Advanced determinant calculus: A complement

Linear Algebra and Its Applications

Volumn 411, Issue 1-3, 2005, Pages 68-166

  Krattenthaler, C ^a  

^a UNIVERSITÉ LYON 1   (France)

Author keywords

Cauchy's double alternant; Chebyshev polynomials; Confluent alternant; Confluent Cauchy determinant; Determinants; Hankel determinants; Laguerre polynomials; Meixner polynomials; Orthogonal polynomials; Pfaffian; Skew circulant matrix; Vandermonde determinant

Indexed keywords

CHEBYSHEV APPROXIMATION; LINEAR ALGEBRA; NUMBER THEORY; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING;

CAUCHY'S DOUBLE ALTERNANT; CONFLUENT ALTERNANT; CONFLUENT CAUCHY DETERMINANT; DETERMINANTS; HANKEL DETERMINANTS; MEIXNER POLYNOMIALS; ORTHOGONAL POLYNOMIALS; PFAFFIAN; SKEW CIRCULANT MATRIX; VANDERMONDE DETERMINANT;

DIFFERENTIATION (CALCULUS);

EID: 27644456268     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2005.06.042     Document Type: Article

References (195)

1. R.M. Adin, F. Brenti, and Y. Roichman Descent numbers and major indices for the hyperoctahedral group Adv. Appl. Math. 27 2001 210 224 math.CO/0012111 (p. 134, 135)
2. R.M. Adin, F. Brenti, and Y. Roichman Descent representations and multivariate statistics Trans. Amer. Math. Soc. 357 2005 3051 3082 math.CO/0112073 (p. 135)
3. R.M. Adin, and Y. Roichman The flag major index and group actions on polynomial rings Europ. J. Combin. 22 2001 431 446 math.CO/0002245 (p. 132, 133, 134)
4. M. Aigner Catalan-like numbers and determinants J. Combin. Theory Ser. A 87 1999 33 51 (p. 122)
5. G. Almkvist, C. Krattenthaler, and J. Petersson Some new formulas for π Experiment. Math. 12 2003 441 456 math.NT/0110238 (p. 78, 83, 84, 99)
6. E. Altinişik, B.E. Sagan, N. Tuǧlu, GCD matrices, posets and non-intersecting paths, Linear Multilinear Algebra, in press, math.CO/0406155 (p. 140).
7. T. Amdeberhan A determinant of the Chudnovskys generalizing the elliptic Frobenius-Stickelberger-Cauchy determinantal identity Electron. J. Combin. 7 1 2000 Article #N6, 3 pp. (p. 150)
8. T. Amdeberhan, Lewis strikes again, and again!, unpublished manuscript. Available from: (p. 87).
9. T. Amdeberhan, and D. Zeilberger Determinants through the looking glass Adv. Appl. Math. 27 2001 225 230 (p. 100, 101, 129)
10. G.E. Andrews Plane partitions (III): The weak Macdonald conjecture Invent. Math. 53 1979 193 225 (p. 122, 124, 125)
11. G.E. Andrews Macdonald's conjecture and descending plane partitions T.V. Narayana R.M. Mathsen J.G. Williams Combinatorics, Representation Theory and Statistical Methods in Groups, Young Day Proceedings Lecture Notes in Pure Math. vol. 57 1980 Marcel Dekker New York, Basel 91 106 (p. 122, 123, 125)
12. G.E. Andrews, R.A. Askey, and R. Roy Special Functions The Encyclopedia of Mathematics and Its Applications vol. 71 1999 Cambridge University Press Cambridge (p. 81)
13. G.E. Andrews, and D.W. Stanton Determinants in plane partition enumeration Europ. J. Combin. 19 1998 273 282 (p. 157)
14. G.E. Andrews, and J. Wimp Some q-orthogonal polynomials and related Hankel determinants Rocky Mountain J. Math. 32 2002 429 442 (p. 122)
15. L.F.A. Arbogast Du calcul des dérivations 1800 Levrault Strasbourg (p. 113)
16. n SIAM J. Discrete Math. 18 2004 397 417 (p. 139)
17. R. Bacher Determinants of matrices related to the Pascal triangle J. Théor. Nombres Bordeaux 14 2002 19 41 math.CO/0109013 (p. 128, 129)
18. E. Bagno Euler-Mahonian parameters on colored permutation groups Séminaire Lotharingien Combin. 51 2004 Article B51f, 16 pp. (p. 135)
19. D. Bailey, P. Borwein, and S. Plouffe On the rapid computation of various polylogarithmic constants Math. Comp. 66 1997 903 913 (p. 78)
20. F. Bellard, π page. Available from: (p. 78).
21. N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and coinvariants of the symmetric group in noncommuting variables, preprint, math.CO/0502082 (p. 141, 142).
22. D. Bernstein MacMahon-type identities for signed even permutations Electron. J. Combin. 11 1 2004 Article #R83, 18 pp, math.CO/0405346 (p. 135)
23. n, preprint, math.CO/0412091 (p. 135).
24. C. Bessenrodt, and J.B. Olsson On character tables related to the alternating groups Séminaire Lotharingien Combin. 52 2004 Article B52c, 8 pp. (p. 149)
25. C. Bessenrodt, J.B. Olsson, and R.P. Stanley Properties of some character tables related to the symmetric groups J. Algebraic Combin. 21 2005 163 177 math.CO/0403110 (p. 149)
26. R. Biagioli Major and descent statistics for the even-signed permutation group Adv. Appl. Math. 31 2003 163 179 (p. 135)
27. R. Biagioli, and F. Caselli Invariant algebras and major indices for classical Weyl groups Proc. Lond. Math. Soc. 88 2004 603 631 (p. 135)
28. D.M. Bressoud Proofs and Confirmations-The Story of the Alternating Sign Matrix Conjecture 1999 Cambridge University Press Cambridge (p. 70, 87, 121)
29. J.M. Brunat, C. Krattenthaler, A. Montes, Some composition determinants, preprint math.CO/0509157 (p. 141).
30. J.M. Brunat, and A. Montes The power-compositions determinant and its application to global optimization SIAM J. Matrix Anal. Appl. 23 2001 459 471 (p. 140)
31. J.M. Brunat, A. Montes, A polynomial generalization of the power-compositions determinant, Linear Multilinear Algebra, in press. Available from: (p. 141).
32. C.-O. Chow, Noncommutative symmetric functions of type B, Ph.D. thesis, Brandeis University, 2001 (p. 135).
33. W.C. Chu, The Faà di Bruno formula and determinant identities, Linear Multilinear Algebra, in press (p. 113, 114).
34. F. Chyzak, and B. Salvy Non-commutative elimination in Ore algebras proves multivariate identities J. Symbolic Comput. 11 1996 187 227 (p. 92)
35. J. Cigler Operatormethoden für q-Identitäten VII: q-Catalan-Determinanten Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 208 1999 123 142 (p. 122)
36. J. Cigler Eine Charakterisierung der q-Exponentialpolynome Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 208 1999 123 142 (p. 122)
37. J. Cigler A new class of q-Fibonacci polynomials Electron. J. Combin. 10 2003 Article #R19, 15 pp. (p. 122)
38. M. Ciucu, Plane partitions I: A generalization of MacMahon's formula, Memoirs Amer. Math. Soc., in press, math.CO/9808017 (p. 77).
39. M. Ciucu, The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions, preprint, math-ph/0501071 (p. 110).
40. M. Ciucu, T. Eisenkölbl, C. Krattenthaler, and D. Zare Enumeration of lozenge tilings of hexagons with a central triangular hole J. Combin. Theory Ser. A 95 2001 251 334 math.CO/9912053 (p. 73, 74, 75, 76, 77, 124, 125, 127)
41. M. Ciucu, and C. Krattenthaler The number of centered lozenge tilings of a symmetric hexagon J. Combin. Theory Ser. A 86 1999 103 126 math.CO/9712209 (p. 76, 77)
42. M. Ciucu, and C. Krattenthaler Plane partitions II: 5 1 2 symmetry classes M. Kashiwara K. Koike S. Okada I. Terada H. Yamada Combinatorial Methods in Representation Theory Advanced Studies in Pure Mathematics vol. 28 2000 RIMS Kyoto 83 103 math.CO/9808018 (p. 77)
43. M. Ciucu, and C. Krattenthaler Enumeration of lozenge tilings of hexagons with cut off corners J. Combin. Theory Ser. A 100 2002 201 231 math.CO/0104058 (p. 76, 77, 127)
44. M. Ciucu, and C. Krattenthaler A non-automatic (!) application of Gosper's algorithm evaluates a determinant from tiling enumeration Rocky Mountain J. Math. 32 2002 589 605 math.CO/0011047 (p. 76, 77)
45. H. Cohen A Course in Computational Algebraic Number Theory second ed. 1995 Springer-Verlag (p. 79)
46. S. Colton, A. Bundy, T. Walsh, Automatic invention of integer sequences, in: Proceedings of the Seventeenth National Conference on Artificial Intelligence AAAI-2000, Austin, 2000. Available from: (p. 71).
47. A. Copeland, F. Schmidt, and R. Simion On two determinants with interesting factorizations Discrete Math. 256 2002 449 458 (p. 139)
48. A.D.D. Craik Prehistory of Faà di Bruno's formula Amer. Math. Monthly 112 2005 119 130 (p. 113)
49. A. Cvetković, P. Rajković, and M. Ivković Catalan numbers, and Hankel transform, and Fibonacci numbers J. Integer Seq. 5 2002 Article 02.1.3, 8 pp. (p. 122)
50. R. Dahab, The Birkhoff-Lewis equations, Ph.D. Dissertation, University of Waterloo, 1993 (p. 136).
51. G. David, and C. Tomei The problem of the calissons Amer. Math. Monthly 96 1989 429 431 (p. 73)
52. J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B.D. Saunders, W.J. Turner, and G. Villard Linbox: A Generic Library for Exact Linear Algebra A. Cohen X.-S. Gao N. Takayama Mathematical Software: ICMS 2002 (Proceedings of the first International Congress of Mathematical Software, Beijing, China) 2002 World Scientific 40 50 (p. 132)
53. Ö. Eǧecioǧlu, T. Redmond, and C. Ryavec From a polynomial Riemann hypothesis to alternating sign matrices Electron. J. Combin. 8 1 2001 Article #R36, 51 pp. (p. 118, 121)
54. R. Ehrenborg Determinants involving q-Stirling numbers Adv. Appl. Math. 31 2003 630 642 (p. 122)
55. T. Eisenkölbl Rhombus tilings of a hexagon with three missing border tiles J. Combin. Theory Ser. A 88 1999 368 378 math.CO/9712261 (p. 76, 77)
56. T. Eisenkölbl Rhombus tilings of a hexagon with two triangles missing on the symmetry axis Electron. J. Combin. 6 1 1999 Article #R30, 19 pp. math.CO/9810019 (p. 76, 77)
57. T. Eisenkölbl (-1)-enumeration of self-complementary plane partitions Electron. J. Combin. 12 1 2005 Article #R7, 22 pp. math.CO/0412118 (p. 76, 77, 126)
58. T. Eisenkölbl, A Schur function identity and the (-1)-enumeration of self-complementary plane partitions, preprint (p. 126).
59. M. Elkies, G. Kuperberg, M. Larsen, and J. Propp Alternating sign matrices and domino tilings, II J. Alg. Combin. 1 1992 219 234 (p. 101)
60. J.D. Fay Theta functions on Riemann surfaces Lecture Notes in Math. vol. 352 1973 Springer-Verlag (p. 150)
61. I. Fischer Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the centre J. Combin. Theory Ser. A 96 2001 31 88 math.CO/9906102 (p. 76, 77)
62. M.E. Fisher Walks, walls, wetting, and melting J. Statist. Phys. 34 1984 667 729 (p. 76)
63. D. Foata, G.-N. Han, q-series in combinatorics; permutation statistics, Lecture Notes, Strasbourg, 2003 (p. 108).
64. D. Foata, G.-N. Han, Signed words and permutations, I: a fundamental transformation, Proc. Amer. Math. Soc., in press. Available from: (p. 135).
65. D. Foata, G.-N. Han, Signed words and permutations, II: the Euler-Mahonian polynomials, Electron. J. Combin. 11(2) (2004/05), to appear. Available from: (p. 135).
66. D. Foata, and G.-N. Han Signed words and permutations, III: the MacMahon Verfahren Séminaire Lotharingien Combin. 54 2005 Article B54a, 20 pp. (p. 135)
67. P.J. Forrester Exact solution of the lock step model of vicious walkers J. Phys. A: Math. Gen. 23 1990 1259 1273 (p. 107)
68. P. Di Francesco Meander determinants Comm. Math. Phys. 191 1998 543 583 hep-th/9612026 (p. 138)
69. F.G. Frobenius Über die elliptischen Funktionen zweiter Art J. reine angew. Math. 93 1882 53 68 (p. 150)
70. M. Fulmek Nonintersecting lattice paths on the cylinder Séminaire Lotharingien Combin. 52 2004 Article #B52b, 16 pp. math.CO/0311331 (p. 107)
71. M. Fulmek, and C. Krattenthaler The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I Ann. Combin. 2 1998 19 40 math.CO/9712244 (p. 77, 128)
72. G. Gasper, and M. Rahman second ed. Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications vol. 96 2004 Cambridge University Press Cambridge (p. 108, 149)
73. I.M. Gessel, and X. Viennot Binomial determinants, paths, and hook length formulae Adv. in Math. 58 1985 300 321 (p. 76, 77)
74. I.M. Gessel, X. Viennot, Determinants, paths, and plane partitions, preprint, 1989. Available from: (p. 76, 77).
75. I.M. Gessel, G. Xin, The generating functions of ternary trees and continued fractions, preprint, math.CO/0505217 (p. 118).
76. S.R. Ghorpade, and C. Krattenthaler The Hilbert series of Pfaffian rings C. Christensen G. Sundaram A. Sathaye C. Bajaj Algebra, Arithmetic and Geometry with Applications 2004 Springer-Verlag New York 337 356 math.AC/0105151 (p. 121)
77. R. Wm. Gosper, unpublished research announcement, 1974 (p. 78).
78. H.-D.O.F. Gronau, W. Just, W. Schade, P. Scheffler, and J. Wojciechowski Path systems in acyclic directed graphs Zastos. Mat. 19 1988 399 411 (p. 76)
79. J. Haglund, N. Loehr, and J. Remmel Statistics on wreath products, perfect matchings and signed words Europ. J. Combin. 26 2005 835 868 Available from: (p. 135)
80. G.-N. Han, and C. Krattenthaler Rectangular Scott-type permanents Séminaire Lotharingien Combin. 43 2000 Article B43g, 25 pp. math.RA/0003072 (p. 106, 111)
81. K. Hasegawa Ruijsenaars' commuting difference operators as commuting transfer matrices Commun. Math. Phys. 187 1997 289 325 q-alg/9512029 (p. 150)
82. Q.-H. Hou, A. Lascoux, and Y.-P. Mu Continued fractions for Rogers-Szego polynomials Numer. Algorithms 35 2004 81 90 (p. 122)
83. Q.-H. Hou, A. Lascoux, and Y.-P. Mu Evaluation of some Hankel determinants Adv. Appl. Math. 34 2005 845 852 Available from: (p. 122)
84. J.E. Humphreys Reflection groups and Coxeter groups 1990 Cambridge University Press Cambridge (p. 130, 131, 151)
85. M. Ishikawa, S. Okada, H. Tagawa, J. Zeng, Generalizations of Cauchy's Determinant and Schur's Pfaffian, preprint, math.CO/0411280 (p. 103).
86. M.E.H. Ismail, and D. Stanton Classical orthogonal polynomials as moments Can. J. Math. 49 1997 520 542 (p. 116, 117)
87. M.E.H. Ismail, and D. Stanton More orthogonal polynomials as moments B.E. Sagan R.P. Stanley Mathematical Essays in Honor of Gian-Carlo Rota Progress in Math. vol. 161 1998 Birkhäuser Boston 377 396 (p. 116, 117)
88. A.G. Izergin Partition function of the six-vertex model in a finite volume Soviet Phys. Dokl. 32 1987 878 879 (p. 101)
89. G.D. James The representation theory of the symmetric groups Lecture Notes in Math. vol. 682 1978 Springer-Verlag Berlin (p. 147)
90. G.D. James, and A. Kerber The representation theory of the symmetric group Encyclopedia of Mathematics and its Applications vol. 16 1981 Addison-Wesley Reading, Mass (p. 147)
91. P. John, H. Sachs, Wegesysteme und Linearfaktoren in hexagonalen und quadratischen Systemen, in: Graphs in Research and Teaching (Kiel, 1985), Franzbecker, Bad Salzdetfurth, 1985, pp. 85-101 (p. 76).
92. W.P. Johnson The curious history of Faà di Bruno's formula Amer. Math. Monthly 109 2002 217 234 (p. 113)
93. W.P. Johnson, Confluent q-extensions of some classical determinants, Linear Algebra Appl., in press, doi:10.1016/j.laa.2004.06.008 (p. 109).
94. W.B. Jones, and W.J. Thron Continued Fractions 1980 Addison-Wesley Reading, Massachusetts (p. 117)
95. Y. Kajihara, and M. Noumi Multiple elliptic hypergeometric series. An approach from the Cauchy determinant Indag. Math. (N.S.) 14 2003 395 421 math.CA/0306219 (p. 150)
96. S. Karlin, and J.L. McGregor Coincidence properties of birth-and-death processes Pacific J. Math. 9 1959 1109 1140 (p. 76)
97. S. Karlin, and J.L. McGregor Coincidence probabilities Pacific J. Math. 9 1959 1141 1164 (p. 76)
98. P.W. Kasteleyn Graph theory and crystal physics F. Harary Graph Theory and Theoretical Physics 1967 Academic Press San Diego 43 110 (p. 75)
99. K.S. Kedlaya Another combinatorial determinant J. Combin. Theory Ser. A 90 2000 221 223 math.CO/9810127 (p. 113)
100. D.E. Knuth, Overlapping pfaffians, Electron. J. Combin. 3 (no. 2, "The Foata Festschrift) (1996), Article #R5, 13 pp. (p. 87).
101. K.H. Ko, and L. Smolinsky A combinatorial matrix in 3-manifold theory Pacific J. Math. 149 1991 319 336 (p. 138)
102. R. Koekoek R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 1998, TU Delft, The Netherlands. Available from: (p. 117).
103. A. Kohnert, and S. Veigneau Using Schubert basis to compute with multivariate polynomials Adv. Appl. Math. 19 1997 45 60 (p. 146)
104. T.H. Koornwinder Special functions and q-commuting variables M.E.H. Ismail D.R. Masson M. Rahman Special Functions, q-Series and Related Topics Fields Institute Communications vol. 14 1997 Amer. Math. Soc. Providence, RI 131 166 (p. 108)
105. C. Krattenthaler Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions Electron. J. Combin. 4 1 1997 #R27, 62 pp. math.CO/9712202 (p. 77, 88, 92)
106. C. Krattenthaler An involution principle-free bijective proof of Stanley's hook-content formula Discrete Math. Theor. Comput. Sci. 3 1998 011 032 (p. 74)
107. C. Krattenthaler An alternative evaluation of the Andrews-Burge determinant B.E. Sagan R.P. Stanley Mathematical Essays in Honor of Gian-Carlo Rota Progress in Math. vol. 161 1998 Birkhäuser Boston 263 270 (p. 89)
108. C. Krattenthaler Another involution principle-free bijective proof of Stanley's hook-content formula J. Combin. Theory Ser. A 88 1999 66 92 math.CO/9807068 (p. 74)
109. C. Krattenthaler Advanced determinant calculus Séminaire Lotharingien Combin. 42 1999 ("The Andrews Festschrift"), Article B42q, 67 pp. math.CO/9902004 (p. 69, 70, 71, 77, 78, 86, 87, 88, 100, 101, 104, 106, 107, 108, 109, 111, 113, 115, 116, 117, 121, 122, 124, 128, 129, 130, 131, 135, 136, 138, 139, 143, 147, 149, 150, 151, 153, 154, 155, 157)
110. C. Krattenthaler, Review of 191, Zentralblatt, Zbl 0956.05016, 2001 (p. 122).
111. C. Krattenthaler Evaluations of some determinants of matrices related to the Pascal triangle Séminaire. Lotharingien Combin. 47 2002 Article B47g, 19 pp. math.CO/0111328 (p. 129, 130)
112. C. Krattenthaler, Descending plane partitions and rhombus tilings of a hexagon with triangular hole, Discrete Math., in press, math.CO/0310188 (p. 76).
113. C. Krattenthaler, T. Rivoal, Hypergéométrie et fonction zêta de Riemann, preliminary version, math.NT/0311114v1 (p. 142).
114. C. Krattenthaler, T. Rivoal, Hypergéométrie et fonction zêta de Riemann, Memoirs Amer. Math. Soc., in press, math.NT/0311114 (p. 142, 144).
115. G. Kuperberg Another proof of the alternating sign matrix conjecture Math. Res. Lett. 3 1996 139 150 math.CO/9712207 (p. 101)
116. G. Kuperberg An exploration of the permanent-determinant method Electron. J. Combin. 5 1998 Article #R46, 34 pp. math.CO/9810091 (p. 75)
117. G. Kuperberg Symmetry classes of alternating-sign matrices under one roof Ann. Math. 156 2002 835 866 math.CO/0008184 (p. 71, 101, 102, 121)
118. S.-F. Lacroix, Traité du calcul différentiel et du calcul intégral, vol. 3, 2ème édition, Mme. Courcier, Paris, 1819 (p.113).
119. D. Laksov, A. Lascoux, and A. Thorup On Giambelli's theorem on complete correlations Acta Math. 162 1989 163 199 (p. 102)
120. T.Y. Lam Representations of finite groups: a hundred years Notices Amer. Math. Soc. 45 1998 361 372 465-474 (p. 135)
121. A. Lascoux Symmetric functions and combinatorial operators on polynomials CBMS Regional Conference Series in Mathematics vol. 99 2003 Amer. Math. Soc. Providence, RI (p. 122, 142, 144, 147)
122. A. Lascoux, Pfaffians and representations of the symmetric group, preprint (p. 103).
123. A.K. Lenstra, H.W. Lenstra Jr., and L. Lovász Factoring polynomials with rational coefficients Math. Ann. 261 1982 515 534 (p. 79)
124. W.B.R. Lickorish Invariants for 3-manifolds from the combinatorics of the Jones polynomial Pacific J. Math. 149 1991 337 347 (p. 138)
125. B. Lindström Determinants on semilattices Proc. Amer. Math. Soc. 20 1969 207 208 (p. 139)
126. B. Lindström On the vector representations of induced matroids Bull. London Math. Soc. 5 1973 85 90 (p. 76)
127. J.-G. Luque, and J.-Y. Thibon Hankel hyperdeterminants and Selberg integrals J. Phys. A 36 2003 5267 5292 math-ph/0211044 (p. 122)
128. I.G. Macdonald Affine root systems and Dedekind's η-function Invent. Math. 15 1972 91 143 (p. 151, 152)
129. I.G. Macdonald, Notes on Schubert Polynomials, Publ. du LaBRI, Université de Québec à Montréal, 1991 (p. 145).
130. I.G. Macdonald Symmetric Functions and Hall Polynomials Second ed. 1995 Oxford University Press New York/London (p. 142, 144, 148)
131. P.A. MacMahon Combinatory Analysis vol. 2 1916 Cambridge University Press reprinted by Chelsea, New York, 1960 (p. 73)
132. W.H. Mills, D.H. Robbins, and H. Rumsey Enumeration of a symmetry class of plane partitions Discrete Math. 67 1987 43 55 (p. 121, 124)
133. S.C. Milne Continued fractions, Hankel determinants, and further identities for powers of classical theta functions F. Garvan M.E.H. Ismail Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics Dev. Math. vol. 4 2001 Kluwer Acad. Publ. Dordrecht 171 188 (p. 150, 157)
134. S.C. Milne Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions Ramanujan J. 6 2002 7 149 math.NT/0008068 (p. 150, 157)
135. L. Mina Formule generali delle successive d'una funzione espresse mediante quelle della sua inverse Giornale di Mat. 43 1905 96 112 (p. 113)
136. T. Muir The Theory of Determinants in the Historical Order of Development vol. IV 1923 Macmillan London (p. 101, 107)
137. T. Muir A treatise on the theory of determinants revised and enlarged by W.H. Metzler 1960 Dover New York (p. 111)
138. E. Neuwirth, Threeway Galton arrays and Pascal-like determinants, Adv. Appl. Math., in press (p. 130).
139. S. Okada, Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, preprint, math.CO/0408234 (p. 103).
140. S. Okada, An elliptic generalization of Schur's Pfaffian identity, Adv. Math., in press, math.CA/0412038 (p. 150).
141. S. Okada, and C. Krattenthaler The number of rhombus tilings of a "punctured" hexagon and the minor summation formula Adv. Appl. Math. 21 1998 381 404 math.CO/9712203 (p. 76)
142. J.B. Olsson Regular character tables of symmetric groups Electron. J. Combin. 10 1 2003 Article #N3, 5 pp. (p. 149)
143. M. Petkovšek, and H.S. Wilf A high-tech proof of the Mills-Robbins-Rumsey determinant formula Electron. J. Combin. 3 2 1996 Article #R19, 3 pp. (p. 129)
144. M. Petkovšek, H. Wilf, and D. Zeilberger A = B 1996 A.K. Peters Wellesley (p. 70, 92, 129)
145. S. Plouffe, On the computation of the n'th decimal digit of various transcendental numbers, manuscript, 1996. Available from: http://www.lacim.uqam. ca/~plouffe/ (p. 78).
146. J. Propp Enumeration of matchings: Problems and progress L. Billera A. Björner C. Greene R. Simion R.P. Stanley New Perspectives in Algebraic Combinatorics Mathematical Sciences Research Institute Publications vol. 38 1999 Cambridge University Press 255 291 math.CO/9904150 (p. 72)
147. E.M. Rains, Transformations of elliptic hypergeometric integrals, preprint, math.QA/0309252 (p. 150, 155).
148. A. Regev, and Y. Roichman Permutation statistics on the alternating group Adv. Appl. Math. 33 2004 676 709 math.CO/0302301 (p. 135)
149. A. Regev, Y. Roichman, Statistics on wreath products and generalized binomial-Stirling numbers, preprint, math.CO/0404354 (p. 135).
150. V. Reiner Signed permutation statistics Europ. J. Combin. 14 1993 553 567 (p. 131, 135)
151. V. Reiner Non-crossing partitions for classical reflection groups Discrete Math. 177 1997 195 222 (p. 136, 139)
152. H. Rosengren A proof of a multivariable elliptic summation formula conjectured by Warnaar B.C. Berndt K. Ono q-Series with Applications to Combinatorics, Number Theory, and Physics Contemp. Math. vol. 291 2001 Amer. Math. Soc. Providence 193 202 math.CA/0101073 (p. 150, 154)
153. H. Rosengren Elliptic hypergeometric series on root systems Adv. Math. 181 2004 417 447 math.CA/0207046 (p. 150, 154)
154. H. Rosengren, Sums of triangular numbers from the Frobenius determinant, preprint, math.NT/0504272 (p. 150).
155. H. Rosengren, and M. Schlosser Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations Indag. Math. (N.S.) 14 2003 483 514 math.CA/0304249 (p. 150, 154)
156. H. Rosengren and M. Schlosser, Elliptic determinant evaluations and the Macdonald identities for affine root systems, preprint, math.CA/0505213 (p. 104, 105, 106, 151, 152, 155, 156).
157. S.N.M. Ruijsenaars Complete integrability of relativistic Calogero-Moser systems and elliptic function identities Comm. Math. Phys. 110 1987 191 213 (p. 150)
158. M. Rubey, Extending Rate, preprint (p. 71).
159. B.E. Sagan The Symmetric Group Second ed. 2001 Springer-Verlag New York (p. 147)
160. B.D. Saunders, and Z. Wan Smith normal form of dense integer matrices, fast algorithms into practice Proceedings of ISSAC04, Santander, Spain 2004 ACM Press New York 274 281 (p. 132)
161. L. Schendel Das alternirende Exponentialdifferenzenproduct Zeitschrift Math. Phys. 1891 84 87 (p. 109)
162. r basic hypergeometric series Ramanujan J. 1 1997 243 274 (p. 104, 106, 155)
163. r Ramanujan J. 3 1999 405 461 (p. 106)
164. r Contemp. Math. 254 2000 413 432 (p. 104, 155)
165. F. Schmidt Problems related to type-A and type-B matrices of chromatic joins Adv. Appl. Math. 32 2004 380 390 (p. 135, 137, 138, 139)
166. I. Schur Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen J. reine angew. Math. 139 1911 155 250 (p. 101)
167. R.F. Scott Note on a determinant theorem of Mr. Glaisher's Quart. J. Math. 17 1880 129 132 (p. 107)
168. R. Simion Noncrossing partitions Discrete Math. 217 2000 367 409 (p. 136)
169. V.N. Singh The basic analogues of identities of the Cayley-Orr type J. London Math. Soc. 34 1959 15 22 (p. 92)
170. J.C. Slater The theory of complex spectra Phys. Rev. 34 1929 1293 1322 (p. 76)
171. J.C. Slater Quantum Theory of Matter Second ed. 1968 McGraw-Hill New York (p. 76)
172. N.J.A. Sloane The on-line encyclopedia of integer sequences Notices Amer. Math. Soc. 50 2003 912 915 (p. 137)
173. N.J.A. Sloane, and S. Plouffe The encyclopedia of integer sequences 1995 Academic Press San Diego (p. 137)
174. H.J.S. Smith On the value of a certain arithmetical determinant Proc. London Math. Soc. Ser. 1 7 1876 208 212 (p. 140)
175. L. Solomon A Mackey formula in the group ring of a Coxeter group J. Algebra 41 2 1976 255 264 (p. 135)
176. V. Spiridonov Theta hypergeometric integrals Algebra i Analiz (St. Petersburg Math. J.) 15 2003 161 215 math.CA/0303205 (p. 150, 154)
177. R.P. Stanley Enumerative Combinatorics vol. 1 1986 Wadsworth & Brooks/Cole Pacific Grove, California reprinted by Cambridge University Press, Cambridge, 1998 (p. 136, 139, 140)
178. R.P. Stanley Enumerative Combinatorics vol. 2 1999 Cambridge University Press Cambridge (p. 70, 142)
179. J.R. Stembridge Nonintersecting paths, pfaffians and plane partitions Adv. in Math. 83 1990 96 131 (p. 77, 101, 102)
180. V. Strehl, and H.S. Wilf Five surprisingly simple complexities J. Symbolic Comput. 20 1995 725 729 (p. 113)
181. U. Tamm Some aspects of Hankel matrices in coding theory and combinatorics Electron. J. Combin. 8 1 2001 Article #A1, 31 pp. (p. 117)
182. V. Tarasov, and A. Varchenko Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups Astérisque 246 1997 q-alg/9703044 (p. 156)
183. W.T. Tutte The matrix of chromatic joins J. Combin. Theory Ser. B 57 1993 269 288 (p. 136)
184. A. Varchenko Bilinear form of real configuration of hyperplanes Adv. in Math. 97 1993 110 144 (p. 131)
185. X. Viennot, Une théorie combinatoire des polynômes orthogonaux généraux, UQAM, Montreal, Quebec, 1983 (p. 115, 116, 117).
186. H.S. Wall Analytic Theory of Continued Fractions 1948 Van Nostrand New York (p. 115)
187. S.O. Warnaar Summation and transformation formulas for elliptic hypergeometric series Constr. Approx. 18 2002 479 502 math.QA/0001006 (p. 150, 153, 154, 157)
188. K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma thesis, Johannes Kepler University, Linz, Austria, 1997. Available from: (p. 129).
189. E.T. Whittaker, and G.N. Watson A course of modern analysis reprint of the 4th ed. (1927) 1996 Cambridge University Press Cambridge (p. 149)
190. H.S. Wilf, and D. Zeilberger An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities Invent. Math. 108 1992 575 633 (p. 92, 129)
191. J. Wimp Hankel determinants of some polynomials arising in combinatorial analysis Numer. Algorithms 24 2000 179 193 (p. 122, 162)
192. H. Zakrajšek, and M. Petkovšek Pascal-like determinants are recursive Adv. Appl. Math. 33 2004 431 450 (p. 129)
193. D. Zeilberger Three recitations on holonomic systems and hypergeometric series Séminaire Lotharingien Combin. 24 1990 Article B24a, 28 pp. (p. 70, 71, 92)
194. D. Zeilberger A fast algorithm for proving terminating hypergeometric identities Discrete Math. 80 1990 207 211 (p. 92, 129)
195. D. Zeilberger The method of creative telescoping J. Symbolic Comput. 11 1991 195 204 (p. 92, 129)