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Signed words and permutations, I: A fundamental transformation
Author(s):
Dominique
Foata;
Guo-Niu
Han
Journal:
Proc. Amer. Math. Soc.
135
(2007),
31-40.
MSC (2000):
Primary 05A15, 05A30, 05E15
Posted:
June 29, 2006
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Abstract:
The statistics major index and inversion number, usually defined on ordinary words, have their counterparts in signed words, namely the so-called flag-major index and flag-inversion number. We give the construction of a new transformation on those signed words that maps the former statistic onto the latter one. It is proved that the transformation also preserves two other set-statistics: the inverse ligne of route and the lower records.
References:
-
- 1.
- Ron M. Adin and Yuval Roichman, The flag major index and group actions on polynomial rings, Europ. J. Combin. 22 (2001), 431-446. MR 1829737 (2002d:05004)
- 2.
- Ron M. Adin, Francesco Brenti and Yuval Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. 27 (2001), 210-224. MR 1868962 (2003m:05211)
- 3.
- Riccardo Biagioli, Major and descent statistics for the even-signed permutation group, Adv. in Appl. Math. 31 (2003), 163-179. MR 1985826 (2004c:05007)
- 4.
- Riccardo Biagioli and Fabrizio Caselli, Invariant algebras and major indices for classical Weyl groups, Proc. London Math. Soc. 88 (2004), 603-631. MR 2044051 (2005a:05216)
- 5.
- Anders Björner and Michelle L. Wachs, Permutation statistics and linear extensions of posets, J. Combin. Theory, Ser. A 58 (1991), 85-114. MR 1119703 (92m:06010)
- 6.
- N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4,5 et 6, Hermann, Paris, 1968. MR 0240238 (39:1590)
- 7.
- Francesco Brenti,
 -Eulerian polynomials arising from coxeter groups, Europ. J. Combinatorics 15 (1994), 417-441. MR 1292954 (95i:05013) - 8.
- Dominique Foata, On the Netto inversion number of a sequence, Proc. Amer. Math. Soc. 19 (1968), 236-240. MR 0223256 (36:6304)
- 9.
- Dominique Foata and Guo-Niu Han, Signed words and permutations, II; The Euler-Mahonian polynomials, Electronic J. Combin. 11 (2), #R22 (2005), 18 pages. MR 2195428
- 10.
- Dominique Foata and Marcel-Paul Schützenberger, Major Index and Inversion number of Permutations, Math. Nachr. 83 (1978), 143-159. MR 0506852 (81d:05007)
- 11.
- Herbert O. Foulkes, Tangent and secant numbers and representations of symmetric groups, Discrete Math. 15 (1976), 311-324. MR 0406810 (53:10596)
- 12.
- Herbert O. Foulkes, Eulerian numbers, Newcomb's problem and representations of symmetric groups, Discrete Math. 30 (1980), 3-49. MR 0561763 (81g:05020)
- 13.
- Ira Gessel, Generating functions and enumeration of sequences, Ph.D. thesis, Dept. Math., M.I.T., Cambridge, Mass., 111 pp., 1977.
- 14.
- James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge (Cambridge Studies in Adv. Math., 29), 1990. MR 1066460 (92h:20002)
- 15.
- Donald E. Knuth, The Art of Computer Programming, vol. 3, Sorting and Searching, Addison-Wesley, Reading, 1973. MR 0445948 (56:4281)
- 16.
- M. Lothaire, Combinatorics on Words, Addison-Wesley, London (Encyclopedia of Math. and its Appl., 17), 1983. MR 0675953 (84g:05002)
- 17.
- Percy Alexander MacMahon, The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects, Amer. J. Math. 35 (1913), 314-321. MR 1506186
- 18.
- Percy Alexander MacMahon, Combinatory Analysis, vol. 1 and 2, Cambridge, Cambridge Univ. Press, 1915 (Reprinted by Chelsea, New York, 1995).
- 19.
- Percy Alexander MacMahon, ``Collected Papers, vol. 1" [G.E. Andrews, ed.], Cambridge, Mass., The M.I.T. Press, 1978. MR 0514405 (80k:01065)
- 20.
- V. Reiner, Signed permutation statistics, Europ. J. Combinatorics 14 (1993), 553-567. MR 1248063 (95e:05008)
- 21.
- V. Reiner, Signed permutation statistics and cycle type, Europ. J. Combinatorics 14 (1993), 569-579. MR 1248064 (95f:05009)
- 22.
- V. Reiner, Upper binomial posets and signed permutation statistics, Europ. J. Combinatorics 14 (1993), 581-588. MR 1248065 (94m:05006)
- 23.
- V. Reiner, Descents and one-dimensional characters for classical Weyl groups, Discrete Math. 140 (1995), 129-140. MR 1333715 (96d:05116)
- 24.
- V. Reiner, The distribution of descents and length in a Coxeter group, Electronic J. Combinatorics 2, # R25 (1995). MR 1359412 (96k:05208)
- 25.
- Richard P. Stanley, Ordered structures and partitions, Mem. Amer. Math. Soc., vol. 119, Amer. Math. Soc., Providence, RI, 1972. MR 0332509 (48:10836)
- 26.
- Richard P. Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J. Combinatorial Theory Ser. A 20 (1976), 336-356. MR 0409206 (53:12968)
- 27.
- John Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Math. 99 (1992), 307-320. MR 1158793 (93f:05103)
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Additional Information:
Dominique
Foata
Affiliation:
Institut Lothaire, 1 rue Murner, F-67000 Strasbourg, France
Email:
foata@math.u-strasbg.fr
Guo-Niu
Han
Affiliation:
I.R.M.A. UMR 7501, Université Louis Pasteur et CNRS, 7 rue René-Descartes, F-67084 Strasbourg, France
Email:
guoniu@math.u-strasbg.fr
DOI:
10.1090/S0002-9939-06-08436-X
PII:
S 0002-9939(06)08436-X
Keywords:
Flag-inversion number,
flag-major index,
length function,
signed permutations,
signed words.
Received by editor(s):
April 11, 2005
Received by editor(s) in revised form:
August 4, 2005
Posted:
June 29, 2006
Dedicated:
This paper is dedicated to the memory of Percy Alexander MacMahon
Communicated by:
John R. Stembridge
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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