Non-commutative extensions of the MacMahon Master Theorem

We present several non-commutative extensions of the MacMahon Master Theorem, further extending the results of Cartier?Foata and Garoufalidis?Le?Zeilberger. The proofs are combinatorial and new even in the classical cases. We also give applications to the s-extension and Krattenthaler?Schlosser's q-analogue.

[1]  I. Gel'fand,et al.  A theory of noncommutative determinants and characteristic functions of graphs , 1992 .

[2]  Y. Manin.  Multiparametric quantum deformation of the general linear supergroup , 1989 .

[3]  Igor Pak,et al.  AN ALGEBRAIC EXTENSION OF THE MACMAHON MASTER THEOREM , 2006 .

[4]  M. Lorenz,et al.  Koszul algebras and the quantum MacMahon master theorem , 2006, math/0603169.

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

[6]  Doron Zeilberger,et al.  Laguerre Polynomials, Weighted Derangements, and Positivity , 1988, SIAM J. Discret. Math..

[7]  R. Berger.  Koszulity for Nonquadratic Algebras , 2001 .

[8]  Michael J. Schlosser,et al.  A new multidimensional matrix inverse with applications to multiple q-series , 1999, Discret. Math..

[9]  George E. Andrews.  Identities in combinatorics. II. A $q$-analog of the Lagrange inversion theorem , 1975 .

[10]  Doron Zeilberger,et al.  A combinatorial approach to matrix algebra , 1985, Discret. Math..

[11]  The β-extension of the multivariable Lagrange inversion formula , 1991 .

[12]  Y. Manin.  Some remarks on Koszul algebras and quantum groups , 1987 .

[13]  G. Lallement.  Semigroups and combinatorial applications , 1979 .

[14]  C. Krattenthaler.  Operator methods and Lagrange inversion: a unified approach to Lagrange formulas , 1988 .

[15]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[16]  The quantum MacMahon Master Theorem , 2003, Proceedings of the National Academy of Sciences.

[18]  A basis for the right quantum algebra and the “1=q” principle , 2006, math/0603463.

[19]  A. Kirillov,et al.  Introduction to Superanalysis , 1987 .

[20]  M. Minoux.  Extension of MacMahon's Master Theorem to pre-semi-rings , 2001 .

[21]  Dominique Foata,et al.  A noncommutative version of the matrix inversion formula , 1979 .

[22]  Ira M. Gessel,et al.  Applications of q-lagrange inversion to basic hypergeometric series , 1983 .

[23]  Pierre Cartier,et al.  Problemes combinatoires de commutation et rearrangements , 1969 .

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

[25]  Ira M. Gessel,et al.  A noncommutative generalization and $q$-analog of the Lagrange inversion formula , 1980 .

[26]  I. Good.  A short proof of MacMahon's ‘Master Theorem’ , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  G. Viennot.  Heaps of Pieces, I: Basic Definitions and Combinatorial Lemmas , 1989 .