A noncommutative generalization and 𝑞-analog of the Lagrange inversion formula

Gessel, Ira

doi:10.1090/S0002-9947-1980-0552269-2

A noncommutative generalization and $q$-analog of the Lagrange inversion formula
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Trans. Amer. Math. Soc. 257 (1980), 455-482 Request permission

Abstract:

The Lagrange inversion formula is generalized to formal power series in noncommutative variables. A q-analog is obtained by applying a linear operator to the noncommutative formula before substituting commuting variables.

References

George E. Andrews, Identities in combinatorics. II. A $q$-analog of the Lagrange inversion theorem, Proc. Amer. Math. Soc. 53 (1975), no. 1, 240–245. MR 389610, DOI 10.1090/S0002-9939-1975-0389610-3
George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
L. Carlitz, Sequences, paths, ballot numbers, Fibonacci Quart. 10 (1972), no. 5, 531–549. MR 317949
L. Carlitz and J. Riordan, Two element lattice permutation numbers and their $q$-generalization, Duke Math. J. 31 (1964), 371–388. MR 168490, DOI 10.1215/S0012-7094-64-03136-9
L. Carlitz and Richard Scoville, A note on weighted sequences, Fibonacci Quart. 13 (1975), no. 4, 303–306. MR 387174
Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128, DOI 10.1007/978-94-010-2196-8
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Dominique Foata, La série génératrice exponentielle dans les problèmes d’énumération, Séminaire de Mathématiques Supérieures, No. 54 (Été, vol. 1971, Les Presses de l’Université de Montréal, Montreal, Que., 1974 (French). Avec un chapitre sur les identités probabilistes dérivées de la formule exponentielle, par B. Kittel. MR 0505546
D. Foata and M. P. Schützenberger, On the principle of equivalence of Sparre Anderson, Math. Scand. 28 (1971), 308–316 (1972). MR 308296, DOI 10.7146/math.scand.a-11025

A factorization for formal Laurent series and lattice path enumeration

Ira Gessel and Da Lun Wang, Depth-first search as a combinatorial correspondence, J. Combin. Theory Ser. A 26 (1979), no. 3, 308–313. MR 535161, DOI 10.1016/0097-3165(79)90108-0
Jay Goldman and Gian-Carlo Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Studies in Appl. Math. 49 (1970), 239–258. MR 265181, DOI 10.1002/sapm1970493239
D. Gouyou-Beauchamps, Deux propriétés combinatoires du langage de Łukasiewicz, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Informat. Théor. 9 (1975), no. R-3, 13–24 (French, with English summary). MR 395351, DOI 10.1051/ita/197509R300131

An introduction to the theory of numbers

M. J. Hodel, Weighted sequences of nonnegative integers, Duke Math. J. 40 (1973), 493–510. MR 337753, DOI 10.1215/S0012-7094-73-04043-X
John G. Kemeny and J. Laurie Snell, Mathematical models in the social sciences, MIT Press, Cambridge, Mass.-London, 1972. Reprint of the 1962 edition. MR 0363521
Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 0378456

Note on the number of pairs of non-intersecting routes

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C. L. Mallows and John Riordan, The inversion enumerator for labeled trees, Bull. Amer. Math. Soc. 74 (1968), 92–94. MR 218254, DOI 10.1090/S0002-9904-1968-11888-9
T. V. Narayana, A partial order and its applications to probability theory, Sankhyā 21 (1959), 91–98. MR 106498
G. Pólya, On the number of certain lattice polygons, J. Combinatorial Theory 6 (1969), 102–105. MR 236031, DOI 10.1016/S0021-9800(69)80113-4
George N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc. 94 (1960), 441–451. MR 114765, DOI 10.1090/S0002-9947-1960-0114765-9
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
John Riordan, A note on a $q$-extension of ballot numbers, J. Combinatorial Theory 4 (1968), 191–193. MR 221958, DOI 10.1016/S0021-9800(68)80041-9
Issai Schur, Identities in the theory of power series, Amer. J. Math. 69 (1947), 14–26. MR 19752, DOI 10.2307/2371650
M. P. Schützenberger, On a factorisation of free monoids, Proc. Amer. Math. Soc. 16 (1965), 21–24. MR 170971, DOI 10.1090/S0002-9939-1965-0170971-9

Le théorème de Lagrange selon G. N. Raney

L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83–90. MR 387069, DOI 10.1016/0012-365X(76)90009-1
J. G. Wendel, Left-continuous random walk and the Lagrange expansion, Amer. Math. Monthly 82 (1975), 494–499. MR 381000, DOI 10.2307/2319745