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Abstract: 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. (Update)
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BibTeX entry: (Update)
C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (1999), to appear. http://citeseer.ist.psu.edu/article/krattenthaler99advanced.html More
@misc{ krattenthaler99advanced,
author = "C. Krattenthaler",
title = "Advanced determinant calculus",
text = "C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien
Combin. 42 (1999), to appear.",
year = "1999",
url = "citeseer.ist.psu.edu/article/krattenthaler99advanced.html" }
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36 Generating functions for plane partitions of a given shape (context) - Krattenthaler - 1990
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34 Determinant identities and a generalization of the number of.. - Krattenthaler - 1997
33 The number of rhombus tilings of a symmetric hexagon which c.. - Fulmek, Krattenthaler - 1998
33 The number of rhombus tilings of a symmetric hexagon which c.. - Fulmek, Krattenthaler
32 Twenty open problems on enumeration of matchings (context) - Propp - 1996
32 The problem of the calissons (context) - David, Tomei - 1989
32 Alternating sign matrices and descending plane partitions (context) - Mills, Robbins et al. - 1983
30 American Mathematical Society Colloquium Publications Vol (context) - Szego, Polynomials - 1959
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28 pfaffians and plane partitions (context) - Stembridge, paths - 1990
28 Wilf and D (context) - Petkovsek - 1996
27 A treatise on the theory of determinants (context) - Muir - 1960
25 analogues of determinant identities which arose in plane par.. (context) - Krattenthaler, q- - 1996
24 Cambridge University Press (context) - MacMahon, Analysis
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21 Noncommutative symmetric functions III: Deformations of Cauc.. - Duchamp, Klyachko et al. - 1997
20 Odd symplectic groups (context) - Proctor - 1988
20 Noncommutative symmetric functions II: Transformations of al.. - Krob, Leclerc et al. - 1997
18 conjecture --- including a generalization (context) - Krattenthaler, proof et al.
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18 Proof of the Macdonald conjecture (context) - Mills, Robbins et al. - 1982
17 Determinant identities (context) - Andrews, Burge - 1993
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16 Another proof of the alternating sign matrix conjecture (context) - Kuperberg - 1996
15 Determinants in plane partition enumeration - Andrews, Stanton - 1998
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15 parking functions and tree inversions (context) - Stanley, arrangements - 1998
15 The enumeration of totally symmetric plane partitions (context) - Stembridge - 1995
15 Multidimensional hypergeometric functions and representation.. (context) - Varchenko - 1995
15 An alternative evaluation of the Andrews--Burge determinant - Krattenthaler - 1998
15 Realizability of a model in infinite statistics (context) - Zagier - 1992
14 New symmetric plane partition identities from invariant theo.. (context) - Proctor - 1990
14 Applications of minor summation formulas to rectangular-shap.. (context) - Okada - 1998
14 A baker's dozen of conjectures concerning plane partitions (context) - Stanley - 1986
14 Further determinants with the averaging property of AndrewsB.. (context) - Goulden, Jackson - 1996
13 On identities satisfied by minors of a matrix (context) - Leclerc - 1993
13 The weak Macdonald conjecture (context) - Andrews, partitions - 1979
13 Proofs and confirmations --- The story of the alternating si.. (context) - Bressoud - 1999
13 Determinants in the study of Thue's method and curves with p.. - Bombieri, Hunt et al. - 1995
12 A determinant evaluation and some enumeration results for pl.. - Krattenthaler - 1990
12 hexagon and the minor summation formula (context) - Okada, Krattenthaler et al. - 1998
11 Condensation of determinants (context) - Dodgson
11 The equivalence of the Bender--Knuth and the MacMahon conjec.. (context) - Andrews, partitions - 1977
11 Multidimensional matrix inversion and A r and D r basic hype.. (context) - Schlosser - 1997
11 Bilinear form of real configuration of hyperplanes (context) - Varchenko - 1993
11 A proof of the Bender--Knuth conjecture (context) - Gordon - 1983
10 Plane partitions I: A generalization of MacMahon's formula - Ciucu
10 Holonomic systems and automatic proofs of identities - Chyzak - 1994
10 Plane partitions II: 5 1 2 symmetry classes - Ciucu, Krattenthaler
10 Determinants and alternating sign matrices (context) - Robbins, Rumsey - 1986
9 On certain determinants whose elements are orthogonal polyno.. (context) - Karlin, Szego - 1961
9 the generating functions for certain classes of plane partit.. (context) - Okada - 1989
9 Proof of a determinant evaluation conjectured by Bombieri - Krattenthaler, Zeilberger - 1997
8 Enumeration of lozenge tilings of hexagons with a central tr.. - Ciucu, Eisenkolbl et al.
8 Noncrossing partitions (context) - Simion
8 Continued Fractions (context) - Jones, Thron - 1980
8 The Hahn and Meixner polynomials of an imaginary argument an.. (context) - Atakishiyev, Suslov - 1985
8 Proof of the refined alternating sign matrix conjecture - Zeilberger - 1996
8 and branching rules for classical groups (context) - Proctor, tableaux et al. - 1994
7 translated from the Russian (context) - Gantmacher, of - 1959
7 deformation of a trivial symmetric group action (context) - Hanlon, Stanley et al. - 1999
7 The determinant formula for a matroid bilinear form (context) - Brylawski, Varchenko - 1997
7 On Giambelli's theorem on complete correlations (context) - Laksov, Thorup et al. - 1989
6 Summation theorems for multidimensional basic hypergeometric.. - Schlosser
6 Sur la r'eduction en fraction continue d'une s'erie pr'ec'ed.. (context) - Stieltjes
6 Generating functions for shifted plane partitions - Krattenthaler - 1993
6 Exact enumeration of tilings of diamonds and hexagons with d.. (context) - Helfgott, Gessel - 1999
6 Minor summation formula for pfaffians (context) - Ishikawa, Wakayama - 1995
6 Continuous Hahn polynomials (context) - Askey - 1985
6 Enumeration of rhombus tilings of a hexagon which contain a .. - Fischer
5 A new multidimensional matrix inversion in A r (context) - Schlosser
5 a bilinear form on the universal enveloping algebra of a com.. (context) - Shapovalov - 1972
5 Plane partitions (context) - Andrews - 1994
5 Die Determinanten (context) - Pascal - 1900
5 Study of Gram matrices in Fock representation of multiparame.. (context) - Meljanac, Svrtan - 1996
5 Sur quelques int'egrales d'efinies et leur d'eveloppement en.. (context) - Stieltjes
5 Meander determinants (context) - Di Francesco - 1998
5 meander determinants (context) - Di Francesco - 1997
5 the representation of certain asymptotic series as convergen.. (context) - Rogers - 1907
5 Calculation of norms of Bethe wave functions (context) - Korepin - 1982
5 Classical orthogonal polynomials as moments (context) - Ismail, Stanton - 1997
5 New infinite families of exact sums of squares formulas (context) - Milne - 1996
5 A conjecture of Mills--Robbins--Rumsey (context) - Andrews, partitions - 1987
5 London and Glasgow (context) - Turnbull, of et al. - 1929
5 Some algebraic properties of the Schechtman--Varchenko bilin.. (context) - Denham, Hanlon - 1998
4 Editions de Physique (context) - Mehta, theory - 1989
4 The Wilf Festschrift (context) - Zeilberger, rule et al. - 1997
4 Macdonald's conjecture and descending plane partitions (context) - Andrews - 1980
4 Notes on plane partitions V (context) - Gordon - 1971
4 Orthogonal functions from Gram determinants (context) - Wilson - 1991
4 hypergeometric functions as a bridge between Yangians and qu.. (context) - Tarasov, Varchenko et al. - 1997
4 On some determinants with combinatorial numbers (context) - Ostrowski - 1964
4 Some determinants of Bernoulli (context) - Al-Salam, Carlitz - 1959
4 as of Karlin and Szego, S'eminaire Lotharingien Combin (context) - Leclerc, formul - 1998
4 A combinatorial matrix in 3-manifold theory (context) - Ko, Smolinsky - 1991
4 Rhombus tilings of a hexagon with two triangles missing on t.. (context) - Eisenkolbl
4 quantum affine algebras and elliptic quantum groups (context) - Tarasov, Varchenko et al. - 1997
4 Enumeration of lozenge tilings of hexagons with cut off corn.. - Ciucu, Krattenthaler
4 A powerful determinant (context) - van der Poorten
4 Nombres de Bell et polynomes de Charlier (context) - Delsarte - 1978
4 the Smith normal form of the Varchenko bilinear form of a hy.. - Denham, Hanlon - 1997
4 unpublished research announcement (context) - Proctor - 1984
3 The Foata Festschrift (context) - Knuth, pfaffians et al. - 1996
3 The Andrews Festschrift (context) - Lascoux, enumeration et al. - 1999
3 Contributions to the history of determinants (context) - Muir
3 Formulas for the evaluation of Toeplitz determinants with ra.. (context) - Basor, Forrester - 1994
3 Bestimmung der symmetrischen Verbindungen ihrer erzeugenden .. (context) - Borchardt
3 Partition function of the six-vertex model in a finite volum.. (context) - Izergin - 1987
3 Reverend Charles to the aid of Major Percy and Fields medali.. (context) - Zeilberger - 1996
3 Asymptotic redundancies for universal quantum coding - Krattenthaler, Slater
3 More orthogonal polynomials as moments - Ismail, Stanton - 1998
3 Representations of finite groups: A hundred years (context) - Lam - 1998
3 The matrix of chromatic joins (context) - Tutte - 1993
3 The Foata Festschrift (context) - Petkovsek, Wilf et al. - 1996
3 nombres de Catalan et q-polynomes de Genocchi (context) - Randrianarivony, continues - 1997
3 analogue des nombres de Catalan (context) - Randrianarivony, p- - 1998
3 A note on generalized Vandermonde determinants (context) - Flowe, Harris - 1993
3 Meanders and the Temperley--Lieb algebra (context) - Di Francesco, Golinelli et al. - 1997
3 The MacMahon conjecture (context) - Andrews, partitions - 1978
3 Five surprisingly simple complexities (context) - Strehl, Wilf - 1995
2 Some extensions of Cauchy's determinant (context) - Milne
2 The Foata Festschrift (context) - Zeilberger, the et al. - 1996
2 Another combinatorial determinant (context) - Kedlaya
2 Some continued fractions related to elliptic functions (context) - Ismail, Masson
2 Invariants for 3-manifold from the combinatorics of the Jone.. (context) - Lickorish - 1991
2 Sur l'apolarit'e des formes binaires (context) - Voigt - 1901
2 difference equation from connection matrix for Selberg type .. (context) - Aomoto, Kato et al. - 1998
2 Enumeration of higher-dimensional paths under restrictions (context) - Handa, Mohanty - 1979
2 Polynomes de Ghandi et statistique de Denert (context) - Han, Zeng
2 a family of orthogonal polynomials related to elliptic funct.. (context) - Ismail, Valent - 1998
2 Determinant formula for Selberg type integrals (context) - Varchenko - 1991
2 A diagonal form for the incidence matrices of t-subsets vers.. (context) - Wilson - 1990
2 The Andrews Festschrift (context) - Han, Randrianarivony et al. - 1999
2 and critical points of linear function on a configuration of.. (context) - Varchenko, beta et al.
2 Hankel determinants of Eisenstein series (context) - Milne
2 extensions of Heine's 2 OE 1 -transformations (context) - Gustafson, Krattenthaler et al. - 1997
2 and further identities for powers of classical theta functio.. (context) - Milne, fractions et al.
2 Evaluation of some Steck determinants with applications (context) - Steck - 1974
2 Ballot sequences and a determinant of Good's (context) - Goulden, Jackson - 1985
2 The Andrews Festschrift (context) - Garvan, tutorial et al. - 1999
2 Symmetrieklassen von Plane Partitions (context) - Schlosser - 1993
2 difference system of the 1st order associated with a Jackson.. (context) - Mimachi, q- - 1994
2 the Birkhoff--Lewis equation (context) - Tutte - 1991
2 A generalized Vandermonde determinant (context) - Buck, Coley et al. - 1992
2 Gould Anniversary Volume (context) - Chu, convolutions et al. - 1999
2 Das alternirende Exponentialdifferenzenproduct (context) - Schendel
2 The general symbolic notation for the principal of duality (context) - Turnbull - 1923
2 Rhombus tilings of a hexagon with three missing border tiles (context) - Eisenkolbl
2 Determinants of partition matrices (context) - Reinhart - 1996
2 A family of polynomials interpolating several classical seri.. (context) - Randrianarivony, Zeng - 1996
2 Some determinants that should be better known (context) - van der Poorten - 1976
2 Extension of Euler numbers and records of up-down permutatio.. (context) - Randrianarivony, Zeng - 1994
2 combinatoire et extensions de nombres classiques (context) - Randrianarivony, continues - 1994
2 The determinant of a hypergeometric period matrix (context) - Douai, Terao - 1997
2 Infinite families of exact sums of squares formulas (context) - Milne
2 Product evaluations of Lefschetz determinants for Grassmanni.. (context) - Proctor - 1990
2 A combinatorial determinant - Wilf
2 The lattice of noncrossing partitions and the Birkhoff--Lewi.. (context) - Jackson - 1994
2 Factoring the Shapovalov determinant (context) - Coleman, Cao - 1997
2 Symmetry patterns in the mass spectra of dual string models (context) - Curtright, Thorn - 1986
2 The Birkhoff--Lewis equations (context) - Dahab - 1993
2 hypergeometric functions and another proof of the Askey conj.. (context) - Terasoma, q- - 1997
2 The proofs of the alternating sign matrix conjecture (context) - Bressoud, Propp
2 invariant inhomogeneous Ising model (context) - Au-Yang, Perk et al. - 1987
2 Eigenspace decompositions with respect to symmetrized incide.. (context) - Kramer - 1998
1 319--333 (context) - Zeng, propri'etes et al. - 1996
1 Solution to problem (context) - Robbins - 1997
1 ADVANCED DETERMINANT CALCULUS (context) - Carlitz, of et al. - 1967
http://www.math.temple.edu/~tewodros/programs/kratdet.html
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