×

Advanced determinant calculus: a complement. (English) Zbl 1079.05008

This paper is a sequel to C. Krattenthaler’s Advanced determinant calculus [Sémin. Lothar. Comb. 42, B42q, electronic only (1999; Zbl 0923.05007)]. The paper continues the survey of advanced determinant evaluations, which have important applications in enumerative combinatorics and number theory. An infinite sequence of identities expressing \(\pi\) is the most important result of the determinant evaluations in this paper; see G. Almkvist, C. Krattenthaler and J. Petersson [Exp. Math. 12, No. 4, 441–456 (2003; Zbl 1161.11419)].

MSC:

05A19 Combinatorial identities, bijective combinatorics
15A15 Determinants, permanents, traces, other special matrix functions
05A15 Exact enumeration problems, generating functions
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
05A10 Factorials, binomial coefficients, combinatorial functions
05A17 Combinatorial aspects of partitions of integers
05A18 Partitions of sets
05A30 \(q\)-calculus and related topics
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
11C20 Matrices, determinants in number theory
11Y60 Evaluation of number-theoretic constants
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33E05 Elliptic functions and integrals

Software:

LinBox; OEIS; DODGSON
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adin, R. M.; Brenti, F.; Roichman, Y., Descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math., 27, 210-224 (2001), (p. 134, 135) · Zbl 0995.05008
[2] Adin, R. M.; Brenti, F.; Roichman, Y., Descent representations and multivariate statistics, Trans. Amer. Math. Soc., 357, 3051-3082 (2005), (p. 135) · Zbl 1059.05105
[3] Adin, R. M.; Roichman, Y., The flag major index and group actions on polynomial rings, Europ. J. Combin., 22, 431-446 (2001), (p. 132, 133, 134) · Zbl 1058.20031
[4] Aigner, M., Catalan-like numbers and determinants, J. Combin. Theory Ser. A, 87, 33-51 (1999), (p. 122) · Zbl 0929.05004
[5] Almkvist, G.; Krattenthaler, C.; Petersson, J., Some new formulas for \(π\), Experiment. Math., 12, 441-456 (2003), (p. 78, 83, 84, 99) · Zbl 1161.11419
[6] E. Altinişik, B.E. Sagan, N. Tuğlu, GCD matrices, posets and non-intersecting paths, Linear Multilinear Algebra, in press,; E. Altinişik, B.E. Sagan, N. Tuğlu, GCD matrices, posets and non-intersecting paths, Linear Multilinear Algebra, in press,
[7] Amdeberhan, T., 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) · Zbl 0961.05002
[8] T. Amdeberhan, Lewis strikes again, and again!, unpublished manuscript. Available from: <; T. Amdeberhan, Lewis strikes again, and again!, unpublished manuscript. Available from: <
[9] Amdeberhan, T.; Zeilberger, D., Determinants through the looking glass, Adv. Appl. Math., 27, 225-230 (2001), (p. 100, 101, 129) · Zbl 0994.05018
[10] Andrews, G. E., Plane partitions (III): The weak Macdonald conjecture, Invent. Math., 53, 193-225 (1979), (p. 122, 124, 125) · Zbl 0421.10011
[11] Andrews, G. E., Macdonald’s conjecture and descending plane partitions, (Narayana, T. V.; Mathsen, R. M.; Williams, J. G., Combinatorics, Representation Theory and Statistical Methods in Groups, Young Day Proceedings. Combinatorics, Representation Theory and Statistical Methods in Groups, Young Day Proceedings, Lecture Notes in Pure Math., vol. 57 (1980), Marcel Dekker: Marcel Dekker New York, Basel), 91-106, (p. 122, 123, 125) · Zbl 0441.05005
[12] Andrews, G. E.; Askey, R. A.; Roy, R., Special Functions. Special Functions, The Encyclopedia of Mathematics and Its Applications, vol. 71 (1999), Cambridge University Press: Cambridge University Press Cambridge, (p. 81)
[13] Andrews, G. E.; Stanton, D. W., Determinants in plane partition enumeration, Europ. J. Combin., 19, 273-282 (1998), (p. 157) · Zbl 0908.05007
[14] Andrews, G. E.; Wimp, J., Some \(q\)-orthogonal polynomials and related Hankel determinants, Rocky Mountain J. Math., 32, 429-442 (2002), (p. 122) · Zbl 1039.33007
[15] Arbogast, L. F.A., Du calcul des dérivations (1800), Levrault: Levrault Strasbourg, (p. 113)
[16] Athanasiadis, C. A.; Reiner, V., Noncrossing partitions for the group \(D_n\), SIAM J. Discrete Math., 18, 397-417 (2004), (p. 139) · Zbl 1085.06001
[17] Bacher, R., Determinants of matrices related to the Pascal triangle, J. Théor. Nombres Bordeaux, 14, 19-41 (2002), (p. 128, 129) · Zbl 1023.11011
[18] Bagno, E., Euler-Mahonian parameters on colored permutation groups, Séminaire Lotharingien Combin., 51 (2004), Article B51f, 16 pp. (p. 135) · Zbl 1062.05004
[19] Bailey, D.; Borwein, P.; Plouffe, S., On the rapid computation of various polylogarithmic constants, Math. Comp., 66, 903-913 (1997), (p. 78) · Zbl 0879.11073
[20] F. Bellard, \(π\)<; F. Bellard, \(π\)<
[21] N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and coinvariants of the symmetric group in noncommuting variables, preprint,; N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and coinvariants of the symmetric group in noncommuting variables, preprint, · Zbl 1180.16025
[22] Bernstein, D., MacMahon-type identities for signed even permutations, Electron. J. Combin., 11, 1 (2004), Article #R83, 18 pp, (p. 135) · Zbl 1065.05004
[23] D. Bernstein, Euler-Mahonian polynomials for \(C_aS_{n\)
[24] Bessenrodt, C.; Olsson, J. B., On character tables related to the alternating groups, Séminaire Lotharingien Combin., 52 (2004), Article B52c, 8 pp. (p. 149) · Zbl 1068.20012
[25] Bessenrodt, C.; Olsson, J. B.; Stanley, R. P., Properties of some character tables related to the symmetric groups, J. Algebraic Combin., 21, 163-177 (2005), (p. 149) · Zbl 1062.05144
[26] Biagioli, R., Major and descent statistics for the even-signed permutation group, Adv. Appl. Math., 31, 163-179 (2003), (p. 135) · Zbl 1020.05006
[27] Biagioli, R.; Caselli, F., Invariant algebras and major indices for classical Weyl groups, Proc. Lond. Math. Soc., 88, 603-631 (2004), (p. 135) · Zbl 1067.05077
[28] Bressoud, D. M., Proofs and Confirmations—The Story of the Alternating Sign Matrix Conjecture (1999), Cambridge University Press: Cambridge University Press Cambridge, (p. 70, 87, 121) · Zbl 0944.05001
[29] J.M. Brunat, C. Krattenthaler, A. Montes, Some composition determinants, preprint; J.M. Brunat, C. Krattenthaler, A. Montes, Some composition determinants, preprint · Zbl 1097.65060
[30] Brunat, J. M.; Montes, A., The power-compositions determinant and its application to global optimization, SIAM J. Matrix Anal. Appl., 23, 459-471 (2001), (p. 140) · Zbl 1004.15013
[31] J.M. Brunat, A. Montes, A polynomial generalization of the power-compositions determinant, Linear Multilinear Algebra, in press. Available from: <; J.M. Brunat, A. Montes, A polynomial generalization of the power-compositions determinant, Linear Multilinear Algebra, in press. Available from: < · Zbl 1116.11016
[32] C.-O. Chow, Noncommutative symmetric functions of type \(B\); C.-O. Chow, Noncommutative symmetric functions of type \(B\)
[33] W.C. Chu, The Faà di Bruno formula and determinant identities, Linear Multilinear Algebra, in press (p. 113, 114).; W.C. Chu, The Faà di Bruno formula and determinant identities, Linear Multilinear Algebra, in press (p. 113, 114).
[34] Chyzak, F.; Salvy, B., Non-commutative elimination in Ore algebras proves multivariate identities, J. Symbolic Comput., 11, 187-227 (1996), (p. 92) · Zbl 0944.05006
[35] Cigler, J., Operatormethoden für \(q\)-Identitäten VII: \(q\)-Catalan-Determinanten, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 208, 123-142 (1999), (p. 122) · Zbl 1003.05005
[36] Cigler, J., Eine Charakterisierung der \(q\)-Exponentialpolynome, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 208, 123-142 (1999), (p. 122) · Zbl 1003.05005
[37] Cigler, J., A new class of \(q\)-Fibonacci polynomials, Electron. J. Combin., 10 (2003), Article #R19, 15 pp. (p. 122) · Zbl 1027.05006
[38] M. Ciucu, Plane partitions I: A generalization of MacMahon’s formula, Memoirs Amer. Math. Soc., in press,; M. Ciucu, Plane partitions I: A generalization of MacMahon’s formula, Memoirs Amer. Math. Soc., in press,
[39] M. Ciucu, The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions, preprint,; M. Ciucu, The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions, preprint, · Zbl 1177.82001
[40] Ciucu, M.; Eisenkölbl, T.; Krattenthaler, C.; Zare, D., Enumeration of lozenge tilings of hexagons with a central triangular hole, J. Combin. Theory Ser. A, 95, 251-334 (2001), (p. 73, 74, 75, 76, 77, 124, 125, 127) · Zbl 0997.52010
[41] Ciucu, M.; Krattenthaler, C., The number of centered lozenge tilings of a symmetric hexagon, J. Combin. Theory Ser. A, 86, 103-126 (1999), (p. 76, 77) · Zbl 0915.05038
[42] Ciucu, M.; Krattenthaler, C., Plane partitions II: \(5 \frac{1}{2}\) symmetry classes, (Kashiwara, M.; Koike, K.; Okada, S.; Terada, I.; Yamada, H., Combinatorial Methods in Representation Theory. Combinatorial Methods in Representation Theory, Advanced Studies in Pure Mathematics, vol. 28 (2000), RIMS: RIMS Kyoto), 83-103, (p. 77)
[43] Ciucu, M.; Krattenthaler, C., Enumeration of lozenge tilings of hexagons with cut off corners, J. Combin. Theory Ser. A, 100, 201-231 (2002), (p. 76, 77, 127) · Zbl 1015.05006
[44] Ciucu, M.; Krattenthaler, C., A non-automatic (!) application of Gosper’s algorithm evaluates a determinant from tiling enumeration, Rocky Mountain J. Math., 32, 589-605 (2002), (p. 76, 77) · Zbl 1031.05008
[45] Cohen, H., A Course in Computational Algebraic Number Theory (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: <; 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: <
[47] Copeland, A.; Schmidt, F.; Simion, R., On two determinants with interesting factorizations, Discrete Math., 256, 449-458 (2002), (p. 139) · Zbl 1009.15004
[48] Craik, A. D.D., Prehistory of Faà di Bruno’s formula, Amer. Math. Monthly, 112, 119-130 (2005), (p. 113) · Zbl 1088.01008
[49] Cvetković, A.; Rajković, P.; Ivković, M., Catalan numbers, and Hankel transform, and Fibonacci numbers, J. Integer Seq., 5 (2002), Article 02.1.3, 8 pp. (p. 122) · Zbl 1041.11014
[50] R. Dahab, The Birkhoff-Lewis equations, Ph.D. Dissertation, University of Waterloo, 1993 (p. 136).; R. Dahab, The Birkhoff-Lewis equations, Ph.D. Dissertation, University of Waterloo, 1993 (p. 136).
[51] David, G.; Tomei, C., The problem of the calissons, Amer. Math. Monthly, 96, 429-431 (1989), (p. 73) · Zbl 0723.05037
[52] Dumas, J.-G.; Gautier, T.; Giesbrecht, M.; Giorgi, P.; Hovinen, B.; Kaltofen, E.; Saunders, B. D.; Turner, W. J.; Villard, G., Linbox: A Generic Library for Exact Linear Algebra, (Cohen, A.; Gao, X.-S.; Takayama, N., Mathematical Software: ICMS 2002 (Proceedings of the first International Congress of Mathematical Software, Beijing, China) (2002), World Scientific), 40-50, (p. 132) · Zbl 1011.68182
[53] Eğecioğlu, Ö.; Redmond, T.; Ryavec, C., From a polynomial Riemann hypothesis to alternating sign matrices, Electron. J. Combin., 8, 1 (2001), Article #R36, 51 pp. (p. 118, 121) · Zbl 0996.05120
[54] Ehrenborg, R., Determinants involving \(q\)-Stirling numbers, Adv. Appl. Math., 31, 630-642 (2003), (p. 122) · Zbl 1071.05011
[55] Eisenkölbl, T., Rhombus tilings of a hexagon with three missing border tiles, J. Combin. Theory Ser. A, 88, 368-378 (1999), (p. 76, 77) · Zbl 0943.05004
[56] Eisenkölbl, T., Rhombus tilings of a hexagon with two triangles missing on the symmetry axis, Electron. J. Combin., 6, 1 (1999), Article #R30, 19 pp. (p. 76, 77) · Zbl 0921.05002
[57] Eisenkölbl, T., (−1)-enumeration of self-complementary plane partitions, Electron. J. Combin., 12, 1 (2005), Article #R7, 22 pp. (p. 76, 77, 126) · Zbl 1063.05008
[58] T. Eisenkölbl, A Schur function identity and the (−1)-enumeration of self-complementary plane partitions, preprint (p. 126).; T. Eisenkölbl, A Schur function identity and the (−1)-enumeration of self-complementary plane partitions, preprint (p. 126).
[59] Elkies, M.; Kuperberg, G.; Larsen, M.; Propp, J., Alternating sign matrices and domino tilings, II, J. Alg. Combin., 1, 219-234 (1992), (p. 101) · Zbl 0788.05017
[60] Fay, J. D., Theta functions on Riemann surfaces, Lecture Notes in Math., vol. 352 (1973), Springer-Verlag, (p. 150) · Zbl 0281.30013
[61] Fischer, I., Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the centre, J. Combin. Theory Ser. A, 96, 31-88 (2001), (p. 76, 77) · Zbl 1012.05011
[62] Fisher, M. E., Walks, walls, wetting, and melting, J. Statist. Phys., 34, 667-729 (1984), (p. 76) · Zbl 0589.60098
[63] D. Foata, G.-N. Han, \(q\); D. Foata, G.-N. Han, \(q\)
[64] D. Foata, G.-N. Han, Signed words and permutations, I: a fundamental transformation, Proc. Amer. Math. Soc., in press. Available from: <; D. Foata, G.-N. Han, Signed words and permutations, I: a fundamental transformation, Proc. Amer. Math. Soc., in press. Available from: < · Zbl 1107.05004
[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: <; 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: < · Zbl 1078.05004
[66] Foata, D.; Han, G.-N., Signed words and permutations, III: the MacMahon Verfahren, Séminaire Lotharingien Combin., 54 (2005), Article B54a, 20 pp. (p. 135) · Zbl 1085.05003
[67] Forrester, P. J., Exact solution of the lock step model of vicious walkers, J. Phys. A: Math. Gen., 23, 1259-1273 (1990), (p. 107) · Zbl 0706.60079
[68] Di Francesco, P., Meander determinants, Comm. Math. Phys., 191, 543-583 (1998), (p. 138) · Zbl 0923.57002
[69] Frobenius, F. G., Über die elliptischen Funktionen zweiter Art, J. reine angew. Math., 93, 53-68 (1882), (p. 150) · JFM 14.0389.01
[70] Fulmek, M., Nonintersecting lattice paths on the cylinder, Séminaire Lotharingien Combin., 52 (2004), Article #B52b, 16 pp. (p. 107) · Zbl 1064.05013
[71] Fulmek, M.; Krattenthaler, C., The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I, Ann. Combin., 2, 19-40 (1998), (p. 77, 128) · Zbl 0917.05004
[72] Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 96 (2004), Cambridge University Press: Cambridge University Press Cambridge, (p. 108, 149)
[73] Gessel, I. M.; Viennot, X., Binomial determinants, paths, and hook length formulae, Adv. in Math., 58, 300-321 (1985), (p. 76, 77) · Zbl 0579.05004
[74] I.M. Gessel, X. Viennot, Determinants, paths, and plane partitions, preprint, 1989. Available from: <; I.M. Gessel, X. Viennot, Determinants, paths, and plane partitions, preprint, 1989. Available from: <
[75] I.M. Gessel, G. Xin, The generating functions of ternary trees and continued fractions, preprint,; I.M. Gessel, G. Xin, The generating functions of ternary trees and continued fractions, preprint, · Zbl 1098.05006
[76] Ghorpade, S. R.; Krattenthaler, C., The Hilbert series of Pfaffian rings, (Christensen, C.; Sundaram, G.; Sathaye, A.; Bajaj, C., Algebra, Arithmetic and Geometry with Applications (2004), Springer-Verlag: Springer-Verlag New York), 337-356, (p. 121) · Zbl 1083.13504
[77] R. Wm. Gosper, unpublished research announcement, 1974 (p. 78).; R. Wm. Gosper, unpublished research announcement, 1974 (p. 78).
[78] Gronau, H.-D. O.F.; Just, W.; Schade, W.; Scheffler, P.; Wojciechowski, J., Path systems in acyclic directed graphs, Zastos. Mat., 19, 399-411 (1988), (p. 76) · Zbl 0718.05031
[79] Haglund, J.; Loehr, N.; Remmel, J., Statistics on wreath products, perfect matchings and signed words, Europ. J. Combin., 26, 835-868 (2005), (p. 135) · Zbl 1063.05009
[80] Han, G.-N.; Krattenthaler, C., Rectangular Scott-type permanents, Séminaire Lotharingien Combin., 43 (2000), Article B43g, 25 pp. (p. 106, 111) · Zbl 0959.15005
[81] Hasegawa, K., Ruijsenaars’ commuting difference operators as commuting transfer matrices, Commun. Math. Phys., 187, 289-325 (1997), (p. 150) · Zbl 0891.47021
[82] Hou, Q.-H.; Lascoux, A.; Mu, Y.-P., Continued fractions for Rogers-Szegő polynomials, Numer. Algorithms, 35, 81-90 (2004), (p. 122) · Zbl 1039.05010
[83] Hou, Q.-H.; Lascoux, A.; Mu, Y.-P., Evaluation of some Hankel determinants, Adv. Appl. Math., 34, 845-852 (2005), (p. 122) · Zbl 1066.05150
[84] Humphreys, J. E., Reflection groups and Coxeter groups (1990), Cambridge University Press: Cambridge University Press Cambridge, (p. 130, 131, 151) · Zbl 0725.20028
[85] M. Ishikawa, S. Okada, H. Tagawa, J. Zeng, Generalizations of Cauchy’s Determinant and Schur’s Pfaffian, preprint,; M. Ishikawa, S. Okada, H. Tagawa, J. Zeng, Generalizations of Cauchy’s Determinant and Schur’s Pfaffian, preprint, · Zbl 1134.15005
[86] Ismail, M. E.H.; Stanton, D., Classical orthogonal polynomials as moments, Can. J. Math., 49, 520-542 (1997), (p. 116, 117) · Zbl 0882.33012
[87] Ismail, M. E.H.; Stanton, D., More orthogonal polynomials as moments, (Sagan, B. E.; Stanley, R. P., Mathematical Essays in Honor of Gian-Carlo Rota. Mathematical Essays in Honor of Gian-Carlo Rota, Progress in Math., vol. 161 (1998), Birkhäuser: Birkhäuser Boston), 377-396, (p. 116, 117) · Zbl 0905.05083
[88] Izergin, A. G., Partition function of the six-vertex model in a finite volume, Soviet Phys. Dokl., 32, 878-879 (1987), (p. 101) · Zbl 0875.82015
[89] James, G. D., The representation theory of the symmetric groups, Lecture Notes in Math., vol. 682 (1978), Springer-Verlag: Springer-Verlag Berlin, (p. 147) · Zbl 0393.20009
[90] James, G. D.; Kerber, A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16 (1981), Addison-Wesley: 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).; 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). · Zbl 0562.05039
[92] Johnson, W. P., The curious history of Faà di Bruno’s formula, Amer. Math. Monthly, 109, 217-234 (2002), (p. 113) · Zbl 1024.01010
[93] W.P. Johnson, Confluent \(q\) doi:10.1016/j.laa.2004.06.008; W.P. Johnson, Confluent \(q\) doi:10.1016/j.laa.2004.06.008 · Zbl 1086.15006
[94] Jones, W. B.; Thron, W. J., Continued Fractions (1980), Addison-Wesley: Addison-Wesley Reading, Massachusetts, (p. 117) · Zbl 0162.09903
[95] Kajihara, Y.; Noumi, M., Multiple elliptic hypergeometric series. An approach from the Cauchy determinant, Indag. Math. (N.S.), 14, 395-421 (2003), (p. 150) · Zbl 1051.33009
[96] Karlin, S.; McGregor, J. L., Coincidence properties of birth-and-death processes, Pacific J. Math., 9, 1109-1140 (1959), (p. 76) · Zbl 0097.34102
[97] Karlin, S.; McGregor, J. L., Coincidence probabilities, Pacific J. Math., 9, 1141-1164 (1959), (p. 76) · Zbl 0092.34503
[98] Kasteleyn, P. W., Graph theory and crystal physics, (Harary, F., Graph Theory and Theoretical Physics (1967), Academic Press: Academic Press San Diego), 43-110, (p. 75) · Zbl 0205.28402
[99] Kedlaya, K. S., Another combinatorial determinant, J. Combin. Theory Ser. A, 90, 221-223 (2000), (p. 113) · Zbl 0945.05008
[100] D.E. Knuth, Overlapping pfaffians, Electron. J. Combin. 3 (no. 2, ”The Foata Festschrift) (1996), Article #R5, 13 pp. (p. 87).; D.E. Knuth, Overlapping pfaffians, Electron. J. Combin. 3 (no. 2, ”The Foata Festschrift) (1996), Article #R5, 13 pp. (p. 87). · Zbl 0862.15007
[101] Ko, K. H.; Smolinsky, L., A combinatorial matrix in 3-manifold theory, Pacific J. Math., 149, 319-336 (1991), (p. 138) · Zbl 0728.57010
[102] R. Koekoek R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 1998, TU Delft, The Netherlands. Available from: <http://aw.twi.tudelft.nl/ koekoek/research.html>; R. Koekoek R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 1998, TU Delft, The Netherlands. Available from: <http://aw.twi.tudelft.nl/ koekoek/research.html>
[103] Kohnert, A.; Veigneau, S., Using Schubert basis to compute with multivariate polynomials, Adv. Appl. Math., 19, 45-60 (1997), (p. 146) · Zbl 0898.05082
[104] Koornwinder, T. H., Special functions and \(q\)-commuting variables, (Ismail, M. E.H.; Masson, D. R.; Rahman, M., Special Functions, \(q\)-Series and Related Topics. Special Functions, \(q\)-Series and Related Topics, Fields Institute Communications, vol. 14 (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 131-166, (p. 108) · Zbl 0882.33014
[105] Krattenthaler, C., Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions, Electron. J. Combin., 4, 1 (1997), #R27, 62 pp. (p. 77, 88, 92) · Zbl 0885.05008
[106] Krattenthaler, C., An involution principle-free bijective proof of Stanley’s hook-content formula, Discrete Math. Theor. Comput. Sci., 3, 011-032 (1998), (p. 74)
[107] Krattenthaler, C., An alternative evaluation of the Andrews-Burge determinant, (Sagan, B. E.; Stanley, R. P., Mathematical Essays in Honor of Gian-Carlo Rota. Mathematical Essays in Honor of Gian-Carlo Rota, Progress in Math., vol. 161 (1998), Birkhäuser: Birkhäuser Boston), 263-270, (p. 89) · Zbl 0907.15005
[108] Krattenthaler, C., Another involution principle-free bijective proof of Stanley’s hook-content formula, J. Combin. Theory Ser. A, 88, 66-92 (1999), (p. 74) · Zbl 0936.05087
[109] Krattenthaler, C., Advanced determinant calculus, Séminaire Lotharingien Combin., 42 (1999), (“The Andrews Festschrift”), Article B42q, 67 pp. (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) · Zbl 0923.05007
[110] C. Krattenthaler, Review of 191; C. Krattenthaler, Review of 191
[111] Krattenthaler, C., Evaluations of some determinants of matrices related to the Pascal triangle, Séminaire. Lotharingien Combin., 47 (2002), Article B47g, 19 pp. (p. 129, 130) · Zbl 1049.33006
[112] C. Krattenthaler, Descending plane partitions and rhombus tilings of a hexagon with triangular hole, Discrete Math., in press,; C. Krattenthaler, Descending plane partitions and rhombus tilings of a hexagon with triangular hole, Discrete Math., in press, · Zbl 1110.05012
[113] C. Krattenthaler, T. Rivoal, Hypergéométrie et fonction zêta de Riemann, preliminary version,; C. Krattenthaler, T. Rivoal, Hypergéométrie et fonction zêta de Riemann, preliminary version,
[114] C. Krattenthaler, T. Rivoal, Hypergéométrie et fonction zêta de Riemann, Memoirs Amer. Math. Soc., in press,; C. Krattenthaler, T. Rivoal, Hypergéométrie et fonction zêta de Riemann, Memoirs Amer. Math. Soc., in press,
[115] Kuperberg, G., Another proof of the alternating sign matrix conjecture, Math. Res. Lett., 3, 139-150 (1996), (p. 101) · Zbl 0859.05027
[116] Kuperberg, G., An exploration of the permanent-determinant method, Electron. J. Combin., 5 (1998), Article #R46, 34 pp. (p. 75) · Zbl 0906.05055
[117] Kuperberg, G., Symmetry classes of alternating-sign matrices under one roof, Ann. Math., 156, 835-866 (2002), (p. 71, 101, 102, 121) · Zbl 1010.05014
[118] S.-F. Lacroix, Traité du calcul différentiel et du calcul intégral, vol. 3, 2ème édition, Mme. Courcier, Paris, 1819 (p.113).; S.-F. Lacroix, Traité du calcul différentiel et du calcul intégral, vol. 3, 2ème édition, Mme. Courcier, Paris, 1819 (p.113).
[119] Laksov, D.; Lascoux, A.; Thorup, A., On Giambelli’s theorem on complete correlations, Acta Math., 162, 163-199 (1989), (p. 102) · Zbl 0695.14023
[120] Lam, T. Y., Representations of finite groups: a hundred years, Notices Amer. Math. Soc., 45, 361-372 (1998), 465-474 (p. 135) · Zbl 0924.01008
[121] Lascoux, A., Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, vol. 99 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, (p. 122, 142, 144, 147) · Zbl 1039.05066
[122] A. Lascoux, Pfaffians and representations of the symmetric group, preprint (p. 103).; A. Lascoux, Pfaffians and representations of the symmetric group, preprint (p. 103). · Zbl 1230.05280
[123] Lenstra, A. K.; Lenstra, H. W.; Lovász, L., Factoring polynomials with rational coefficients, Math. Ann., 261, 515-534 (1982), (p. 79) · Zbl 0488.12001
[124] Lickorish, W. B.R., Invariants for 3-manifolds from the combinatorics of the Jones polynomial, Pacific J. Math., 149, 337-347 (1991), (p. 138) · Zbl 0728.57011
[125] Lindström, B., Determinants on semilattices, Proc. Amer. Math. Soc., 20, 207-208 (1969), (p. 139) · Zbl 0165.02902
[126] Lindström, B., On the vector representations of induced matroids, Bull. London Math. Soc., 5, 85-90 (1973), (p. 76) · Zbl 0262.05018
[127] Luque, J.-G.; Thibon, J.-Y., Hankel hyperdeterminants and Selberg integrals, J. Phys. A, 36, 5267-5292 (2003), (p. 122) · Zbl 1058.33019
[128] Macdonald, I. G., Affine root systems and Dedekind’s \(η\)-function, Invent. Math., 15, 91-143 (1972), (p. 151, 152) · Zbl 0244.17005
[129] I.G. Macdonald, Notes on Schubert Polynomials, Publ. du LaBRI, Université de Québec à Montréal, 1991 (p. 145).; I.G. Macdonald, Notes on Schubert Polynomials, Publ. du LaBRI, Université de Québec à Montréal, 1991 (p. 145).
[130] Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford University Press: Oxford University Press New York/London, (p. 142, 144, 148) · Zbl 0899.05068
[131] MacMahon, P. A., Combinatory Analysis, vol. 2 (1916), Cambridge University Press, reprinted by Chelsea, New York, 1960 (p. 73) · JFM 46.0118.07
[132] Mills, W. H.; Robbins, D. H.; Rumsey, H., Enumeration of a symmetry class of plane partitions, Discrete Math., 67, 43-55 (1987), (p. 121, 124) · Zbl 0656.05006
[133] Milne, S. C., Continued fractions, Hankel determinants, and further identities for powers of classical theta functions, (Garvan, F.; Ismail, M. E.H., Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Dev. Math., vol. 4 (2001), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 171-188, (p. 150, 157)
[134] Milne, S. C., Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Ramanujan J., 6, 7-149 (2002), (p. 150, 157) · Zbl 1125.11315
[135] Mina, L., Formule generali delle successive d’una funzione espresse mediante quelle della sua inverse, Giornale di Mat., 43, 96-112 (1905), (p. 113) · JFM 36.0356.01
[136] Muir, T., The Theory of Determinants in the Historical Order of Development, vol. IV (1923), Macmillan: Macmillan London, (p. 101, 107) · JFM 49.0078.01
[137] Muir, T., A treatise on the theory of determinants (1960), Dover: Dover New York, (p. 111)
[138] E. Neuwirth, Threeway Galton arrays and Pascal-like determinants, Adv. Appl. Math., in press (p. 130).; 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,; S. Okada, Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, preprint, · Zbl 1088.05012
[140] S. Okada, An elliptic generalization of Schur’s Pfaffian identity, Adv. Math., in press,; S. Okada, An elliptic generalization of Schur’s Pfaffian identity, Adv. Math., in press, · Zbl 1099.33015
[141] Okada, S.; Krattenthaler, C., The number of rhombus tilings of a “punctured” hexagon and the minor summation formula, Adv. Appl. Math., 21, 381-404 (1998), (p. 76) · Zbl 0922.05018
[142] Olsson, J. B., Regular character tables of symmetric groups, Electron. J. Combin., 10, 1 (2003), Article #N3, 5 pp. (p. 149) · Zbl 1036.20014
[143] Petkovšek, M.; Wilf, H. S., A high-tech proof of the Mills-Robbins-Rumsey determinant formula, Electron. J. Combin., 3, 2 (1996), Article #R19, 3 pp. (p. 129) · Zbl 0851.05005
[144] Petkovšek, M.; Wilf, H.; Zeilberger, D., \(A=B (1996)\), A.K. Peters: A.K. Peters Wellesley, (p. 70, 92, 129)
[145] S. Plouffe, On the computation of the \(n\); S. Plouffe, On the computation of the \(n\)
[146] Propp, J., Enumeration of matchings: Problems and progress, (Billera, L.; Björner, A.; Greene, C.; Simion, R.; Stanley, R. P., New Perspectives in Algebraic Combinatorics. New Perspectives in Algebraic Combinatorics, Mathematical Sciences Research Institute Publications, vol. 38 (1999), Cambridge University Press), 255-291, (p. 72) · Zbl 0937.05065
[147] E.M. Rains, Transformations of elliptic hypergeometric integrals, preprint,; E.M. Rains, Transformations of elliptic hypergeometric integrals, preprint, · Zbl 1209.33014
[148] Regev, A.; Roichman, Y., Permutation statistics on the alternating group, Adv. Appl. Math., 33, 676-709 (2004), (p. 135) · Zbl 1057.05004
[149] A. Regev, Y. Roichman, Statistics on wreath products and generalized binomial-Stirling numbers, preprint,; A. Regev, Y. Roichman, Statistics on wreath products and generalized binomial-Stirling numbers, preprint, · Zbl 1126.05008
[150] Reiner, V., Signed permutation statistics, Europ. J. Combin., 14, 553-567 (1993), (p. 131, 135) · Zbl 0793.05005
[151] Reiner, V., Non-crossing partitions for classical reflection groups, Discrete Math., 177, 195-222 (1997), (p. 136, 139) · Zbl 0892.06001
[152] Rosengren, H., A proof of a multivariable elliptic summation formula conjectured by Warnaar, (Berndt, B. C.; Ono, K., \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics. \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics, Contemp. Math., vol. 291 (2001), Amer. Math. Soc.: Amer. Math. Soc. Providence), 193-202, (p. 150, 154) · Zbl 1040.33014
[153] Rosengren, H., Elliptic hypergeometric series on root systems, Adv. Math., 181, 417-447 (2004), (p. 150, 154) · Zbl 1066.33017
[154] H. Rosengren, Sums of triangular numbers from the Frobenius determinant, preprint,; H. Rosengren, Sums of triangular numbers from the Frobenius determinant, preprint, · Zbl 1133.11031
[155] Rosengren, H.; Schlosser, M., Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations, Indag. Math. (N.S.), 14, 483-514 (2003), (p. 150, 154) · Zbl 1045.33011
[156] H. Rosengren and M. Schlosser, Elliptic determinant evaluations and the Macdonald identities for affine root systems, preprint,; H. Rosengren and M. Schlosser, Elliptic determinant evaluations and the Macdonald identities for affine root systems, preprint, · Zbl 1104.15009
[157] Ruijsenaars, S. N.M., Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys., 110, 191-213 (1987), (p. 150) · Zbl 0673.58024
[158] M. Rubey, Extending Rate, preprint (p. 71).; M. Rubey, Extending Rate, preprint (p. 71). · Zbl 1189.65040
[159] Sagan, B. E., The Symmetric Group (2001), Springer-Verlag: Springer-Verlag New York, (p. 147)
[160] Saunders, B. D.; Wan, Z., Smith normal form of dense integer matrices, fast algorithms into practice, (Proceedings of ISSAC04, Santander, Spain (2004), ACM Press: ACM Press New York), 274-281, (p. 132) · Zbl 1134.65345
[161] Schendel, L., Das alternirende Exponentialdifferenzenproduct, Zeitschrift Math. Phys., 84-87 (1891), (p. 109)
[162] Schlosser, M., Multidimensional matrix inversions and \(A_r\) and \(D_r\) basic hypergeometric series, Ramanujan J., 1, 243-274 (1997), (p. 104, 106, 155) · Zbl 0934.33006
[163] Schlosser, M., Some new applications of matrix inversions in \(A_r\), Ramanujan J., 3, 405-461 (1999), (p. 106) · Zbl 0944.33016
[164] Schlosser, M., A new multidimensional matrix inversion in \(A_r\), Contemp. Math., 254, 413-432 (2000), (p. 104, 155) · Zbl 0954.33007
[165] Schmidt, F., Problems related to type-\(A\) and type-\(B\) matrices of chromatic joins, Adv. Appl. Math., 32, 380-390 (2004), (p. 135, 137, 138, 139) · Zbl 1050.06003
[166] Schur, I., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math., 139, 155-250 (1911), (p. 101) · JFM 42.0154.02
[167] Scott, R. F., Note on a determinant theorem of Mr. Glaisher’s, Quart. J. Math., 17, 129-132 (1880), (p. 107) · JFM 12.0111.01
[168] Simion, R., Noncrossing partitions, Discrete Math., 217, 367-409 (2000), (p. 136) · Zbl 0959.05009
[169] Singh, V. N., The basic analogues of identities of the Cayley-Orr type, J. London Math. Soc., 34, 15-22 (1959), (p. 92) · Zbl 0088.27702
[170] Slater, J. C., The theory of complex spectra, Phys. Rev., 34, 1293-1322 (1929), (p. 76) · JFM 55.0535.04
[171] Slater, J. C., Quantum Theory of Matter (1968), McGraw-Hill: McGraw-Hill New York, (p. 76)
[172] Sloane, N. J.A., The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., 50, 912-915 (2003), (p. 137) · Zbl 1044.11108
[173] Sloane, N. J.A.; Plouffe, S., The encyclopedia of integer sequences (1995), Academic Press: Academic Press San Diego, (p. 137) · Zbl 0845.11001
[174] Smith, H. J.S., On the value of a certain arithmetical determinant, Proc. London Math. Soc. Ser. 1, 7, 208-212 (1876), (p. 140) · JFM 08.0074.03
[175] Solomon, L., A Mackey formula in the group ring of a Coxeter group, J. Algebra, 41, 2, 255-264 (1976), (p. 135)
[176] Spiridonov, V., Theta hypergeometric integrals, Algebra i Analiz (St. Petersburg Math. J.), 15, 161-215 (2003), (p. 150, 154)
[177] Stanley, R. P., Enumerative Combinatorics, vol. 1 (1986), Wadsworth & Brooks/Cole: Wadsworth & Brooks/Cole Pacific Grove, California, reprinted by Cambridge University Press, Cambridge, 1998 (p. 136, 139, 140)
[178] Stanley, R. P., Enumerative Combinatorics, vol. 2 (1999), Cambridge University Press: Cambridge University Press Cambridge, (p. 70, 142)
[179] Stembridge, J. R., Nonintersecting paths, pfaffians and plane partitions, Adv. in Math., 83, 96-131 (1990), (p. 77, 101, 102) · Zbl 0790.05007
[180] Strehl, V.; Wilf, H. S., Five surprisingly simple complexities, J. Symbolic Comput., 20, 725-729 (1995), (p. 113) · Zbl 0851.68053
[181] Tamm, U., Some aspects of Hankel matrices in coding theory and combinatorics, Electron. J. Combin., 8, 1 (2001), Article #A1, 31 pp. (p. 117) · Zbl 0981.05007
[182] Tarasov, V.; Varchenko, A., Geometry of \(q\)-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque, 246 (1997), (p. 156) · Zbl 0938.17012
[183] Tutte, W. T., The matrix of chromatic joins, J. Combin. Theory Ser. B, 57, 269-288 (1993), (p. 136) · Zbl 0793.05030
[184] Varchenko, A., Bilinear form of real configuration of hyperplanes, Adv. in Math., 97, 110-144 (1993), (p. 131) · Zbl 0777.52006
[185] X. Viennot, Une théorie combinatoire des polynômes orthogonaux généraux, UQAM, Montreal, Quebec, 1983 (p. 115, 116, 117).; X. Viennot, Une théorie combinatoire des polynômes orthogonaux généraux, UQAM, Montreal, Quebec, 1983 (p. 115, 116, 117).
[186] Wall, H. S., Analytic Theory of Continued Fractions (1948), Van Nostrand: Van Nostrand New York, (p. 115) · Zbl 0035.03601
[187] Warnaar, S. O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx., 18, 479-502 (2002), (p. 150, 153, 154, 157) · Zbl 1040.33013
[188] K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma thesis, Johannes Kepler University, Linz, Austria, 1997. Available from: <; K. Wegschaider, Computer generated proofs of binomial multi-sum identities, diploma thesis, Johannes Kepler University, Linz, Austria, 1997. Available from: <
[189] Whittaker, E. T.; Watson, G. N., A course of modern analysis (1996), Cambridge University Press: Cambridge University Press Cambridge, (p. 149) · Zbl 0951.30002
[190] Wilf, H. S.; Zeilberger, D., An algorithmic proof theory for hypergeometric (ordinary and “\(q\)”) multisum/integral identities, Invent. Math., 108, 575-633 (1992), (p. 92, 129) · Zbl 0739.05007
[191] Wimp, J., Hankel determinants of some polynomials arising in combinatorial analysis, Numer. Algorithms, 24, 179-193 (2000), (p. 122, 162) · Zbl 0956.05016
[192] Zakrajšek, H.; Petkovšek, M., Pascal-like determinants are recursive, Adv. Appl. Math., 33, 431-450 (2004), (p. 129) · Zbl 1059.05009
[193] Zeilberger, D., Three recitations on holonomic systems and hypergeometric series, Séminaire Lotharingien Combin., 24 (1990), Article B24a, 28 pp. (p. 70, 71, 92) · Zbl 0981.05514
[194] Zeilberger, D., A fast algorithm for proving terminating hypergeometric identities, Discrete Math., 80, 207-211 (1990), (p. 92, 129) · Zbl 0701.05001
[195] Zeilberger, D., The method of creative telescoping, J. Symbolic Comput., 11, 195-204 (1991), (p. 92, 129) · Zbl 0738.33002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.