars after a trailer, Australas. J. Combin., 70 (2018), 402--406, with A. Happ. pdf
  34. q-Stirling identities revisited, Electron J. Combin., 25 (2018), 18 pp. with Y. Cai and M. Readdy. pdf
  35. Box polynomials and the excedance matrix, with A. Happ, D. Hedmark and C. Hettle. pdf
  36. Catalan-Spitzer Permutations, Enumerative Combinatorics and Applications, 4 (2024), S2R15, with G. Hetyei and M. Readdy. pdf
  37. Cyclotomic enumeration of polynomials, Discrete Math., 349 (2026), 114688. pdf
  38. On the structure of the d-indivisible noncrossing partition posets, with G. Hetyei. pdf

• Flag vectors of polytopes
  39. 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
  40. The c-2d-index of oriented matroids, J. Combin. Theory Ser. A 80 (1997), 79--105, with L. Billera and M. Readdy. pdf
  41. Coproducts and the cd-index, J. Algebraic Combin. 8 (1998), 273--299, with M. Readdy. pdf
  42. On flag vectors, the Dowling lattice and braid arrangements, Discrete Comput. Geom. 21 (1999), 389--403, with M. Readdy. pdf
  43. Cutting polytopes and flag f-vectors, Discrete Comput. Geom. 23 (2000), 261--271, with D. Johnston, R. Rajagopalan and M. Readdy. pdf
  44. Flags and shellings of Eulerian cubical posets, Ann. Comb. 4 (2000), 199--226, with G. Hetyei. pdf
  45. Monotonicity of the cd-index for polytopes, Math. Z. 233 (2000), 421--441, with L. Billera. pdf
  46. The toric h-vectors of partially ordered sets, Trans. Amer. Math. Soc. 352 (2000), 4515--4531, with M. Bayer. pdf
  47. Inequalities for cd-indices of joins and products of polytopes, Combinatorica 23 (2003), 427--452, with H. Fox. pdf
  48. Lifting inequalities for polytopes, Adv. Math. 193 (2005), 205--222. pdf
  49. 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
  50. Decomposition theorem for the cd-index of Gorenstein* posets, J. Algebraic Combin. 26 (2007), 225--251, with K. Karu. pdf
  51. The Tchebyshev transforms of the first and second kind, Ann. Comb. 14 (2010), 211--244, with M. Readdy. pdf
  52. On the non-existence of an R-labeling, Order, 28 (2011), 437--442, with M. Readdy. pdf
  53. Level Eulerian posets, Graphs and Combinatorics, 28 (2013), 857--882, with G. Hetyei and M. Readdy. pdf
  54. Balanced and Bruhat graphs, Ann. Comb., 24 (2020), 587--617, with M. Readdy.
  55. Generating functions for the cd-indices of simplices and cubes, Enumerative Combinatorics and Applications, 4 (2024), S2R18. pdf
  56. Two classes of level Eulerian posets, Discrete Math., 347 (2024), 114127. pdf

• Flag enumeration on manifolds
  57. Affine and toric arrangements, Discrete Comput. Geom. 41 (2009), 481--512, with M. Readdy and M. Slone, pdf
  58. Euler flag enumeration of Whitney stratified spaces, Adv. Math. 268 (2015), 85--128, with M. Goresky and M. Readdy, pdf
  59. Manifold arrangements, J. Combin. Theory Ser. A, 125 (2014), 214--239, with M. Readdy, pdf
  60. Prisms and pyramids of shelling components, Ann. Comb. 22 (2018), 769--779.

• Polytopes
  61. The f-vector of the descent polytope, Discrete Comput. Geom. 45 (2011), 410--424, with D. Chebikin. pdf
  62. 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
  63. A bijective answer to a question of Simion, J. Integer Seq., 22 (2019), Art. 19.1.2, with G. Hetyei and M. Readdy. pdf
  64. The boustrophedon transform for descent polytopes, Ann. Comb. 23 (2019), 67--72, with Alex Happ.
  65. 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
  66. Apolarity and canonical forms for homogeneous polynomials, European J. Combin. 14 (1993), 157--181, with G.-C. Rota. pdf
  67. On apolarity and generic canonical forms, J. Algebra 213 (1999), 167--194. pdf
  68. Canonical forms of two by two by two matrices, J. Algebra 213 (1999), 195--224. pdf

• Hyperplane arrangements
  69. On valuations, the characteristic polynomial and complex subspace arrangements, Adv. Math. 134 (1998), 32--42, with M. Readdy. pdf
  70. The Dowling transform of subspace arrangements, J. Combin. Theory Ser. A 91 (2000), 322--333, with M. Readdy. pdf
  71. A geometric approach to acyclic orientations, Order 26 (2009), 283--288, with M. Slone. pdf
  72. 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
  73. Mixed volumes and slices of the cube, J. Combin. Theory Ser. A 81 (1998), 121--126, with M. Readdy and E. Steingrímsson. pdf
  74. Maximizing the descent statistic, Ann. Comb. 2 (1998), 111--129, with S. Mahajan. pdf
  75. The excedance set of a permutation, Adv. in Appl. Math. 24 (2000), 284--299, with E. Steingrímsson. pdf
  76. 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
  77. Yet another triangle for the Genocchi numbers, European J. Combin. 21 (2000), 593--600, with E. Steingrímsson. pdf
  78. The asymptotics of almost alternating permutations, Adv. in Appl. Math. 28 (2002), 421--437. pdf
  79. 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
  80. Excedances of affine permutations, Adv. in Appl. Math. special issue in honor of Dennis Stanton, 46 (2011), 175--191, with E. Clark. pdf
  81. The descent set polynomial revisited, European J. Combin., 51 (2016), 47--68, with N. B. Fox. pdf
  82. The signed descent set polynomial revisited, Discrete Math. 339 (2016), 2263--2266, with N. B. Fox.
  83. On the powers of the descent set statistic, Adv. in Appl. Math. 96 (2018), 1--17, with A. Happ. pdf
  84. A determinant for cyclically consecutive 123-avoiding permutations, J. Comb. 15 (2024), 307--317.

  See also papers listed under the heading Enumerative combinatorics and Partially ordered sets.

• Analytic approaches to permutation enumeration
  85. A probabilistic approach to the descent statistic, J. Combin. Theory Ser. A 98 (2002), 150--162 with M. Levin and M. Readdy. pdf
  86. Asymptotics of the Euler number of bipartite graphs, Adv. in Appl. Math. 44 (2010), 155--167, with Y. Farjoun. pdf
  87. A spectral approach to consecutive pattern-avoiding permutations, J. Comb. 2 (2011), 305--353, with S. Kitaev and P. Perry. pdf
  88. Descent pattern avoidance, Adv. in Appl. Math. 49 (2012), 375--390, with J. Jung. pdf
  89. On the generating function for consecutively weighted permutations, European J. Combin. 41 (2014), 262--265. pdf
  90. Cyclically consecutive permutation avoidance, SIAM J. Discrete Math. 30 (2016), 1385--1390. pdf
  91. The expectation of the Vandermonde product squared for uniform random variables, Adv. in Appl. Math. 118 (2020), 102030.

• Combinatorics of Shimura varieties
  92. 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
  93. 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
  94. Heron's formula from a Pythagoras-type theorem, The Mathematical Gazette 68 (1984), 124--126.
  95. Ehrhart-Macdonald reciprocity extended, preprint 2005, with M. Beck. pdf
  96. Hamiltonian cycles on Archimedean solids are twisting free, Amer. Math. Monthly 121 (2014), 158--161. pdf
      • See also youtube and the link for a dodecahedral calendar demonstrating this result.
  97. Coxeter arrangements in three dimensions, Beitr. Algebra Geom. 57 (2016), 1--7, with C. Klivans and N. Reading. pdf
  98. The Wästlund partition in higher dimensions, Amer. Math. Monthly 127 (2020), 175--178.
  99. Sharing pizza in n dimensions, Trans. Amer. Math. Soc., 375 (2022), 5829--5857, with S. Morel and M. Readdy. pdf
  100. Pizza and 2-structures, Discrete Comput. Geom., 70 (2023), 1221--1244, with S. Morel and M. Readdy. pdf
  101. Sharing pizza among a two power number of friends, Enumer. Comb. Appl., 5:3 (2025), Article S2R23, with C. Ehrenborg, D. Ehrenborg and M. Ehrenborg. pdf
  102. Conjectures for cutting pizza with Coxeter arrangements, Exp. Math., 34 (2025), 422--431. pdf

• Graph Theory
  103. The average reliability of a graph, Discrete Appl. Math. 177 (2014), 19--33, with J. Brown and D. Cox.
  104. 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
  105. An unbiased marriage theorem, Amer. Math. Monthly 122 (2015), 59.
  106. The number of spanning trees of the Bruhat graph, Adv. in Appl. Math. 125 (2021), 102150.
  107. Bounding monochromatic triangles using squares, Math. Magazine 94 (2021), 383--386.

• Graph Theory in Higher Dimensions
  108. 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
  109. 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.
       
       • See also The apparent ordering of foams.

• Number Theory
  110. The Frobenius coin problem -- A cylindrical approach, Math. Intelligencer, 42 (2020) 78--79. https://doi.org/10.1007/s00283-019-09948-x
  111. 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
  112. The blind bartender's problem, J. Combin. Theory Ser. A 70 (1995), 249--266, with C. Skinner. pdf
  113. Playing nim on a simplicial complex, Electron J. Combin. 3 (1996), 33 pp. with E. Steingrímsson. pdf

• Linear Algebra
  114. A conceptual proof of Cramer's rule, Math. Magazine 77 (2004), 308. pdf

• Magic
  115. 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.
  116. Fano, Galois, Hamming and a card trick, The College Mathematics Journal, 52 (2021), 274--280.

• Proofs without words
  117. Game of Parts, The College Mathematics Journal, 55 (2024), page 428. See the front cover.

• Miscellaneous
  118. Advertise! FOCUS 26 (2006), August/September issue, number 6, pages 28--29. pdf

I have had the pleasure and honor to work with the following people: Margaret Bayer, Matthias Beck, Frits Beukers, Louis Billera, Miklós Bóna, Benjamin Braun, Jason I. Brown, Yue Cai, Karthik Chandrasekhar, Denis Chebikin, Eric Clark, Danielle Cox, Christian Ehrenborg, Didrik Ehrenborg, Michaël Ehrenborg, Yossi Farjoun, Bradley Fox, Harold Fox, Mark Goresky, Likith Govindaiah, Masahiro Hachimori, Alex Happ, Dustin Hedmark, Cyrus Hettle, Gábor Hetyei, Dan Johnston, JiYoon Jung, Kalle Karu, Sergey Kitaev, Carly Klivans, Menachem Lazar, Michael Levin, Swapneel Mahajan, Jeremy Mason, Miguel Méndez, Sophie Morel, Peter S. Park, Peter Perry, Pavlo Pylyavskyy, Rajmohan Rajagopalan, Margaret Readdy, Nathan Reading, Gian-Carlo Rota, Chris Skinner, Michael Slone, Einar Steingrímsson and Stephanie van Willigenburg.

See also at my listing at MathSciNet.

I co-organized the Kentucky Math Carnival April 2023.

Margaret Readdy and I held the 2019 Julia Robinson Math Festival at the Central Public Library in Lexington (November 2019). See pictures.

We coorganizer the first and second Julia Robinson Math Festival in Kentucky, 2017 and 2018.

Margaret Readdy and I are the founders and organizers of the Math Movie of the Month.

I also help out at the Henry Clay High School Math Club.

Michael Levin received Honorable Mention for the Morgan Prize for his work in the paper, "A probabilistic approach to the descent statistic." pdf

Here is a page with links to departments and institutes where I have worked or visited.

Here is a collection of math related links.

Here is a family tree of mathematical descendents of Gian-Carlo Rota.

Finally, my coauthors are from the following countries: