Algebraic combinatorial Fourier and Legendre transform with applications in perturbative quantum field theory.
(temporary title) (with A.Kempf and A.Morales)
ß
II) In preparation:
Papers
.
Notes
"Linear algebra with advanced application" (temporary title).
(with G.W.Bennett)
"An introduction to Quantum and Vassiliev Invariants of Knots".
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:
An Introduction to Quantum and Vassiliev Knot Invariants,
(with I. Moffatt).
CMS Books in Mathematics, Springer.
To appear in June 2019.
See II(2) above for comments on content.
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.
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:
2018
Combinatorial Aspects of the Quantized Universal Enveloping Algebra of sl_(n+1).
Annals of Combinatorics,
22 (2018), 681--710.
(with R. Chang and G.J. Stanley).
2016
A robust generalization of the Legendre transform.
arXiv : 1612.004 v1 [hep:th] 1 Dec 2016
(with A.Kempf and A.H.Morales).
Transitive factorizations of permutations and geometry.
(with I.P.Goulden).
In: The Mathematical Legacy of Richard P.Stanley,
Eds. P.Hersh, T.Lam, P.Pylyavskyy, V.Reiner.
American Mathematical Society, 2016. (189 -- 201).
2015
How to (Path-) integrate by differentiating.
J. Phys.: Conf. Ser. 626, 012015 (2015)
(with A.Kempf and A.H.Morales).
2014
New Dirac delta function based methods with applications to perturbative expansions in quantum field theory.
J. Phys. A: Math. Theor.
47 (2014) 415204 (12pp).
(with A.Kempf and A.H.Morales).
2012
Near-central permutation factorization and Strahov's generalized Murnaghan-Nakayama rule.
J. Combinatorial Theory (A),
119 (2012), 1856--1874.
(with C.A.Sloss).
Character-theoretic techniques for near-central enumerative problems.
J. Combinatorial Theory (A),
119 (2012), 1665--1679.
(with C.Sloss).
2011
The moduli space of curves, double Hurwitz numbers,
and Faber's intersection number conjecture.
Ann. Comb.,
15 (2011) 381--436.
(with I.P.Goulden and R.Vakil).
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
Transitive factorizations in the hypoctahedral group,
Canad. J. Math.,
60 (2008), 297--312.
(with G.Bini and I.P.Goulden).
The KP hierarchy, branched covers and triangulations,
Adv. Math.,
219 2008, 932--951.
(with I.P.Goulden).
On the structure of QFT in the Particle Picture of the Path Integral Formulation.
arXiv:0810.4293 [hep-th].
(with A.Kempf and A.Morales).
2007
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
A symmetric function resolution of the number of permutations with respect to
block-stable elements.
Annals. Combinatorics,
10 (2006) 463--480.
(with M. Yip).
A rooted map invariant, non-orientability and Jack symmetric functions,
J. Combinatorial Theory (B),
97 (2007) 430--452.
(with D.R.L.Brown).
2005
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
The matrix of chromatic joins and the Temperley-Lieb algebra,
J. Combinatorial Theory (B)
89 (2003) 109--155.
(with S.Cautis).
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
The quadrangulation conjecture for orientable surfaces,
J. Combinatorial Theory (B)
86 (2002) 54--79.
(with D.R.L.Brown)
2001
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)
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)
Inequivalent transitive factorizations into transpositions.
Canad. J. Math.
53 (2001) 758--779
(with I.P.Goulden and F.G.Latour)
2000
Transitive factorisations in the symmetric group, and combinatorial
aspects of singularity theory.
Europ. J. Combinatorics
21 (2000) 1001--1016.
(with I.P.Goulden)
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
A combinatorial relationship between Eulerian maps and hypermaps
in orientable surfaces,
J. Combinatorial Theory (A)
78 (1999) 120--150.
(with T.I.Visentin)
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)
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
Transitive factorisations into transpositions and holomorphic
mappings on the sphere,
Proc. AMS
125 (1997) 51--60.
(with I.P.Goulden)
Maps in locally orientable surfaces and integrals over real
symmetric matrices,
Canad. J. Math.
48 (1997) 865--882.
(with I.P.Goulden)
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
Factorisations for partition functions of random Hermitian matrix models,
Comm. Math. Phys.
179 (1996) 25--60.
(with M.J.Perry and T.I.Visentin)
Maps in locally orientable surfaces, the double coset algebra,
and zonal polynomials,
Canad. J. Math.
48 (1996) 569--584.
(with I.P.Goulden)
Connection coefficients, matchings, maps, and combinatorial
conjectures for Jack symmetric functions,
Trans. AMS
348 (1996) 873--892.
(with I.P.Goulden)
A formulation for the genus series for regular maps,
J. Combinatorial Theory (A)
74 (1996) 14--32.
(with T.I.Visentin)
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.
Further determinants with the averaging property of Andrews-Burge,
J. Combinatorial Theory (A)
73 (1996) 368--375.
(with I.P.Goulden)
An umbral relation between pattern and commutation in strings,
Discrete Math.
158 (1996) 77-85.
(with I.P.Goulden, and C.Springer)
Research Problems 246, 247, 248.
In: Research Problems,
Discrete Math.
157 (1996) 375--388.
1995
Combinatorial constructions for integrals over normally distributed
random matrices,
Proc. AMS
123 (1995) 995--1003.
(with I.P.Goulden)
The genus series for maps,
J. Pure and Applied Algebra
105 (1995) 293--297.
1994
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)
On an integral representation for the genus series for 2-cell
embeddings,
Trans. AMS
344 (1994) 755--772.
Symmetric functions and Macdonald's result for top connexion
coefficients in the symmetric group,
J. Algebra
166 (1994) 364--378.
(wth I.P.Goulden)
The lattice of non-crossing partitions and the Birkhoff-Lewis equations,
Europ. J. Combinatorics
15 (1994) 245--250.
1992
Immanants of combinatorial matrices,
J. Algebra
148 (1992) 305--324.
(with I.P.Goulden)
Immanants, Schur functions and the MacMahon Master theorem,
Proc. AMS
115 (1992) 605--612.
(with I.P.Goulden)
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)
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)
A D-finiteness result for products of permutations,
Discrete Math.
99 (1992) 25-30.
(with E.R.Canfield)
1990
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)
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)
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)
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
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
Counting semiregular permutations which are products of a full cycle
and an involution,
Trans. AMS
305 (1988) 317--331.
Some combinatorial problems associated with products of conjugacy classes
of the symmetric group,
J. Combinatorial Theory (A)
49 (1988) 363--369.
Generalisations of Cauchy's summation theorem for Schur functions,
Trans. AMS
310 (1988) 805--820.
(with G.E.Andrews and I.P.Goulden)
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
Counting cycles in permutations by group characters, with an application
to a topological problem,
Trans. AMS
299 (1987) 785--801.
Labelled graphs with vertices of degree at least three,
J. Graph Theory
11 (1987) 177--189.
(with I.P.Goulden)
1986
Distributions, continued fractions and the Ehrenfest urn model,
J. Combinatorial Theory (A)
41 (1986) 21--31.
(with I.P.Goulden)
Path generating functions and continued fractions,
J. Combinatorial Theory (A)
41 (1986) 1--10.
(with I.P.Goulden)
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)
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)
Labelled graphs with small vertex degrees and P-recursiveness,
SIAM J. Alg. Discrete Meth.
7 (1986) 6--66.
(with I.P.Goulden)
1985
Ballot sequences and a determinants of Good's,
J. Combinatorial Theory (A)
40 (1985) 55--61.
(with I.P.Goulden)
1984
A logarithmic connexion for circular permutation enumeration,
Studies Appl. Math.
70 (1984) 121--139.
(with I.P.Goulden)
"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
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)
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)
An inversion model for q-identities,
Europ. J. Combinatorics
4 (1983) 225--230.
(with I.P.Goulden)
1982
The enumeration of covers of a finite set,
J. London Math. Soc.
25 (1982) 1--6.
(with J.S.Devitt)
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
A correspondence between plane planted chromatic trees and generalised
derangements,
Bull. London Math. Soc.
13 (1981) 28-32.
(with I.P.Goulden)
The generalisation of Tutte's result for chromatic trees, by
Lagrangian methods,
Canad. J. Math.
33 (1981) 12--19.
(with I.P.Goulden)
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)
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).
Algebraic metods for permutations with prescribed patterns,
Advances in Math.
42 (1981) 113--135.
(with I.P.Goulden)
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
A formal calculus for the enumerative system of sequences - III.
Further developments,
Studies Appl. Math.
62 (1980) 113--141.
(with I.P.Goulden)
An algebraic theory of sequence enumeration,
J. Combinatorial Theory (A)
28 (1980) 191--218.
(with B. Jeffcott and W.T.Spears)
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
A formal calculus for the enumerative system of sequences - II.
Applications,
Studies Appl. Math.
61 (1979) 245--277.
(with I.P.Goulden)
A formal calculus for the enumerative system of sequences - I.
Combinatorial Theorems,
Studies Appl. Math.
61 (1979) 141--178.
(with I.P.Goulden)
An inversion theorem for cluster decompositions with
distinguished subsequences,
J. London Math. Soc.
20 (1979) 567--576.
(with I.P.Goulden)
Sequence enumeration and the de Bruijn, van Aardenne-Ehrenfest,
Smith, Tutte theorem,
Canad. J. Math.
31 (1979) 488--495.
(with I.P.Goulden)
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
Some results on product weighted lead codes,
J. Combinatorial Theory (A)
25 (1978) 181--187.
A note on permutations without runs of given length,
Aequat. Math.
17 (1978) 336-343.
(with R.C.Read)
The enumeration of generalised alternating subsets with congruences,
Discrete Math.
22 (1978) 99--104.
(with I.P.Goulden)
1977
The combinatorial interpretation of the Jacobi Identity from Lie algebras,
J. Combinatorial Theory (A)
23 (1977) 23--256.
Decomposition based generating functions for sequences,
Canad. J. Math.
77 (1977) 971--1009.
(with R. Aleliunas)
The unification of certain enumeration problems for sequences,
J. Combinatorial Theory (A)
22 (1977) 92--96.
1976
Laguerre polynomials and derangements,
Math. Proc. Camb. Philos. Soc.
80 (1976) 213-214.
Permutations with a prescribed number of p-runs,
Ars Combinatoria
1 (1976) 297--305.
(with J.W.Reilly)
1975
The enumeration of homeomorphically irreducible labelled graphs,
J. Combinatorial Theory
19 (1975) 272--286.
(with J.W.Reilly)
The enumeration of doubly stochastic non-negative integer square matrices,
SIAM J. Comput.
4 (1975) 474--477.
(with G.H.J. van Rees)
1973
Closed form approximations for random errors in distance functions,
Computer J.
16 (1973) 52-56.
1972
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.