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| Mahonian statistics | |
|---|---|
| Name | Mahonian statistics |
| Field | Algebraic combinatorics |
| Introduced | 1915–1950s |
| Key figures | Percy A. MacMahon, Dominique Foata, Richard P. Stanley, Donald Knuth, Gian-Carlo Rota |
Mahonian statistics are integer-valued permutation statistics that enumerate permutations according to statistics equidistributed with the number of inversions. Originating in early 20th-century enumerative work, they connect classical permutation statistics to q-analogs, generating functions, and deep bijective correspondences. They play a central role in the study of symmetric functions, Coxeter groups, and q-enumeration problems.
Mahonian statistics are statistics on the symmetric group S_n whose distribution over S_n equals the distribution of the inversion number. The notion derives its name from Percy A. MacMahon, whose enumerative work influenced later developments by Dominique Foata, Richard P. Stanley, and others. Historical developments involve correspondence among mathematicians associated with the University of Cambridge, the École Normale, and institutions such as the Massachusetts Institute of Technology and Princeton University, and intersect with work by Srinivasa Ramanujan and George Pólya.
Two prototypical Mahonian statistics are the inversion number and the major index. The inversion number counts pairs (i,j) with i
Combinatorial interpretations of Mahonian statistics use bijections between permutations, words, tableaux, and lattice paths. Key bijective techniques originate in the work of Foata, Marcel-Paul Schützenberger, and François Bergeron, and draw on tableau theory by Alfred Young, Carl Gustav Jacob Jacobi, and Isaacs-style combinatorics. Bijections relate permutations to inversion sequences studied by Herb Wilf, David Callan, and Miklós Bóna, and to standard Young tableaux investigated by William Fulton, George Lusztig, and Israel Gelfand. Connections to the Robinson–Schensted correspondence appear in research by C. S. Sagan, Bruce Sagan, and Alexandre Kirillov.
Mahonian distributions give rise to q-analogs of factorials and binomial coefficients, such as the q-factorial and q-binomial coefficients. Generating functions studied by MacMahon, Leonard Carlitz, and Ira Gessel produce q-analog identities that interact with the theory of basic hypergeometric series developed by George Gasper, Mizan Rahman, and Srinivasa Ramanujan. Algebraic structures explored by Richard Stanley, Dennis Stanton, and George Andrews reveal connections to the Rogers–Ramanujan identities and to Hecke algebras studied by Iwahori and Kazhdan–Lusztig.
Generalizations extend Mahonian statistics from symmetric groups to Coxeter groups, hyperoctahedral groups, and complex reflection groups. Foundational work involves Coxeter group theory by H. S. M. Coxeter, Bourbaki-type treatments, and developments by Louis Solomon, T. A. Springer, and Jim Humphreys. Investigations in data by John Stembridge, Victor Reiner, and Nathan Reading analyze Mahonian phenomena on Weyl groups, while research by Francesco Brenti, Christian Stump, and Ezra Miller explores statistics on hyperplane arrangements and Artin groups.
Mahonian statistics interface with representation theory through symmetric functions, quasisymmetric functions, and modules for the symmetric group. Influential contributors include Alain Lascoux, Marcel-Paul Schützenberger, Richard P. Stanley, and James Haglund, whose work ties Mahonian distributions to Macdonald polynomials and Hilbert series of coinvariant algebras studied by George Lusztig and Victor Ginzburg. Connections to the representation theory of Hecke algebras and Cherednik algebras were pursued by I. G. Macdonald, Ivan Cherednik, and Nolan Wallach, while applications in Schubert calculus engage work by Anders Buch, William Fulton, and Sara Billey.