Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson
Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1205-1226.

In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, Combinatorial Enumeration, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collaborative work.

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DOI: 10.5802/alco.269
Classification: 05A05, 05A10, 05E05, 05A15, 05E10, 05E16, 81Q30
Keywords: combinatorial enumeration, factorizations, Lagrange inversion, Hurwitz numbers, KP hierarchy, maps, symmetric functions, immanants

Foley, Angèle M. 1; Morales, Alejandro H. 2; Rattan, Amarpreet 3; Yeats, Karen 4

1 Department of Physics and Computer Science Wilfrid Laurier University 75 University Avenue West Waterloo ON N2L 3C5 (Canada)
2 Department of Mathematics and Statistics, University of Massachusetts, Amherst 710 N. Pleasant Street Amherst MA 01003 (USA)
3 Department of Mathematics, Simon Fraser University 8888 University Dr Burnaby BC V5A 1S6 (Canada)
4 Department of Combinatorics and Optimization, University of Waterloo 200 University Ave W Waterloo ON N2L 3G1 (Canada)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Foley, Angèle M.; Morales, Alejandro H.; Rattan, Amarpreet; Yeats, Karen. Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson. Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1205-1226. doi : 10.5802/alco.269. https://alco.centre-mersenne.org/articles/10.5802/alco.269/

[1] Berkolaiko, G.; Irving, J. Inequivalent factorizations of permutations, J. Combin. Theory Ser. A, Volume 140 (2016), pp. 1-37 | DOI | MR | Zbl

[2] Bernardi, O. An analogue of the Harer-Zagier formula for unicellular maps on general surfaces, Adv. in Appl. Math., Volume 48 (2012) no. 1, pp. 164-180 | DOI | MR | Zbl

[3] Bernardi, O.; Morales, A. H. Bijections and symmetries for the factorizations of the long cycle, Adv. in Appl. Math., Volume 50 (2013) no. 5, pp. 702-722 | DOI | MR | Zbl

[4] Bernardi, O.; Morales, A. H. Some probabilistic trees with algebraic roots, Electron. J. Combin., Volume 23 (2016) no. 2, Paper no. 2.36, 24 pages | DOI | MR | Zbl

[5] Bessis, D. Finite complex reflection arrangements are K(π,1), Ann. of Math. (2), Volume 181 (2015) no. 3, pp. 809-904 | DOI | MR

[6] Bialas, P.; Burda, Z.; Johnston, D. Condensation in the Backgammon model, Nuclear Physics B, Volume 493 (1997) no. 3, pp. 505-516 | DOI | Zbl

[7] Biane, P. Parking functions of types A and B, Electron. J. Combin., Volume 9 (2002) no. 1, Paper no. 7, 5 pages | MR | Zbl

[8] Bini, G.; Goulden, I. P.; Jackson, D. M. Transitive factorizations in the hyperoctahedral group, Canad. J. Math., Volume 60 (2008) no. 2, Paper no. B51a, pp. 297-312 | DOI | MR | Zbl

[9] Bonzom, V.; Chapuy, G.; Dołęga, M. b-Monotone Hurwitz Numbers: Virasoro Constraints, BKP Hierarchy, and O(N)-BGW Integral, preprint, arXiv:2109.01499, 2021 (to appear in Int. Math. Res. Not. IMRN) | arXiv | DOI | MR | Zbl

[10] Bousquet-Mélou, M.; Schaeffer, G. Enumeration of planar constellations, Adv. in Appl. Math., Volume 24 (2000) no. 4, pp. 337-368 | arXiv | DOI | MR

[11] Brown, D. R. L.; Jackson, D. M. A rooted map invariant, non-orientability and Jack symmetric functions, J. Combin. Theory Ser. B, Volume 97 (2007) no. 3, pp. 430-452 | arXiv | DOI | MR

[12] Carrell, S. R.; Chapuy, G. Simple recurrence formulas to count maps on orientable surfaces, J. Combin. Theory Ser. A, Volume 133 (2015), pp. 58-75 | DOI | MR | Zbl

[13] Cartier, P.; Foata, D. Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Mathematics, No. 85, Springer-Verlag, Berlin-New York, 1969, iv+88 pages | DOI | MR | Zbl

[14] Cavalieri, R.; Johnson, P.; Markwig, H. Wall crossings for double Hurwitz numbers, Adv. Math., Volume 228 (2011) no. 4, pp. 1894-1937 | DOI | MR | Zbl

[15] Cavalieri, R.; Miles, E. Riemann surfaces and algebraic curves, London Mathematical Society Student Texts, 87, Cambridge University Press, Cambridge, 2016, xii+183 pages (A first course in Hurwitz theory) | DOI | MR | Zbl

[16] Chapuy, G. A new combinatorial identity for unicellular maps, via a direct bijective approach, Adv. in Appl. Math., Volume 47 (2011) no. 4, pp. 874-893 | DOI | MR

[17] Chapuy, G.; Dołęga, M. Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions, Adv. Math., Volume 409 (2022), Paper no. 108645, 72 pages | DOI | MR | Zbl

[18] Chapuy, G.; Féray, V.; Fusy, É. A simple model of trees for unicellular maps, J. Combin. Theory Ser. A, Volume 120 (2013) no. 8, pp. 2064-2092 | DOI | MR

[19] Chapuy, G.; Stump, C. Counting factorizations of Coxeter elements into products of reflections, J. Lond. Math. Soc. (2), Volume 90 (2014) no. 3, pp. 919-939 | DOI | MR | Zbl

[20] Cooper, B. Review of the book “An introduction to quantum and Vassiliev knot invariants” by D. M. Jackson and I. Moffatt, Mathematical Reviews, Volume MR3931694 (2019), Paper no. 108645 | DOI | MR | Zbl

[21] Dali, H. B. Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra, 2022, Paper no. 108645 | DOI | MR | Zbl

[22] Dénes, J. The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl., Volume 4 (1959), pp. 63-71 | DOI | MR | Zbl

[23] Dołęga, M.; Féray, V. Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J., Volume 165 (2016) no. 7, pp. 1193-1282 | DOI | MR

[24] Dołęga, M.; Féray, V. Cumulants of Jack symmetric functions and the b-conjecture, Trans. Amer. Math. Soc., Volume 369 (2017) no. 12, pp. 9015-9039 | DOI | MR

[25] Douvropoulos, T. Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element, preprint, arXiv:1808.10395, 2018 | arXiv | MR

[26] Douvropoulos, T.; Lewis, J. B.; Morales, A. H. Hurwitz numbers for reflection groups I: Generatingfunctionology, Enumer. Comb. Appl., Volume 2 (2022) no. 3, Paper no. S2R20, 21 pages | DOI | MR | Zbl

[27] Douvropoulos, T.; Lewis, J. B.; Morales, A. H. Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements, preprint, arXiv:2209.00066, 2022 | arXiv | DOI | MR | Zbl

[28] Dubrovin, B.; Yang, D.; Zagier, D. Classical Hurwitz numbers and related combinatorics, Mosc. Math. J., Volume 17 (2017) no. 4, pp. 601-633 | arXiv | DOI | MR

[29] Duchi, E.; Poulalhon, D.; Schaeffer, G. Bijections for simple and double Hurwitz numbers, preprint, arXiv:1410.6521, 2014, Paper no. S2R20 | arXiv | DOI | MR

[30] Duchi, E.; Poulalhon, D.; Schaeffer, G., Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (2014), Paper no. S2R20, pp. 294-304 | arXiv | DOI | MR

[31] Farahat, H. K.; Higman, G. The centres of symmetric group rings, Proc. Roy. Soc. London Ser. A, Volume 250 (1959), pp. 212-221 | arXiv | DOI | MR | Zbl

[32] Féray, V.; Kortchemski, I. The geometry of random minimal factorizations of a long cycle via biconditioned bitype random trees, Ann. H. Lebesgue, Volume 1 (2018), pp. 149-226 | arXiv | DOI | MR

[33] Féray, V.; Kortchemski, I. Trajectories in random minimal transposition factorizations, ALEA Lat. Am. J. Probab. Math. Stat., Volume 16 (2019) no. 1, pp. 759-785 | arXiv | DOI | MR | Zbl

[34] FPSAC/SFCA International Conference Series on Formal Power Series and Algebraic Combinatorics Conference (FPSAC), http://fpsac.org | DOI | MR | Zbl

[35] Frobenius, F. G. Uber gruppencharacktere, Gesammelte Abhandlungen. Bände III, Springer-Verlag, Berlin-New York, 1968 (Herausgegeben von J.-P. Serre) | DOI | Numdam | MR | Zbl

[36] Gewurz, D. A.; Merola, F. Some factorisations counted by Catalan numbers, European J. Combin., Volume 27 (2006) no. 6, pp. 990-994 | DOI | MR | Zbl

[37] Goulden, I. P. A differential operator for symmetric functions and the combinatorics of multiplying transpositions, Trans. Amer. Math. Soc., Volume 344 (1994) no. 1, pp. 421-440 | DOI | MR

[38] Goulden, I. P.; Greene, C. A new tableau representation for supersymmetric Schur functions, J. Algebra, Volume 170 (1994) no. 2, pp. 687-703 | DOI | MR

[39] Goulden, I. P.; Guay-Paquet, M.; Novak, J. Monotone Hurwitz numbers in genus zero, Canad. J. Math., Volume 65 (2013) no. 5, pp. 1020-1042 | DOI | MR | Zbl

[40] Goulden, I. P.; Guay-Paquet, M.; Novak, J. Polynomiality of monotone Hurwitz numbers in higher genera, Adv. Math., Volume 238 (2013), pp. 1-23 | DOI | MR | Zbl

[41] Goulden, I. P.; Guay-Paquet, M.; Novak, J. Monotone Hurwitz numbers and the HCIZ integral, Ann. Math. Blaise Pascal, Volume 21 (2014) no. 1, pp. 71-89 | DOI | MR | Zbl

[42] Goulden, I. P.; Guay-Paquet, M.; Novak, J. Toda equations and piecewise polynomiality for mixed double Hurwitz numbers, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 12 (2016), Paper no. 040, 10 pages | DOI | MR | Zbl

[43] Goulden, I. P.; Guay-Paquet, M.; Novak, J. On the convergence of monotone Hurwitz generating functions, Ann. Comb., Volume 21 (2017) no. 1, pp. 73-81 | DOI | MR | Zbl

[44] Goulden, I. P.; Jackson, D. M. An inversion theorem for cluster decompositions of sequences with distinguished subsequences, J. London Math. Soc. (2), Volume 20 (1979) no. 3, pp. 567-576 | DOI | Numdam | MR | Zbl

[45] Goulden, I. P.; Jackson, D. M. Combinatorial enumeration, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983, Paper no. 040, xxiv+569 pages | DOI | MR | Zbl

[46] Goulden, I. P.; Jackson, D. M. The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group, European J. Combin., Volume 13 (1992) no. 5, Paper no. 040, pp. 357-365 | DOI | MR | Zbl

[47] Goulden, I. P.; Jackson, D. M. Immanants of combinatorial matrices, J. Algebra, Volume 148 (1992) no. 2, pp. 305-324 | DOI | MR | Zbl

[48] Goulden, I. P.; Jackson, D. M. Immanants, Schur functions, and the MacMahon master theorem, Proc. Amer. Math. Soc., Volume 115 (1992) no. 3, pp. 605-612 | DOI | MR

[49] Goulden, I. P.; Jackson, D. M. Symmetric functions and Macdonald’s result for top connexion coefficients in the symmetric group, J. Algebra, Volume 166 (1994) no. 2, pp. 364-378 | DOI | MR | Zbl

[50] Goulden, I. P.; Jackson, D. M. Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc., Volume 348 (1996) no. 3, pp. 873-892 | DOI | MR | Zbl

[51] Goulden, I. P.; Jackson, D. M. Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc., Volume 125 (1997) no. 1, pp. 51-60 | DOI | MR | Zbl

[52] Goulden, I. P.; Jackson, D. M. A proof of a conjecture for the number of ramified coverings of the sphere by the torus, J. Combin. Theory Ser. A, Volume 88 (1999) no. 2, pp. 246-258 | DOI | MR | Zbl

[53] Goulden, I. P.; Jackson, D. M. Combinatorial enumeration, Dover Publications, Inc., Mineola, NY, 2004, xxvi+569 pages | DOI | MR | Zbl

[54] Goulden, I. P.; Jackson, D. M. The KP hierarchy, branched covers, and triangulations, Adv. Math., Volume 219 (2008) no. 3, pp. 932-951 | DOI | MR | Zbl

[55] Goulden, I. P.; Jackson, D. M. Transitive powers of Young-Jucys-Murphy elements are central, J. Algebra, Volume 321 (2009) no. 7, pp. 1826-1835 | DOI | MR | Zbl

[56] Goulden, I. P.; Jackson, D. M. Transitive factorizations of permutations and geometry, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 189-201 | DOI | MR | Zbl

[57] Goulden, I. P.; Jackson, D. M.; Vainshtein, A. The number of ramified coverings of the sphere by the torus and surfaces of higher genera, Ann. Comb., Volume 4 (2000) no. 1, pp. 27-46 | DOI | MR

[58] Goulden, I. P.; Jackson, D. M.; Vakil, R. The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc. (3), Volume 83 (2001) no. 3, pp. 563-581 | DOI | MR | Zbl

[59] Goulden, I. P.; Jackson, D. M.; Vakil, R. Towards the geometry of double Hurwitz numbers, Adv. Math., Volume 198 (2005) no. 1, pp. 43-92 | DOI | MR | Zbl

[60] Goulden, I. P.; Nica, A. A direct bijection for the Harer-Zagier formula, J. Combin. Theory Ser. A, Volume 111 (2005) no. 2, pp. 224-238 | DOI | MR | Zbl

[61] Goulden, I. P.; Pepper, S. Labelled trees and factorizations of a cycle into transpositions, Discrete Math., Volume 113 (1993) no. 1-3, pp. 263-268 | DOI | MR | Zbl

[62] Goulden, I. P.; Rattan, A. An explicit form for Kerov’s character polynomials, Trans. Amer. Math. Soc., Volume 359 (2007) no. 8, pp. 3669-3685 | DOI | MR | Zbl

[63] Goulden, I. P.; Yong, A. Tree-like properties of cycle factorizations, J. Combin. Theory Ser. A, Volume 98 (2002) no. 1, pp. 106-117 | DOI | MR | Zbl

[64] Goupil, A.; Schaeffer, G. Factoring n-cycles and counting maps of given genus, European J. Combin., Volume 19 (1998) no. 7, pp. 819-834 | DOI | MR | Zbl

[65] Haiman, M. Hecke algebra characters and immanant conjectures, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 569-595 | DOI | MR | Zbl

[66] Hamel, A. M.; Goulden, I. P. Planar decompositions of tableaux and Schur function determinants, European J. Combin., Volume 16 (1995) no. 5, pp. 461-477 | DOI | MR | Zbl

[67] Hanlon, P. J.; Stanley, P. P.; Stembridge, J. R. Some combinatorial aspects of the spectra of normally distributed random matrices, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) (Contemp. Math.), Volume 138, Amer. Math. Soc., Providence, RI, 1992, pp. 151-174 | DOI | MR | Zbl

[68] Harer, J.; Zagier, D. The Euler characteristic of the moduli space of curves, Invent. Math., Volume 85 (1986) no. 3, pp. 457-485 | DOI | MR | Zbl

[69] Hurwitz, A. Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891) no. 1, pp. 1-60 | DOI | MR | Zbl

[70] Irving, J.; Rattan, A. Minimal factorizations of permutations into star transpositions, Discrete Math., Volume 309 (2009) no. 6, pp. 1435-1442 | DOI | MR | Zbl

[71] Itzykson, C.; Zuber, J.-B. Matrix integration and combinatorics of modular groups, Comm. Math. Phys., Volume 134 (1990) no. 1, pp. 197-207 | DOI | MR | Zbl

[72] Jackson, D. M. Counting cycles in permutations by group characters, with an application to a topological problem, Trans. Amer. Math. Soc., Volume 299 (1987) no. 2, pp. 785-801 | DOI | MR | Zbl

[73] Jackson, D. M. Some combinatorial problems associated with products of conjugacy classes of the symmetric group, J. Combin. Theory Ser. A, Volume 49 (1988) no. 2, pp. 363-369 | DOI | MR | Zbl

[74] Jackson, D. M. On an integral representation for the genus series for 2-cell embeddings, Trans. Amer. Math. Soc., Volume 344 (1994) no. 2, pp. 755-772 | DOI | MR | Zbl

[75] Jackson, D. M.; Aleliunas, R. Decomposition based generating functions for sequences, Canadian J. Math., Volume 29 (1977) no. 5, pp. 971-1009 | DOI | MR | Zbl

[76] Jackson, D. M.; Goulden, I. P. A formal calculus for the enumerative system of sequences. I. Combinatorial theorems, Stud. Appl. Math., Volume 61 (1979) no. 2, pp. 141-178 | DOI | MR | Zbl

[77] Jackson, D. M.; Goulden, I. P. A formal calculus for the enumerative system of sequences. II. Applications, Stud. Appl. Math., Volume 61 (1979) no. 3, pp. 245-277 | DOI | MR | Zbl

[78] Jackson, D. M.; Goulden, I. P. A formal calculus for the enumerative system of sequences. III. Further developments, Stud. Appl. Math., Volume 62 (1980) no. 2, pp. 113-141 | DOI | MR | Zbl

[79] Jackson, D. M.; Goulden, I. P. Algebraic methods for permutations with prescribed patterns, Adv. in Math., Volume 42 (1981) no. 2, pp. 113-135 | DOI | MR | Zbl

[80] Jackson, D. M.; Kempf, A.; Morales, A. H. On the Structure of QFT in the Particle Picture of the Path Integral Formulation, preprint, arXiv:0810.4293, 2008 | arXiv | DOI | MR | Zbl

[81] Jackson, D. M.; Kempf, A.; Morales, A. H. A robust generalization of the Legendre transform for QFT, J. Phys. A, Volume 50 (2017) no. 22, p. 225201, 16 | DOI | MR | Zbl

[82] Jackson, D. M.; Kempf, A.; Morales, A. H. Towards a more algebraic footing for quantum field theory, J. Phys. A, Volume 54 (2021) no. 28, Paper no. 285201, 46 pages | DOI | MR | Zbl

[83] Jackson, D. M.; Moffatt, I. An introduction to quantum and Vassiliev knot invariants, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2019, xx+422 pages | DOI | MR | Zbl

[84] Jackson, D. M.; Perry, M. J.; Visentin, T. I. Factorisations for partition functions of random Hermitian matrix models, Comm. Math. Phys., Volume 179 (1996) no. 1, pp. 25-59 | DOI | MR | Zbl

[85] Jackson, D. M.; Visentin, T. I. A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus, Trans. Amer. Math. Soc., Volume 322 (1990) no. 1, pp. 343-363 | DOI | MR | Zbl

[86] Jackson, D. M.; Visentin, T. I. Character theory and rooted maps in an orientable surface of given genus: face-colored maps, Trans. Amer. Math. Soc., Volume 322 (1990) no. 1, pp. 365-376 | arXiv | DOI | MR

[87] Jackson, D. M.; Visentin, T. I. An atlas of the smaller maps in orientable and nonorientable surfaces, CRC Press Series on Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2001, viii+279 pages | arXiv | DOI | MR | Zbl

[88] Jia, D.; Tang, E.; Kempf, A. Integration by differentiation: new proofs, methods and examples, J. Phys. A, Volume 50 (2017) no. 23, Paper no. 285201, p. 235201, 25 | DOI | MR

[89] Kanunnikov, A. L.; Promyslov, V. V.; Vassilieva, E. A. A labelled variant of the matchings-Jack and hypermap-Jack conjectures, Sém. Lothar. Combin., Volume 80B (2018), Paper no. 45, 12 pages | DOI | MR

[90] Kanunnikov, A. L.; Vassilieva, E. A. On the matchings-Jack conjecture for Jack connection coefficients indexed by two single part partitions, Electron. J. Combin., Volume 23 (2016) no. 1, Paper no. 1.53, 30 pages | DOI | MR | Zbl

[91] Kempf, A.; Jackson, D. M.; Morales, A. H. New Dirac delta function based methods with applications to perturbative expansions in quantum field theory, J. Phys. A, Volume 47 (2014) no. 41, p. 415204, 12 | DOI | MR | Zbl

[92] Kempf, A.; Jackson, D. M.; Morales, A. H. How to (path-) integrate by differentiating, J. Phys.: Conf. Ser., Volume 626 (2015), p. 012015 | DOI | MR

[93] Kerov, S. V. Rook placements on Ferrers boards, and matrix integrals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 240 (1997), p. 136-146, 293 | DOI | MR

[94] Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Volume 147 (1992) no. 1, pp. 1-23 | DOI | MR | Zbl

[95] La Croix, M. A. The combinatorics of the Jack parameter and the genus series for topological maps, Ph. D. Thesis, University of Waterloo (2009), Paper no. 45 | DOI | MR | Zbl

[96] Lando, S. K.; Zvonkin, A. K. Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer-Verlag, Berlin, 2004, Paper no. 45, xvi+455 pages | DOI | MR | Zbl

[97] Lass, B. Démonstration combinatoire de la formule de Harer-Zagier, C. R. Acad. Sci. Paris Sér. I Math., Volume 333 (2001) no. 3, Paper no. 1.53, pp. 155-160 | DOI | MR | Zbl

[98] Lewis, J. B.; Morales, A. H. GL n (F q )-analogues of factorization problems in the symmetric group, European J. Combin., Volume 58 (2016), pp. 75-95 | DOI | MR

[99] Lewis, J. B.; Morales, A. H. Factorization problems in complex reflection groups, Canad. J. Math., Volume 73 (2021) no. 4, pp. 899-946 | DOI | MR | Zbl

[100] Lewis, J. B.; Reiner, V.; Stanton, D. Reflection factorizations of Singer cycles, J. Algebraic Combin., Volume 40 (2014) no. 3, pp. 663-691 | DOI | MR | Zbl

[101] Lin, X. S.; Willmot, G. E. Analysis of a defective renewal equation arising in ruin theory, Insurance Math. Econom., Volume 25 (1999) no. 1, pp. 63-84 | DOI | MR

[102] Macdonald, I. G. Schur functions: theme and variations, Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992) (Publ. Inst. Rech. Math. Av.), Volume 498, Univ. Louis Pasteur, Strasbourg, 1992, pp. 5-39 | DOI | MR

[103] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | DOI | MR | Zbl

[104] Marciniak, M. Quadratic coefficients of Goulden–Rattan character polynomials, preprint, arXiv:2104.13512, 2021 | arXiv | DOI | MR

[105] Matsumoto, S.; Novak, J. Jucys-Murphy elements and unitary matrix integrals, Int. Math. Res. Not. IMRN (2013) no. 2, pp. 362-397 | DOI | MR | Zbl

[106] Michel, J. Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element, Proc. Amer. Math. Soc., Volume 144 (2016) no. 3, pp. 937-941 | DOI | MR | Zbl

[107] Morales, A. H.; Pak, I.; Panova, G. Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications, SIAM J. Discrete Math., Volume 31 (2017) no. 3, pp. 1953-1989 | DOI | MR | Zbl

[108] Moszkowski, P. A solution to a problem of Dénes: a bijection between trees and factorizations of cyclic permutations, European J. Combin., Volume 10 (1989) no. 1, pp. 13-16 | DOI | MR | Zbl

[109] Okounkov, A. Toda equations for Hurwitz numbers, Math. Res. Lett., Volume 7 (2000) no. 4, pp. 447-453 | DOI | MR

[110] Orlov, A. Y.; Shcherbin, D. M. Hypergeometric solutions of soliton equations, Teoret. Mat. Fiz., Volume 128 (2001) no. 1, pp. 84-108 | arXiv | DOI | MR

[111] Palsberg, J.; Krishnaswamy, S.; Kwon, M.; Ma, D.; Shao, Q.; Zhang, Y., Proceedings 16th Annual Computer Security Applications Conference (ACSAC’00) (2000), pp. 308-316 | arXiv | DOI | MR | Zbl

[112] Penner, R. C. Perturbative series and the moduli space of Riemann surfaces, J. Differential Geom., Volume 27 (1988) no. 1, pp. 35-53 | DOI | MR | Zbl

[113] Polak, E.; Ross, D. Polynomiality of factorizations in reflection groups, preprint, arXiv:2004.13213, 2020 (to appear in Can. J. Math.) | arXiv | DOI | MR | Zbl

[114] Poulalhon, D.; Schaeffer, G. Factorizations of large cycles in the symmetric group, Discrete Math., Volume 254 (2002) no. 1-3, pp. 433-458 | DOI | MR | Zbl

[115] Reiner, V.; Shaw, K. M.; van Willigenburg, S. Coincidences among skew Schur functions, Adv. Math., Volume 216 (2007) no. 1, pp. 118-152 | DOI | MR | Zbl

[116] Schaeffer, G. Planar maps, Handbook of enumerative combinatorics (Discrete Math. Appl. (Boca Raton)), CRC Press, Boca Raton, FL, 2015, pp. 335-395 | DOI | MR | Zbl

[117] Schaeffer, G.; Vassilieva, E. A bijective proof of Jackson’s formula for the number of factorizations of a cycle, J. Combin. Theory Ser. A, Volume 115 (2008) no. 6, pp. 903-924 | DOI | MR

[118] Stanley, R. P. Factorization of permutations into n-cycles, Discrete Math., Volume 37 (1981) no. 2-3, pp. 255-262 | DOI | MR | Zbl

[119] Stanley, R. P. Some combinatorial properties of Jack symmetric functions, Adv. Math., Volume 77 (1989) no. 1, pp. 76-115 | DOI | MR

[120] Stanley, R. P. A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math., Volume 111 (1995) no. 1, pp. 166-194 | DOI | MR

[121] Stanley, R. P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl

[122] Stanley, R. P. Catalan numbers, Cambridge University Press, New York, 2015, viii+215 pages | arXiv | DOI | MR

[123] Stanley, R. P. Publications, with commentary by the author, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 1-37 | arXiv | DOI | MR | Zbl

[124] Stanley, R. P.; Stembridge, J. R. On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A, Volume 62 (1993) no. 2, pp. 261-279 | DOI | MR | Zbl

[125] Steel, M. Decompositions of leaf-colored binary trees, Adv. in Appl. Math., Volume 14 (1993) no. 1, pp. 1-24 | DOI | MR | Zbl

[126] Strehl, V. Minimal transitive products of transpositions—the reconstruction of a proof of A. Hurwitz, Sém. Lothar. Combin., Volume 37 (1996), Paper no. S37c, 12 pages | DOI | MR | Zbl

[127] Tenner, B. E. A combinatorial proof of symmetry among minimal star factorizations, Discrete Math., Volume 312 (2012) no. 16, pp. 2482-2490 | DOI | MR | Zbl

[128] Tenner, B. E. Star factorizations and noncrossing partitions, Discrete Math., Volume 344 (2021) no. 7, Paper no. 112428, 9 pages | DOI | MR | Zbl

[129] Thévenin, P. A geometric representation of fragmentation processes on stable trees, Ann. Probab., Volume 49 (2021) no. 5, pp. 2416-2476 | DOI | MR | Zbl

[130] Tutte, W. T. A census of planar maps, Canadian J. Math., Volume 15 (1963), pp. 249-271 | DOI | MR | Zbl

[131] Tutte, W. T. Graph theory, Encyclopedia of Mathematics and its Applications, 21, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984, xxi+333 pages | DOI | MR

[132] Tutte, W. T. Review of the book “An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces” by D. M. Jackson and T. Visentin, Mathematical Reviews, Volume MR1792279 (2001) | DOI | MR

[133] Vakil, R. Genus 0 and 1 Hurwitz numbers: recursions, formulas, and graph-theoretic interpretations, Trans. Amer. Math. Soc., Volume 353 (2001) no. 10, pp. 4025-4038 | DOI | MR | Zbl

[134] Vershik, A. M.; Okounkov, A. Y. A new approach to the representation theory of the symmetric groups. II, J. Math. Sci., Volume 131 (2005) no. 2, pp. 5471-5494 | DOI | MR | Zbl

[135] Walkup, D. W. How many ways can a permutation be factored into two n-cycles?, Discrete Math., Volume 28 (1979) no. 3, pp. 315-319 | DOI | MR | Zbl

[136] Yeats, K. A combinatorial perspective on quantum field theory, SpringerBriefs in Mathematical Physics, 15, Springer, Cham, 2017, Paper no. S37c, ix+120 pages | DOI | MR | Zbl

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