David M. R. Jackson

II) In preparation:

Papers

1. Combinatorial aspects of quantum field theory an the Legendre transform. (with A.Kempf and A.Morales).
2. Derivation of a straightening formua in the quantised universal enveloping algebra of sl_2. (temporary title) (with G.Stanley).
3. A geometric summation theorem for edge-weighted triangulations and its consequences. (temporary title) (with C.-C.Liu).

Notes

1. "Linear algebra with application to mathematics - I" (temporary title). Advanced undergraduate notes with an emphasis on the structure theory of linear transformations. It includes the determinant function, quotient spaces, dual spaces, tensor product, the corresponding isomorphism theorems, adjoints, the primary decomposition theorem, decomposition theorems, simultaneous diagonalisation of commuting linear operators,the Jordan theorem, inner product spaces, orthogonal polynomials, and spectral theory. Throughout there are advanced topics that demonstrate the use of this material. These include linear transformations acting on vector spaces over finite fields, walks on Young's meet semi-lattice, the determination of the modes of the double pendulum, the ring of symmetric functions, some classical questions in plane projective geometry, determining the spectrum of the hypercube, an introduction to finite groups and the representation theory of finite groups, the coverability of the edges of the complete graph on 10 vertices by 5 edge disjoint copies of the Petersen graph (as an example of the use of spectral theory), and the counting of non-isomorphic trees with a given number of edges, as an example of the use of elementary group theory and Polya's Theorem (groups are needed for the representation theory). (with G.W.Bennett)
2. "Linear algebra with application to mathematics - II" (temporary title). Advanced undergraduate and graduate notes about the role played by linear algebra in i) the representation theory of finite groups, and character theory, ii) 2-cell embeddings of graphs in orientable surfaces, and iii) quantum invariants and Vassiliev invariants in knot theory, and the use of group algebras, Lie algebras and the Temperley-Lieb algebra in these topics. There is a strong concentration on the precise way in which quotient spaces, dual spaces, tensor algebra and special algebras are used to deal with explicit topological phenomena. There is also an emphasis on how this material grows out of the linear algebra of Volume I, and how the abstract formalism lends itself in a natural way to these three advanced topics. Included are the group algebra of the symmetric group, construction of the central idempotents, Schur functions as generating series for the characacters of irreducible representations of the symmetric group. (with G.W.Bennett)
3. "An introduction to Quantum and Vassiliev Invariants of Knots" (temporary title). Advanced undergraduate and graduate notes that are also addressed to combinatorialists who are interested in invariants of knots. They are written in a way that shows algebra at work in a context that combines both topology and combinatorial constructions. Emphasis is place on full exposition of the material that is included. Included are Reidemeister's Theorem, Markov's Theorem, the braid group, a representation of the braid group, the Yang-Baxter equation and a solution to it, Turaev's Theorem, Vassiliev invariants, chord diagrams, Jacobi diagrams, the Lie bracket, Jacobi's identity, Penrose's diagrammatic calculus, Kontsevich's universality theorem, the relationship between quantum invariants of knots and Lie algebras, an introduction to invariants of 3-manifolds, the Kirby moves, and the Temperley-Lieb algebra. Also included is the necessary material on Hopf algebras, ribbon Hopf algebras, and Lie algebras. (with I. Moffatt)

III) Books:

1. An Atlas of the Smaller Maps on Orientable and Nonorientable Surfaces, Chapman Hall/CRC, New York, 2000, xi + 279 pages, (with T.I.Visentin). This is a collection of drawings of polygonal representations of maps with a small number of edges in orientable and non-orientable surfaces and tables of map numbers, together with an account of the Quadrangulation Conjecture and the b-Conjecture. There is a 70 page Introduction to maps.

2. Combinatorial Enumeration, Wiley, New York, 1983, xxiv + 569 pages, with a Foreword by Gian-Carlo Rota. (with I.P.Goulden). This is a research level book with a large number of exercises, many of which are based on research papers.
   • Russian translation of this book, published by "Nauka," Moscow, 1990.
   • Unabridged republication in a Dover edition in 2004.

IV) Papers:

2011

1. The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture. (with I.P.Goulden and R.Vakil) (to appear).

   2010

2. On Tutte's chromatic invariant, Trans. AMS., 362 (2010) 491--507. (with S.Cautis).

   2009

3. A short proof of the $\lambda_g$-Conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves. J. reine angew. Math, 637 2009, 175--191. (with I.P.Goulden and R.Vakil). Transitive powers of Young-Jucys-Murphy elements are central. J.Algebra, 321 2009, 1826--1835. (with I.P.Goulden).

   2008

4. Transitive factorizations in the hypoctahedral group, Canad. J. Math., 60 (2008), 297--312. (with G.Bini and I.P.Goulden).
5. The KP hierarchy, branched covers and triangulations, Adv. Math., 219 2008, 932--951. (with I.P.Goulden).

   2007

6. On the group-like behaviour of the Le-Murakami-Ohtsuki invariant. J. Knot Theory and its Ramifications, 16 (2007) 669--718. (with I. Moffatt and M. Morales).

   2006

7. A symmetric function resolution of the number of permutations with respect to block-stable elements. Annals. Combinatorics, 10 (2006) 463--480. (with M. Yip).
8. A rooted map invariant, non-orientability and Jack symmetric functions, J. Combinatorial Theory (B), 97 (2007) 430--452. (with D.R.L.Brown).

   2005

9. Towards the geometry of the double Hurwitz numbers, Advances in Mathematics, Volume dedicated to the 70th Birthday of Michael Artin, 198 (2005) 43--92 (with I.P.Goulden and R. Vakil).

   2003

10. The matrix of chromatic joins and the Temperley-Lieb algebra, J. Combinatorial Theory (B) 89 (2003) 109--155. (with S.Cautis).
11. Determinants of random matrices and Jack polynomials of rectangular shape. Studies Applied Math. 110 (2003) 377--390. (with G.E.Andrews and I.P.Goulden)

    2002

12. The quadrangulation conjecture for orientable surfaces, J. Combinatorial Theory (B) 86 (2002) 54--79. (with D.R.L.Brown)

    2001

13. The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals. Proc. London Math. Soc. 83 (2001) 563--581. (with I.P.Goulden and R.Vakil)
14. A geometric parametrization for the virtual Euler characteristic of the moduli spaces of real and complex algebraic curves, Trans. AMS 353 (2001) 4005--4427. (with I.P.Goulden and J.L.Harer)
15. Inequivalent transitive factorizations into transpositions. Canad. J. Math. 53 (2001) 758--779 (with I.P.Goulden and F.G.Latour)

    2000

16. Transitive factorisations in the symmetric group, and combinatorial aspects of singularity theory. Europ. J. Combinatorics 21 (2000) 1001--1016. (with I.P.Goulden)
17. The number of ramified coverings of the sphere by the torus and surfaces of higher genera, Annals of Combinatorics 4 (2000) 27--46. (with I.P.Goulden and A.Vainshtein)

    1999

18. A combinatorial relationship between Eulerian maps and hypermaps in orientable surfaces, J. Combinatorial Theory (A) 78 (1999) 120--150. (with T.I.Visentin)
19. A proof of a conjecture for the number of ramified coverings of the sphere by the torus. J. Combinatorial Theory (A) 88 (1999) 246--258. (with I.P.Goulden)
20. The number of ramified coverings of the sphere by the double torus, and a general form for higher genera, J. Combinatorial Theory (A) 88 (1999) 259--275. (with I.P.Goulden)

    1997

21. Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. AMS 125 (1997) 51--60. (with I.P.Goulden)
22. Maps in locally orientable surfaces and integrals over real symmetric matrices, Canad. J. Math. 48 (1997) 865--882. (with I.P.Goulden)
23. Connexion coefficients for the symmetric group, free products in operator algebras and random matrices, Fields Institute Communications 12 (1997) 105--125. (with I.P.Goulden)

    1996

24. Factorisations for partition functions of random Hermitian matrix models, Comm. Math. Phys. 179 (1996) 25--60. (with M.J.Perry and T.I.Visentin)
25. Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials, Canad. J. Math. 48 (1996) 569--584. (with I.P.Goulden)
26. Connection coefficients, matchings, maps, and combinatorial conjectures for Jack symmetric functions, Trans. AMS 348 (1996) 873--892. (with I.P.Goulden)
27. A formulation for the genus series for regular maps, J. Combinatorial Theory (A) 74 (1996) 14--32. (with T.I.Visentin)
28. Algebraic and analytic approaches for the genus series for 2-cell embeddings on orientable and nonorientable surfaces, DIMACS Series in Discrete Math. and Theoretical Computer Science, Vol. 24, 1996, 115--132. AMS, Providence, RI.
29. Further determinants with the averaging property of Andrews-Burge, J. Combinatorial Theory (A) 73 (1996) 368--375. (with I.P.Goulden)
30. An umbral relation between pattern and commutation in strings, Discrete Math. 158 (1996) 77-85. (with I.P.Goulden, and C.Springer)
31. Research Problems 246, 247, 248. In: Research Problems, Discrete Math. 157 (1996) 375--388.

    1995

32. Combinatorial constructions for integrals over normally distributed random matrices, Proc. AMS 123 (1995) 995--1003. (with I.P.Goulden)
33. The genus series for maps, J. Pure and Applied Algebra 105 (1995) 293--297.

    1994

34. A hypergeometric analysis of the genus series for a class of 2-cell embeddings in orientable surfaces, SIAM J. Math. Anal. 25 (1994) 243--255. (with G.E.Andrews and T.I.Visentin)
35. On an integral representation for the genus series for 2-cell embeddings, Trans. AMS 344 (1994) 755--772.
36. Symmetric functions and Macdonald's result for top connexion coefficients in the symmetric group, J. Algebra 166 (1994) 364--378. (wth I.P.Goulden)
37. The lattice of non-crossing partitions and the Birkhoff-Lewis equations, Europ. J. Combinatorics 15 (1994) 245--250.

    1992

38. Immanants of combinatorial matrices, J. Algebra 148 (1992) 305--324. (with I.P.Goulden)
39. Immanants, Schur functions and the MacMahon Master theorem, Proc. AMS 115 (1992) 605--612. (with I.P.Goulden)
40. The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group, Europ. J. Combinatorics 12 (1992) 357--365. (with I.P.Goulden)
41. The asymptotic behaviour of three-connected triangulations of the disk, with a reflective symmetry in a line, Combinatorica 12 (1992) 149--153. (with L.B.Richmond)
42. A D-finiteness result for products of permutations, Discrete Math. 99 (1992) 25-30. (with E.R.Canfield)

    1990

43. An algebrically derived q-analogue of a character sum associated with a class of semi-regular permutations, Pac. J. Math. 144 (1990) 207--218. (with G.E.Andrews)
44. A character theoretic approach to embeddings of rooted maps in an orientable surface of given genus, Trans. AMS 322 (1990) 343--363. (with T.I.Visentin)
45. A character theoretic approach to embeddings of rooted maps in an orientable surface of given genus: face-coloured maps, Trans. AMS 322 (1990) 365--376. (with T.I.Visentin)
46. A combinatorial construction for products of linear transformations over a finite field, J. Combinatorial Theory (A) 53 (1990) 157--160. (with I.P.Goulden)

    1989

47. Proceedings of the Oberwolfach Meeting "Kombinatorik", Oberwolfach, January 19 - January 25, 1986, Discrete. Math. Special Double Issue, eds. W.Deuber, D.M.Jackson and D.Jungnickel, 73(1-2) (1988/89)

    1988

48. Counting semiregular permutations which are products of a full cycle and an involution, Trans. AMS 305 (1988) 317--331.
49. Some combinatorial problems associated with products of conjugacy classes of the symmetric group, J. Combinatorial Theory (A) 49 (1988) 363--369.
50. Generalisations of Cauchy's summation theorem for Schur functions, Trans. AMS 310 (1988) 805--820. (with G.E.Andrews and I.P.Goulden)
51. Generating function methods for macromolecules at surfaces. II. One molecule between two planes, J. Chem. Phys. 88 (1988) 5171-5180. (with B.E.Eichinger and B.D.McKay)

    1987

52. Counting cycles in permutations by group characters, with an application to a topological problem, Trans. AMS 299 (1987) 785--801.
53. Labelled graphs with vertices of degree at least three, J. Graph Theory 11 (1987) 177--189. (with I.P.Goulden)

    1986

54. Distributions, continued fractions and the Ehrenfest urn model, J. Combinatorial Theory (A) 41 (1986) 21--31. (with I.P.Goulden)
55. Path generating functions and continued fractions, J. Combinatorial Theory (A) 41 (1986) 1--10. (with I.P.Goulden)
56. Generating function methods for macromolecules at surfaces. I. One molecule at a plane surface, J. Chem. Phys. 85 (1986) 5299--5305. (with B.E.Eichinger and B.D.McKay)
57. Shank's convergence acceleration transform, Pade approximants and partitions, J. Combinatorial Theory (A) 43 (1986) 70--84. (with G.E.Andrews and I.P.Goulden)
58. Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Alg. Discrete Meth. 7 (1986) 6--66. (with I.P.Goulden)

    1985

59. Ballot sequences and a determinants of Good's, J. Combinatorial Theory (A) 40 (1985) 55--61. (with I.P.Goulden)

    1984

60. A logarithmic connexion for circular permutation enumeration, Studies Appl. Math. 70 (1984) 121--139. (with I.P.Goulden)
61. "Enumeration and Design," Proceedings of the Silver Jubilee Conference on Combinatorics, June 14 - July 2, 1982, University of Waterloo, Waterloo, Ontario, Canada; eds. D.M.Jackson and S.A.Vanstone, Academic Press, New York, 1984.

    1983

62. A combinatorial proof of a continued fraction expansion theorem from the Ramanujan Notebooks, In: "Graph Theory and Combinatorics" (Cambridge 1983) Academic Press, New York 1984, 97--116. (with I.P.Goulden)
63. The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Alg. Discrete Meth. 4 (1983) 179--193. (with I.P.Goulden and J.W.Reilly)
64. An inversion model for q-identities, Europ. J. Combinatorics 4 (1983) 225--230. (with I.P.Goulden)

    1982

65. The enumeration of covers of a finite set, J. London Math. Soc. 25 (1982) 1--6. (with J.S.Devitt)
66. The application of Lagrangian methods to the enumeration of labelled trees with respect to edge partition, Canad. J. Math. 34 (1982) 513-518. (with I.P.Goulden)

    1981

67. A correspondence between plane planted chromatic trees and generalised derangements, Bull. London Math. Soc. 13 (1981) 28-32. (with I.P.Goulden)
68. The generalisation of Tutte's result for chromatic trees, by Lagrangian methods, Canad. J. Math. 33 (1981) 12--19. (with I.P.Goulden)
69. The enumeration of directed closed Euler trails and directed Hamiltonian circuits by Lagrangian methods, Europ. J. Combinatorics, 2 (1981) 131--135. (with I.P.Goulden)
70. A combinatorial application of matrix Riccati equations and their q-analogue, Discrete Math. 36 (1981) 139--153. (with C.B.Collins, I.P.Goulden and O.M.Nierstracz).
71. Sequence enumeration, Aequationes Math. 23 (1981) 6--23. (with I.P.Goulden)
72. Algebraic metods for permutations with prescribed patterns, Advances in Math. 42 (1981) 113--135. (with I.P.Goulden)
73. Comma-free codes: an extension of certain enumerative techniques to recursively defined sequences, J. Combinatorial Theory (A) 30 (1981) 1--18. (with J.S. Devitt)

    1980

74. A formal calculus for the enumerative system of sequences - III. Further developments, Studies Appl. Math. 62 (1980) 113--141. (with I.P.Goulden)
75. An algebraic theory of sequence enumeration, J. Combinatorial Theory (A) 28 (1980) 191--218. (with B. Jeffcott and W.T.Spears)
76. The enumeration of sequences with respect to structures on a bipartition, Discrete Math., 28 (1980) 133--149. (with B.Jeffcott and W.T.Spears)

    1979

77. A formal calculus for the enumerative system of sequences - II. Applications, Studies Appl. Math. 61 (1979) 245--277. (with I.P.Goulden)
78. A formal calculus for the enumerative system of sequences - I. Combinatorial Theorems, Studies Appl. Math. 61 (1979) 141--178. (with I.P.Goulden)
79. An inversion theorem for cluster decompositions with distinguished subsequences, J. London Math. Soc. 20 (1979) 567--576. (with I.P.Goulden)
80. Sequence enumeration and the de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte theorem, Canad. J. Math. 31 (1979) 488--495. (with I.P.Goulden)
81. The application of algebraic methods to the enumeration of sequences and their q-analogues, Ars Combinatoria 8 (1979) 13--18. (with I.P.Goulden)

    1978

82. Some results on product weighted lead codes, J. Combinatorial Theory (A) 25 (1978) 181--187.
83. A note on permutations without runs of given length, Aequat. Math. 17 (1978) 336-343. (with R.C.Read)
84. The enumeration of generalised alternating subsets with congruences, Discrete Math. 22 (1978) 99--104. (with I.P.Goulden)

    1977

85. The combinatorial interpretation of the Jacobi Identity from Lie algebras, J. Combinatorial Theory (A) 23 (1977) 23--256.
86. Decomposition based generating functions for sequences, Canad. J. Math. 77 (1977) 971--1009. (with R. Aleliunas)
87. The unification of certain enueratain problems for sequences, J. Combinatorial Theory (A) 22 (1977) 92--96.

    1976

88. Laguerre polynomials and derangements, Math. Proc. Camb. Philos. Soc. 80 (1976) 213-214.
89. Permutations with a prescribed number of p-runs, Ars Combinatoria 1 (1976) 297--305. (with J.W.Reilly)

    1975

90. The enumeration of homeomorphically irreducible labelled graphs, J. Combinatorial Theory 19 (1975) 272--286. (with J.W.Reilly)
91. The enumeration of doubly stochastic non-negative integer square matrices, SIAM J. Comput. 4 (1975) 474--477. (with G.H.J. van Rees)

    1973

92. Closed form approximations for random errors in distance functions, Computer J. 16 (1973) 52-56.

    1972

93. Expectations of functions of sequences over finite alphabets with given transition probabilities by methods independent of sequence length, SIAM J. Computing 1 (1972) 203--217.