ADVANCED DETERMINANT CALCULUS

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

[1]  C. L. Dodgson,et al.  IV. Condensation of determinants, being a new and brief method for computing their arithmetical values , 1867, Proceedings of the Royal Society of London.

[2]  T. Stieltjes,et al.  Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable , 1889 .

[3]  Sur l'apolarité des formes binaires , 1901 .

[4]  T. B.,et al.  The Theory of Determinants , 1904, Nature.

[5]  L. J. Rogers.  On the Representation of Certain Asymptotic Series as Convergent Continued Fractions , 1907 .

[6]  Schrutka.  Die Determinanten , 1912 .

[7]  The theory of determinants in the historical order of development , 1914 .

[8]  J. Shaw.  Combinatory Analysis , 1917, Nature.

[9]  The General Symbolic Notation for the Principle of Duality, and its Application to Determinants , 1924 .

[10]  N. E. Nörlund.  Vorlesungen über Differenzenrechnung , 1924 .

[11]  H. W. Turnbull,et al.  The Theory of Determinants. Matrices, and Invariants , 1929 .

[12]  L. M. M.-T..  The Theory of Determinants, Matrices and Invariants , 1929, Nature.

[13]  A. A..  Contributions to the History of Determinants, 1900–1920 , 1930, Nature.

[14]  Some determinants of Bernoulli, Euler and related numbers , 1959 .

[15]  Samuel Karlin,et al.  On certain determinants whose elements are orthogonal polynomials , 1960 .

[16]  A. Ostrowski,et al.  On some determinants with combinatorial numbers. , 1964 .

[17]  Some determinants of q-binomial coefficients. , 1967 .

[18]  D. Foata.  ON THE NETTO INVERSION NUMBER OF A SEQUENCE , 1968 .

[19]  Basil Gordon,et al.  Notes on plane partitions. I , 1968 .

[20]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[21]  Madan Lal Mehta,et al.  Random Matrices and the Statistical Theory of Energy Levels. , 1970 .

[22]  R. Stanley.  Theory and applications of plane partitions: Part 1 , 1971 .

[23]  I. Gokhberg,et al.  Canonical factorization of continuous operator functions relative to the circle , 1972 .

[24]  N. N. Shapovalov.  On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra , 1972 .

[25]  Evaluation of Some Steck Determinants with Applications , 1974 .

[26]  A. J. Poorten.  Some determinants that should be better known , 1976, Journal of the Australian Mathematical Society.

[27]  George E. Andrews,et al.  The equivalence of the Bender-Knuth and MacMahon conjectures , 1977 .

[28]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[29]  George E. Andrews,et al.  Plane partitions (III): The weak Macdonald conjecture , 1979 .

[30]  B. R. Handa,et al.  Enumeration of higher-dimensional paths under restrictions , 1979, Discret. Math..

[31]  Philippe Flajolet.  Combinatorial aspects of continued fractions , 1980, Discret. Math..

[32]  W. H. Mills,et al.  Proof of the Macdonald conjecture , 1982 .

[33]  Vladimir E. Korepin,et al.  Calculation of norms of Bethe wave functions , 1982 .

[34]  Basil Gordon,et al.  A proof of the Bender-Knuth conjecture. , 1983 .

[35]  David P. Robbins,et al.  Alternating Sign Matrices and Descending Plane Partitions , 1983, J. Comb. Theory, Ser. A.

[36]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[37]  Sergei K. Suslov,et al.  The Hahn and Meixner polynomials of an imaginary argument and some of their applications , 1985 .

[38]  Ian P. Goulden,et al.  Ballot Sequences and a Determinant of Good's , 1985, J. Comb. Theory, Ser. A.

[39]  Richard P. Stanley,et al.  A Baker's dozen of conjectures concerning plane partitions , 1986 .

[40]  Richard P. Stanley,et al.  Symmetries of plane partitions , 1986, J. Comb. Theory A.

[41]  Howard Rumsey,et al.  Determinants and alternating sign matrices , 1986 .

[42]  T. Curtright,et al.  Symmetry patterns in the mass spectra of dual string models , 1986 .

[43]  David P. Robbins,et al.  Enumeration of a symmetry class of plane partitions , 1987, Discret. Math..

[44]  George E. Andrews,et al.  Plane partitions IV: A conjecture of Mills—Robbins—Rumsey , 1987 .

[45]  A. G. Izergin,et al.  Partition function of the six-vertex model in a finite volume , 1987 .

[46]  Helen Au-Yang,et al.  Critical correlations in a Z-invariant inhomogeneous ising model , 1987 .

[47]  Robert A. Proctor.  Odd symplectic groups , 1988 .

[48]  Anders Thorup,et al.  On Giambelli's theorem on complete correlations , 1989 .

[49]  G. David,et al.  The problem of Calissons , 1989 .

[50]  Souichi Okada,et al.  On the generating functions for certain classes of plane partitions , 1989, J. Comb. Theory, Ser. A.

[51]  John R. Stembridge,et al.  Nonintersecting Paths, Pfaffians, and Plane Partitions , 1990 .

[52]  Robert A. Proctor.  New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi , 1990, Eur. J. Comb..

[53]  Robert A. Proctor.  Product evaluations of lefschetz determinants for grassmannians and of determinants of multinomial coefficients , 1990, J. Comb. Theory, Ser. A.

[54]  Richard M. Wilson,et al.  A Diagonal Form for the Incidence Matrices of t-Subsets vs.k-Subsets , 1990, Eur. J. Comb..

[55]  Doron Zeilberger,et al.  A fast algorithm for proving terminating hypergeometric identities , 1990, Discret. Math..

[56]  C. Krattenthaler.  A determinant evaluation and some enumeration results for plane partitions , 1990 .

[57]  D. Zeilberger.  A holonomic systems approach to special functions identities , 1990 .

[58]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[59]  Christian Krattenthaler,et al.  Generating functions for plane partitions of a given shape , 1990 .

[60]  Lawrence J. Smolinsky,et al.  A combinatorial matrix in $3$-manifold theory. , 1991 .

[61]  Determinant formula for Selberg-type integrals , 1991 .

[62]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..

[63]  W. T. Tutte.  On the Birkhoff-Lewis equations , 1991, Discret. Math..

[64]  W. B. R. Lickorish,et al.  Invariants for $3$-manifolds from the combinatorics of the Jones polynomial. , 1991 .

[65]  James A. Wilson.  Orthogonal functions from gram determinants , 1991 .

[67]  David P. Robbins,et al.  The Story of 1, 2, 7, 42, 429, 7436, … , 1991 .

[68]  Vadim Schechtman,et al.  Arrangements of hyperplanes and Lie algebra homology , 1991 .

[69]  David W. Lewis,et al.  Matrix theory , 1991 .

[70]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[71]  Christian Radoux,et al.  Déterminant de Hankel Construit sur des Polynômes Liés aux Nombres de Dérangements , 1991, Eur. J. Comb..

[72]  Don Zagier,et al.  Realizability of a model in infinite statistics , 1992 .

[73]  Doron Zeilberger,et al.  An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities , 1992 .

[74]  G. Harris,et al.  A Note on Generalized Vandermonde Determinants , 1993, SIAM J. Matrix Anal. Appl..

[75]  V. Korepin,et al.  Quantum Inverse Scattering Method and Correlation Functions , 1993, cond-mat/9301031.

[76]  M. Micu.  Continuous Hahn polynomials , 1993 .

[77]  W. T. Tutte.  The Matrix of Chromatic Joins , 1993, J. Comb. Theory, Ser. B.

[78]  Alexandre Varchenko,et al.  Bilinear form of real configuration of hyperplanes , 1993 .

[79]  C. Krattenthaler.  Generating functions for shifted plane partitions , 1993 .

[80]  B. Leclerc,et al.  On Identities Satisfied by Minors of a Matrix , 1993 .

[81]  George E. Andrews,et al.  Plane Partitions V: The TSSCPP Conjecture , 1994, J. Comb. Theory A.

[82]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[83]  Alain Lascoux,et al.  Noncommutative symmetric functions , 1994 .

[84]  Frédéric Chyzak,et al.  Holonomic systems and automatic proofs of identities , 1994 .

[85]  David M. Jackson.  The Lattice of Non-crossing Partitions and the Birkhoff-Lewis Equations , 1994, Eur. J. Comb..

[86]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[87]  Robert A. Proctor.  Young Tableaux, Gelfand Patterns, and Branching Rules for Classical Groups , 1994 .

[88]  Holonomic $q$-difference system of the first order associated with a Jackson integral of Selberg type , 1994 .

[89]  Greg Kuperberg.  Symmetries of Plane Partitions and the Permanent - Determinant Method , 1994, J. Comb. Theory, Ser. A.

[90]  Jiang Zeng,et al.  An extension of Euler numbers and alternating permutation records , 1994 .

[91]  Fractions continues, combinatoire et extensions de nombres classiques , 1994 .

[92]  A. Varchenko.  Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups , 1995 .

[93]  Neil J. A. Sloane,et al.  The encyclopedia of integer sequences , 1995 .

[94]  Masato Wakayama,et al.  Minor summation formula of Pfaffians , 1995 .

[95]  Alfred J. van der Poorten,et al.  Determinants in the Study of Thue's Method and Curves with Prescribed Singularities , 1995, Exp. Math..

[96]  Israel M. Gelfand,et al.  Noncommutative Symmetrical Functions , 1995 .

[97]  Herbert S. Wilf,et al.  Five Surprisingly Simple Complexities , 1995, J. Symb. Comput..

[98]  Doron Zeilberger.  Reverend Charles to the aid of Major Percy and Fields-Medalist Enrico , 1995 .

[99]  John R. Stembridge.  The Enumeration of Totally Symmetric Plane Partitions , 1995 .

[100]  A. Camina,et al.  Line-transitive Automorphism Groups of Linear Spaces , 1995 .

[101]  Peter Paule,et al.  A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities , 1995, J. Symb. Comput..

[102]  Christian Krattenthaler,et al.  The major counting of nonintersecting lattice paths and generating functions for tableaux , 1995 .

[103]  J. Cardy,et al.  QUANTUM INVERSE SCATTERING METHOD AND CORRELATION FUNCTIONS , 1995 .

[104]  Dragutin Svrtan,et al.  Study of Gram Matrices in Fock Representation of Multiparametric Canonical Commutation Relations, Extended Zagier's Conjecture, Hyperplane Arrangements and Quantum Groups , 1996 .

[105]  Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras , 1996, q-alg/9604011.

[106]  Rene F. Swarttouw,et al.  The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report Fac , 1996, math/9602214.

[107]  Donald E. Knuth,et al.  Overlapping Pfaffians , 1995, Electron. J. Comb..

[108]  Ian P. Goulden,et al.  Further Determinants with the Averaging Property of Andrews-Burge , 1996, J. Comb. Theory, Ser. A.

[109]  P. Di Francesco,et al.  Meander Determinants , 1996, hep-th/9612026.

[110]  Marko Petkovsek,et al.  A high-tech proof of the Mills-Robbins-Rumsey determinant formula , 1995, Electron. J. Comb..

[111]  Determinants of Partition Matrices , 1996 .

[112]  C. Krattenthaler.  SOME q-ANALOGUES OF DETERMINANT IDENTITIES WHICH AROSE IN PLANE PARTITION ENUMERATION , 1996 .

[113]  S. Milne,et al.  New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[114]  Jiang Zeng.  Sur quelques propriétés de symétrie des nombres de Genocchi , 1996, Discret. Math..

[115]  Greg Kuperberg,et al.  Another proof of the alternating sign matrix conjecture , 1996 .

[116]  Doron Zeilberger,et al.  Proof of the alternating sign matrix conjecture , 1994, Electron. J. Comb..

[117]  Jiang Zeng,et al.  On some symmetric properties of Genocchi numbers , 1996 .

[118]  D. Zeilberger.  Proof of the refined alternating sign matrix conjecture. , 1996 .

[119]  Jiang Zeng,et al.  Une Famille de Polynomes Qui Interpole Plusieurs Suites Classiques de Nombres , 1996 .

[120]  Graham Denham,et al.  On the Smith normal form of the Varchenko bilinear form of a hyperplane arrangement , 1997 .

[121]  Daniel Krob,et al.  Noncommutative Symmetric Functions II: Transformations of Alphabets , 1997, Int. J. Algebra Comput..

[122]  Doron Zeilberger.  Dodgson's Determinant-Evaluation Rule Proved by TWO-TIMING MEN and WOMEN , 1997, Electron. J. Comb..

[123]  E. Guitter,et al.  Meanders and the Temperley-Lieb algebra , 1997 .

[124]  Daniel Krob,et al.  Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras , 2006, Discret. Math. Theor. Comput. Sci..

[125]  P. Francesco.  SU(N) meander determinants , 1997, hep-th/9702181.

[126]  Christian Krattenthaler.  Determinant Identities and a Generalization of the Number of Totally Symmetric Self-complementary Plane Partitions , 1997, Electron. J. Comb..

[127]  A. J. Coleman,et al.  Factoring the Shapovalov Determinant, II , 1997 .

[128]  Christian Krattenthaler,et al.  The Number of Rhombus Tilings of a “Punctured” Hexagon and the Minor Summation Formula , 1997 .

[129]  Rhombus Tilings of a Hexagon with Three Missing Border Triangles , 1997 .

[130]  Tom Brylawski,et al.  The Determinant Formula for a Matroid Bilinear Form , 1997 .

[131]  THE DETERMINANT OF A HYPERGEOMETRIC PERIOD MATRIX , 1997, alg-geom/9709017.

[132]  Arthur Randrianarivony,et al.  Fractions Continues, q-Nombres de Catalan etq-Polynômes de Genocchi , 1997, Eur. J. Comb..

[133]  Mihai Ciucu,et al.  Enumeration of Perfect Matchings in Graphs with Reflective Symmetry , 1997, J. Comb. Theory, Ser. A.

[134]  George E. Andrews,et al.  Ramanujan's method in q-series congruences , 1996, Electron. J. Comb..

[135]  A. Varchenko,et al.  Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups , 1997, q-alg/9703044.

[136]  Mourad E. H. Ismail,et al.  Classical Orthogonal Polynomials as Moments , 1997, Canadian Journal of Mathematics.

[137]  Michael J. Schlosser,et al.  Multidimensional Matrix Inversions and Ar and Dr Basic Hypergeometric Series , 1997 .

[138]  Determinant Evaluations and U(n) Extensions of Heine's 2 1 -transformations , 1997 .

[139]  Determinants of $q$-hypergeometric functions and another proof of the Askey conjecture , 1997 .

[140]  T. Lam,et al.  Representations of Finite Groups: A Hundred Years, Part II , 1998 .

[141]  Mourad E. H. Ismail,et al.  More Orthogonal Polynomials as Moments , 1998 .

[142]  Ira M. Gessel,et al.  ENUMERATION OF TILINGS OF DIAMONDS AND HEXAGONS WITH DEFECTS , 1998 .

[143]  Arthur Randrianarivony,et al.  q, p-Analogue des nombres de Catalan , 1998, Discret. Math..

[144]  T. Lam.  Representations of Finite Groups : A Hundred Years , Part I , 1998 .

[145]  Bernard Leclerc,et al.  On certain formulas of Karlin and Szegö. , 1998 .

[146]  Twenty Open Problems in Enumeration of Matchings , 1998, math/9801060.

[147]  Rhombus Tilings of a Hexagon with Two Triangles Missing on the Symmetry Axis , 1998, math/9810019.

[148]  Richard P. Stanley,et al.  Hyperplane Arrangements, Parking Functions and Tree Inversions , 1998 .

[149]  M. Ismail,et al.  On a family of orthogonal polynomials related to elliptic functions , 1998 .

[151]  Christian Krattenthaler,et al.  The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I , 1998 .

[152]  Derivation of q-difference equation from connection matrix for selberg type jackson integrals , 1998 .

[153]  Graham Denham,et al.  Some algebraic properties of the Schechtman-Varchenko bilinear forms , 1998 .

[154]  Mihai Ciucu,et al.  Plane partitions II: 5 1/2 symmetry classes , 1998 .

[155]  Gian-Carlo Rota,et al.  Mathematical Essays in honor of Gian-Carlo Rota , 1998 .

[156]  Richard P. Stanley,et al.  A $q$-deformation of a trivial symmetric group action , 1998 .

[157]  Soichi Okada,et al.  Applications of Minor Summation Formulas to Rectangular-Shaped Representations of Classical Groups , 1998 .

[159]  Eigenspace decompositions with respect to symmetrized incidence mappings. , 1998 .

[160]  George E. Andrews,et al.  Determinants in Plane Partition Enumeration , 1998, Eur. J. Comb..

[161]  Christian Krattenthaler,et al.  An Alternative Evaluation of the Andrews-Burge Determinant , 1998 .

[162]  C. Krattenthaler.  A new proof of the M-R-R conjecture-including a generalization , 1999 .

[163]  David M. Bressoud,et al.  How the Alternating Sign Matrix Conjecture Was Solved , 1999 .

[164]  Jiang Zeng,et al.  q-Polynômes de Gandhi et statistique de Denert , 1999, Discret. Math..

[165]  Mourad E. H. Ismail,et al.  Some continued fractions related to elliptic functions , 1999 .

[166]  Bojan Mohar,et al.  The Grötzsch Theorem for the Hypergraph of Maximal Cliques , 1999, Electron. J. Comb..

[167]  Mukarram Ahmad,et al.  Continued fractions , 2019, Quadratic Number Theory.

[168]  Theresia Eisenkölbl.  Rhombus Tilings of a Hexagon with Two Triangles Missing on the Symmetry Axis , 1999, Electron. J. Comb..

[169]  D. Bressoud.  Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture , 1999 .

[170]  Ira M. Gessel,et al.  Tilings of Diamonds and Hexagons with Defects , 1999, Electron. J. Comb..

[171]  Mihai Ciucu,et al.  The Number of Centered Lozenge Tilings of a Symmetric Hexagon , 1999, J. Comb. Theory, Ser. A.

[172]  Gérard D. Cohen,et al.  New Bounds for Codes Identifying Vertices in Graphs , 1999, Electron. J. Comb..

[173]  Wenchang Chu,et al.  Binomial convolutions and determinant identities , 1999, Discret. Math..

[174]  Theresia Eisenkölbl.  Rhombus Tilings of a Hexagon with Three Fixed Border Tiles , 1999, J. Comb. Theory, Ser. A.

[175]  Michael J. Schlosser.  Summation theorems for multidimensional basic hypergeometric series by determinant evaluations , 2000, Discret. Math..

[176]  Rodica Simion,et al.  Noncrossing partitions , 2000, Discret. Math..

[177]  Christian Krattenthaler,et al.  The Number of Rhombus Tilings of a Symmetric Hexagon which Contain a Fixed Rhombus on the Symmetry Axis, II , 2000, Eur. J. Comb..

[178]  Christian Krattenthaler,et al.  Asymptotic redundancies for universal quantum coding , 2000, IEEE Trans. Inf. Theory.

[179]  A New Multidimensional Matrix Inversion in , 2000 .

[180]  Hankel Determinants of Eisenstein Series , 2000, math/0009130.

[181]  H. Wall,et al.  Analytic Theory of Continued Fractions , 2000 .

[182]  Kiran S. Kedlaya.  Another Combinatorial Determinant , 2000, J. Comb. Theory, Ser. A.

[183]  Stephen C. Milne,et al.  Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions , 2000, math/0008068.

[184]  A. Lascoux.  Square-Ice Enumeration , 2001 .

[185]  Alfred J. van der Poorten,et al.  A Powerful Determinant , 2001, Exp. Math..

[186]  Jiang Zeng,et al.  Un autre q-analogue des nombres d’Euler , 2001 .

[187]  Mihai Ciucu,et al.  Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole , 2001, J. Comb. Theory, Ser. A.

[188]  Ilse Fischer.  Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre , 2001, J. Comb. Theory, Ser. A.