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The Joint Distribution of Descent and Major Index over Restricted Sets of Permutations
Sylvie Corteel1, Ira M. Gessel2, Carla D. Savage3, and Herbert S. Wilf4
1LRI, CNRS et Universit´e Paris-Sud, Bat. 490, F-91405 Orsay, France
corteel@lri.fr
2Department of Mathematics, Brandeis University, Waltham, MA 02454-9110, USA
gessel@brandeis.edu
3Department of Computer Science, College of Engineering, North Carolina State University, Raleigh, NC 27695-8206, USA
savage@csc.ncsu.edu
4Department of Mathematics, School of Arts & Sciences, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
wilf@math.upenn.edu
Annals of Combinatorics 11 (3-4) p.375-386 September, 2007
AMS Subject Classification:05A15, 05A05, 05A30
Abstract:
We compute the joint distribution of descent and major index over permutations of {1,…, n} with no descents in positions {n-i,n-i+1,…,n-1} for fixed i≥0. This was motivated by the problem of enumerating symmetrically constrained compositions and generalizes Carlitz's q-Eulerian polynomial.
Keywords: permutation enumeration, q-Eulerian polynomials, P-partitions

References:

1. G.E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly 86 (6) (1979) 477-478.

2. G.E. Andrews, MacMahon's partition analysis: II fundamental theorems, Ann. Combin. 4 (2000) 327-338.

3. G.E. Andrews, P. Paule, and A. Riese, MacMahon's partition analysis: VII constrained compositions, In: q-Series with Applications to Combinatorics, Number Theory, and Physics, Contemporary Mathematics, Vol. 291, American Mathematical Society, Providence, RI, (2001) 11-27.

4. M. Bóna, Combinatorics of Permutations, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004.

5. L. Carlitz, A combinatorial property of q-Eulerian numbers, Amer. Math. Monthly 82 (1975) 51-54.

6. L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332350.

7. A.M. Garsia and I. Gessel, Permutation statistics and partitions, Adv. Math. 31 (3) (1979) 288-305.

8. J.H. Jordan, R. Walch, and R.J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 (8) (1979) 686-689.

9. S. Lee and C.D. Savage, Symmetrically Constrained Compositions, preliminary draft.

10. P.A. MacMahon, Combinatory Analysis, two volumes (bound as one), Chelsea Publishing Co., New York, 1960.

11. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, Volume 49, Cambridge University Press, Cambridge, 1997.