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Volumn 13, Issue 4, 2007, Pages 33-45

Q-eulerian polynomials: Excedance number and major index

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EID: 38149061157     PISSN: None     EISSN: 10796762     Source Type: Journal    
DOI: 10.1090/S1079-6762-07-00172-2     Document Type: Article
Times cited : (40)

References (38)
  • 1
    • 0002632448 scopus 로고    scopus 로고
    • Generalized permutation patterns and a classification of the Mahonian statistics
    • MR1758852 (2002b:05006)
    • E. Babson and E. Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, Sém. Lothar. Combin., B44b (2000), 18 pp. MR1758852 (2002b:05006)
    • (2000) Sém. Lothar. Combin , vol.B44b , pp. 18
    • Babson, E.1    Steingrímsson, E.2
  • 2
    • 0037609018 scopus 로고
    • Permutation enumeration of the symmetric group and the combinatorics of symmetric functions
    • MR1354966 (96j:05113)
    • D. Beck and J.B. Remmel, Permutation enumeration of the symmetric group and the combinatorics of symmetric functions, J. Combin. Theory Ser. A 72 (1995), 1–49. MR1354966 (96j:05113)
    • (1995) J. Combin. Theory Ser. A , vol.72 , pp. 1-49
    • Beck, D.1    Remmel, J.B.2
  • 3
    • 33646886315 scopus 로고
    • Shellable and Cohen-Macaulay partially ordered sets
    • MR570784 (81i:06001)
    • A. Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. AMS 260 (1980), 159–183. MR570784 (81i:06001)
    • (1980) Trans. AMS , vol.260 , pp. 159-183
    • Björner, A.1
  • 4
    • 16844380765 scopus 로고    scopus 로고
    • Segre and Rees products of posets, with ring-theoretic applications
    • MR2132872 (2005m:06006)
    • A. Björner and V. Welker, Segre and Rees products of posets, with ring-theoretic applications, J. Pure Appl. Algebra 198 (2005), 43–55. MR2132872 (2005m:06006)
    • (2005) J. Pure Appl. Algebra , vol.198 , pp. 43-55
    • Björner, A.1    Welker, V.2
  • 5
    • 0002187070 scopus 로고
    • A combinatorial property of q-Eulerian numbers
    • MR0366683 (51:2930)
    • L. Carlitz, A combinatorial property of q-Eulerian numbers, The American Mathematical Monthly 82 (1975), 51–54. MR0366683 (51:2930)
    • (1975) The American Mathematical Monthly , vol.82 , pp. 51-54
    • Carlitz, L.1
  • 6
    • 0000752989 scopus 로고
    • Enumeration of pairs of sequences by rises, falls and levels
    • MR0432472 (55:5460)
    • L. Carlitz, R. Scoville, and T. Vaughan, Enumeration of pairs of sequences by rises, falls and levels, Manuscripta Math. 19 (1976), 211–243. MR0432472 (55:5460)
    • (1976) Manuscripta Math , vol.19 , pp. 211-243
    • Carlitz, L.1    Scoville, R.2    Vaughan, T.3
  • 7
    • 0000022302 scopus 로고    scopus 로고
    • New Euler-Mahonian statistics on permutations and words
    • MR1436481 (97m:05008)
    • R.J. Clarke, E. Steingrímsson, and J. Zeng, New Euler-Mahonian statistics on permutations and words, Adv. in Appl. Math. 18 (1997), 237–270. MR1436481 (97m:05008)
    • (1997) Adv. In Appl. Math , vol.18 , pp. 237-270
    • Clarke, R.J.1    Steingrímsson, E.2    Zeng, J.3
  • 8
    • 38249001068 scopus 로고
    • Descent classes of permutations with a given number of fixed points
    • MR1245164 (95c:05008)
    • J. Désarménien and M.L. Wachs, Descent classes of permutations with a given number of fixed points, J. Combin. Theory Ser. A 64 (1993), no. 2, 311–328. MR1245164 (95c:05008)
    • (1993) J. Combin. Theory Ser. A , vol.64 , Issue.2 , pp. 311-328
    • Désarménien, J.1    Wachs, M.L.2
  • 9
    • 0010034373 scopus 로고
    • A character formula for the representation of a Weyl group in the cohomology of the associated toric variety
    • MR1293622 (95h:20008)
    • I. Dolgachev and V. Lunts, A character formula for the representation of a Weyl group in the cohomology of the associated toric variety, J. Algebra 168 (1994), 741–772. MR1293622 (95h:20008)
    • (1994) J. Algebra , vol.168 , pp. 741-772
    • Dolgachev, I.1    Lunts, V.2
  • 10
    • 33750472293 scopus 로고    scopus 로고
    • Words avoiding a reflexive acyclic relation
    • #R28. MR2224941 (2007b:05013)
    • J. Dollhopf, I. Goulden, and C. Greene, Words avoiding a reflexive acyclic relation, Electon. J. Combin. 11 (2006) #R28. MR2224941 (2007b:05013)
    • (2006) Electon. J. Combin , vol.11
    • Dollhopf, J.1    Goulden, I.2    Greene, C.3
  • 11
    • 38149039222 scopus 로고
    • Sur un énoncé de MacMahon
    • MR0158834 (28:2057)
    • D. Foata, Sur un énoncé de MacMahon, C. R. Acad. Sci. Paris 258 (1964), 1672–1675. MR0158834 (28:2057)
    • (1964) C. R. Acad. Sci. Paris , vol.258 , pp. 1672-1675
    • Foata, D.1
  • 12
    • 84968502303 scopus 로고
    • On the Netto inversion number of a sequence
    • MR0223256 (36:6304)
    • D. Foata, On the Netto inversion number of a sequence, Proc. Amer. Math. Soc. 19 (1968), 236–240. MR0223256 (36:6304)
    • (1968) Proc. Amer. Math. Soc , vol.19 , pp. 236-240
    • Foata, D.1
  • 13
    • 0011004158 scopus 로고
    • Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations
    • D, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, Reidel, Dordrecht-Boston, Mass., MR519777 (80e:05011)
    • D. Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, NATO Adv. Study Inst. Ser., Ser. C: Math. Phys. Sci., 31, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976), pp. 27–49, Reidel, Dordrecht-Boston, Mass., 1977. MR519777 (80e:05011)
    • (1976) NATO Adv. Study Inst. Ser., Ser. C: Math. Phys. Sci , vol.31 , pp. 27-49
  • 15
    • 1842804784 scopus 로고
    • Théorie géométrique des polyn
    • Springer-Verlag, Berlin–New York, MR0272642 (42:7523)
    • D. Foata and M.-P. Schützenberger, Théorie géométrique des polynômes eulériens, Lecture Notes in Mathematics, Vol. 138 Springer-Verlag, Berlin–New York, 1970. MR0272642 (42:7523)
    • (1970) Lecture Notes in Mathematics , vol.138
    • Foata, D.1    Schützenberger, M.-P.2
  • 16
    • 84985407339 scopus 로고
    • Major index and inversion number of permutations
    • MR506852 (81d:05007)
    • D. Foata and M.-P. Schützenberger, Major index and inversion number of permutations, Math. Nachr. 83 (1978), 143–159. MR506852 (81d:05007)
    • (1978) Math. Nachr , vol.83 , pp. 143-159
    • Foata, D.1    Schützenberger, M.-P.2
  • 17
    • 0002151246 scopus 로고
    • Denert’s permutation statistic is indeed Euler-Mahonian
    • MR1061147 (91h:05016)
    • D. Foata and D. Zeilberger, Denert’s permutation statistic is indeed Euler-Mahonian, Stud. Appl. Math. 83 (1990), 31–59. MR1061147 (91h:05016)
    • (1990) Stud. Appl. Math , vol.83 , pp. 31-59
    • Foata, D.1    Zeilberger, D.2
  • 18
    • 0007020509 scopus 로고
    • Permutation statistics and partitions
    • MR532836 (80e:05013)
    • A.M. Garsia and I. Gessel, Permutation statistics and partitions, Adv. in Math. 31 (1979), 288–305. MR532836 (80e:05013)
    • (1979) Adv. In Math. , vol.31 , pp. 288-305
    • Garsia, A.M.1    Gessel, I.2
  • 19
    • 0001347739 scopus 로고
    • Counting permutations with given cycle structure and descent set
    • MR1245159 (95g:05006)
    • I.M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993), 189–215. MR1245159 (95g:05006)
    • (1993) J. Combin. Theory Ser. A , vol.64 , pp. 189-215
    • Gessel, I.M.1    Reutenauer, C.2
  • 20
    • 0000940257 scopus 로고    scopus 로고
    • Q-Rook polynomials and matrices over finite fields
    • MR1612854 (99f:05004)
    • J. Haglund, q-Rook polynomials and matrices over finite fields, Adv. in Appl. Math. 20 (1998), 450-487. MR1612854 (99f:05004)
    • (1998) Adv. In Appl. Math , vol.20 , pp. 450-487
    • Haglund, J.1
  • 21
    • 0003657590 scopus 로고    scopus 로고
    • Sorting and Searching, Second Edition, Addison-Wesley, Reading, MA, MR0445948 (56:4281)
    • D. Knuth, The Art of Computer Programming, Vol. 3. Sorting and Searching, Second Edition, Addison-Wesley, Reading, MA, 1998. MR0445948 (56:4281)
    • (1998) The Art of Computer Programming , vol.3
    • Knuth, D.1
  • 22
    • 0004205061 scopus 로고
    • Cambridge University Press, London, 1915-1916, Reprinted by Chelsea, New York, MR0141605 (25:5003)
    • P.A. MacMahon, Combinatory Analysis, 2 volumes, Cambridge University Press, London, 1915-1916. Reprinted by Chelsea, New York, 1960. MR0141605 (25:5003)
    • (1960) Combinatory Analysis , vol.2
    • Macmahon, P.A.1
  • 23
    • 0001023516 scopus 로고
    • The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects
    • MR1506186
    • P.A. 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), no. 3, 281–322. MR1506186
    • (1913) Amer. J. Math , vol.35 , Issue.3 , pp. 281-322
    • Macmahon, P.A.1
  • 25
    • 25444489076 scopus 로고
    • The toric variety associated to Weyl chambers
    • Lang. Raison. Calc., Hermès, Paris, MR1252661 (94k:14045)
    • C. Procesi, The toric variety associated to Weyl chambers, Mots, 153–161, Lang. Raison. Calc., Hermès, Paris, 1990. MR1252661 (94k:14045)
    • (1990) Mots
    • Procesi, C.1
  • 26
    • 0030295855 scopus 로고    scopus 로고
    • Combinatorics of the q-basis of symmetric functions
    • MR1416016 (98g:05151)
    • A. Ram, J. Remmel, and T. Whitehead, Combinatorics of the q-basis of symmetric functions, J. Combin. Theory Ser. A 76 (1996), 231–271. MR1416016 (98g:05151)
    • (1996) J. Combin. Theory Ser. A , vol.76 , pp. 231-271
    • Ram, A.1    Remmel, J.2    Whitehead, T.3
  • 27
    • 10844259990 scopus 로고
    • Enumeration of permutations by descents, idescents, imajor index, and basic components
    • MR728499 (85k:05013)
    • D. Rawlings, Enumeration of permutations by descents, idescents, imajor index, and basic components, J. Combin. Theory Ser. A 36 (1984), 1–14. MR728499 (85k:05013)
    • (1984) J. Combin. Theory Ser. A , vol.36 , pp. 1-14
    • Rawlings, D.1
  • 28
    • 0000805196 scopus 로고
    • Note sur les inversions, ou derangements produits dans les permutations
    • O. Rodrigues, Note sur les inversions, ou derangements produits dans les permutations, Journal de Mathematiques 4 (1839), 236–240.
    • (1839) Journal De Mathematiques , vol.4 , pp. 236-240
    • Rodrigues, O.1
  • 29
    • 39749192956 scopus 로고    scopus 로고
    • An Eulerian partner for inversions
    • Art. B46d, electronicMR1848722 (2002g:05010)
    • M. Skandera, An Eulerian partner for inversions, Sém. Lothar. Combin. 46 (2001/02), Art. B46d, 19 pp. (electronic). MR1848722 (2002g:05010)
    • (2001) Sém. Lothar. Combin , vol.46 , pp. 19
    • Skandera, M.1
  • 30
    • 0003045152 scopus 로고
    • Ordered structures and partitions
    • MR0332509 (48:10836)
    • R.P. Stanley, Ordered structures and partitions, Memoirs Amer. Math. Soc. 119 (1972). MR0332509 (48:10836)
    • (1972) Memoirs Amer. Math. Soc , pp. 119
    • Stanley, R.P.1
  • 31
    • 0001608362 scopus 로고
    • Binomial posets, Möbius inversion, and permutation enumeration
    • MR0409206 (53:12968)
    • R.P. Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J. Combinatorial Theory Ser. A 20 (1976), 336–356. MR0409206 (53:12968)
    • (1976) J. Combinatorial Theory Ser , vol.A 20 , pp. 336-356
    • Stanley, R.P.1
  • 32
    • 0024830136 scopus 로고
    • Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986), 500–535
    • New York Acad. Sci., New York, MR1110850 (92e:05124)
    • R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986), 500–535, Ann. New York Acad. Sci., 576, New York Acad. Sci., New York, 1989. MR1110850 (92e:05124)
    • (1989) Ann. New York Acad. Sci , vol.576
    • Stanley, R.P.1
  • 33
    • 0003279650 scopus 로고    scopus 로고
    • Enumerative combinatorics
    • Cambridge University Press, Cambridge, MR1676282 (2000k:05026)
    • R.P. Stanley, Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. MR1676282 (2000k:05026)
    • (1999) Cambridge Studies in Advanced Mathematics , vol.2 , pp. 62
    • Stanley, R.P.1
  • 34
    • 0001452386 scopus 로고
    • A symmetric function generalization of the chromatic polynomial of a graph
    • MR1317387 (96b:05174)
    • R.P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166–194. MR1317387 (96b:05174)
    • (1995) Advances in Math , vol.111 , pp. 166-194
    • Stanley, R.P.1
  • 35
    • 38249012015 scopus 로고
    • Eulerian numbers, tableaux, and the Betti numbers of a toric variety
    • MR1158793 (93f:05103)
    • J.R. Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Math. 99 (1992), 307–320. MR1158793 (93f:05103)
    • (1992) Discrete Math , vol.99 , pp. 307-320
    • Stembridge, J.R.1
  • 36
    • 0000743385 scopus 로고
    • Some permutation representations of Weyl groups associated with the cohomology of toric varieties
    • MR1279220 (95f:20011)
    • J.R. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math. 106 (1994), 244–301. MR1279220 (95f:20011)
    • (1994) Adv. Math , vol.106 , pp. 244-301
    • Stembridge, J.R.1
  • 38
    • 38249013331 scopus 로고
    • An involution for signed Eulerian numbers
    • MR1158780 (93e:11030b)
    • M.L. Wachs, An involution for signed Eulerian numbers, Discrete Math. 99 (1992), 59–62. MR1158780 (93e:11030b)
    • (1992) Discrete Math , vol.99 , pp. 59-62
    • Wachs, M.L.1


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