A Statistic on Involutions
Rajendra S. Deodhar
and Murali K. Srinivasan
DOI: 10.1023/A:1011249732234
Abstract
We define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I( n) denote the set of all involutions on [n](={1,2,..., n}) and let F(2 n) denote the set of all fixed point free involutions on [2 n]. For an involution ( \text k \text n ) q \left( {_{\text{k}}^{\text{n}} } \right)q denote the q-binomial coefficient. There is a statistic wt on I( n) such that the following results are true.
(i) We have the expansion
Pages: 187–198
Keywords: permutation statistics; $q$-binomial coefficient; Bruhat order; involutions; fixed point free involutions
Full Text: PDF
References
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2. A. Bj\ddot orner, “Shellable and Cohen-Macaulay partially ordered sets,” Trans. Amer. Math. Soc. 260 (1980), 159-183.
3. A. Bj\ddot orner, “Homology and shellability of matroids and geometric lattices,” in Matroid Applications, N. White (Ed.), Cambridge University Press, 1992.
4. N.G. de Bruijn, C. Tengbergen, and D. Kruyswijk, “On the set of divisors of a number,” Nieuw Arch. Wiskd. 23 (1951), 191-193.
5. G. Danaraj and Victor Klee, “Shellings of spheres and polytopes,” Duke Math. J. 41 (1974), 443-451.
6. P.H. Edelman, “The Bruhat order of the Symmetric group is lexicographically shellable,” Proc. Amer. Math. Soc. 82 (1981), 355-358.
7. J.R. Goldman and G.C. Rota, “On the foundations of combinatorial theory IV: Finite vector spaces and Eulerian generating functions,” Studies in Applied Math. 49(3) (1970), 239-258.
8. J.R. Griggs, “Sufficient conditions for a symmetric chain order,” SIAM J. Applied Math. 32 (1977), 807-809.
9. R. Howe and H. Kraft, “Principal covariants, multiplicity-free actions, and the K-types of holomorphic discrete series,” in Geometry and Representation Theory of Real and p-adic Groups, Cordoba, 1995, pp. 147-161. Progr. Math., 158, Birkhauser Boston, Boston, MA, 1998.
10. W.P. Johnson, “A q-analog of Faá di Bruno's Formula,” J. Combin Theory, Ser. A 76 (1996), 305-314.
11. W.P. Johnson, “Some applications of the q-exponential formula,” Discrete Math. 157 (1996), 207-225.
12. A. Nijenhuis, A. Solow, and H. Wilf, “Bijective methods in the theory of finite vector spaces,” J. Combin. Theory, Ser. A 37 (1984), 80-84.
13. M.K. Srinivasan, “Boolean packings in dowling geometries,” European Journal of Combinatorics 19 (1998), 727-731.
14. R. Stanley, Enumerative Combinatorics-Vol. 1, Wadsworth & Brooks/Cole, Monterey California, 1986.
15. J.R. Stembridge, “Nonintersecting paths, pfaffians and plane partitions,” Advances in Math. 83 (1990), 96- 131.
16. F. Vogt and B. Voigt, “Symmetric chain decompositions of linear lattices,” Combinatorics, Probability and Computing 6 (1997), 231-245.
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