The Homology of Partitions with an Even Number of Blocks
Sheila Sundaram
DOI: 10.1023/A:1022437708487
Abstract
Let Õ 2 n e \prod {_{2n}^e } denote the subposet obtained by selecting even ranks in the partition lattice Õ 2 n \prod {_{2n} } . We show that the homology of Õ 2 n e \prod {_{2n}^e } has dimension \frac(2 n)!2 2 n - 1 E 2 n - 1 \frac{{(2n)!}}{{2^{2n - 1} }}E_{2n - 1} , where E 2 n - 1 E_{2n - 1} is the tangent number. It is thus an integral multiple of both the Genocchi number and an André or simsun number. Using the general theory of rank-selected homology representations developed in [22], we show that, for the special case of Õ 2 n e \prod {_{2n}^e } , the character of the symmetric group S 2 n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers b i( n), 2 S 2 i xS 1 2 n - 2 i S_2^i xS_1^{2n - 2i} , with nonnegative integer multiplicity b i( n). The nonnegativity of the integers b i( n) would imply the existence of new refinements, into sums of powers of 2, of the tangent number and the André or simsun number a n(2 n).
Similarly, the restriction of this homology module to S 2 n-1 yields a family of integers d i( n), 1 Õ 2 n e \prod {_{2n}^e } , 1 
k n - 1. We conjecture that these are all permutation modules for S 2 n .
k n - 1. We conjecture that these are all permutation modules for S 2 n .Pages: 69–92
Keywords: homology representation; permutation module; André permutations; simsun permutation; tangent and Genocchi number
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References
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25. M.L. Wachs, Personal communication 1993.
2. K. Baclawski, "Whitney numbers of geometric lattices," Adv. in Math. 16 (1975), 125-138.
3. K. Baclawski, "Cohen-Macaulay ordered sets," J. Algebra. 63 (1980), 226-258.
4. A. Bjorner, "Shellable and Cohen-Macaulay partially ordered sets," Trans. Amer. Math. Soc. 260 (1980), 159-183.
5. A. Bjorner, "On the homology of geometric lattices," Algebra Universalis 14 (1982), 107-128.
6. A.R. Calderbank, P. Hanlon and R.W. Robinson, "Partitions into even and odd block size and some unusual characters of the symmetric groups," Proc. London Math. Soc. (3), S3 (1986), 288-320.
7. P. Edelman and R. Simion, "Chains in the lattice of non-crossing partitions," Discrete Math. 126 (1994), 107-119.
8. D. Foata and M-P. Schutzenberger, Nombres d'Euler et permutations alternantes, manuscript. University of Florida, Gainesville (1971). First part published in "A Survey of Combinatorial Theory," J.N. Srivastava et al., eds., North-Holland (1973), 173-187.
9. D. Foata and V. Strehl, "Rearrangements of the Symmetric Group and enumerative properties of the tangent and secant numbers," Math. Zeitschrift 137 (1974), 257-264.
10. Foata, D. and Strehl, V., Euler numbers and Variations of Permutations, in Atti Dei Convegni Lincei 17, Colloquio Internazionale sulle Teorie Combinatorie (1973), Tomo I, Roma, Accademia Nazionale dei Lincei, 1976, 119-131.
11. A.M. Garsia and C. Procesi, "On certain graded Sn-modules and the q-Kostka polynomials," Adv. in Math. 94(1992), 82-138.
12. I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979.
13. J.R. Munkres, "Topological Results in Combinatorics," Mich. Math. J. 31 (1984), 113-128.
14. L. Solomon, "A decomposition of the group algebra of a finite Coxeter group," J. Algebra 9 (1968), 220-239.
15. R.P. Stanley, "Exponential Structures," Stud. Appl. Math. 59 (1978), 73-82.
16. R.P. Stanley, "Balanced Cohen-Macaulay complexes," Trans. Amer. Math. Soc. 249 (1979), 361-371.
17. R.P. Stanley, "Some aspects of groups acting on finite posets," J. Comb. Theory (A) 32 (1982), 132-161.
18. R.P. Stanley, "Enumerative Combinatorics," Vol. 1, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986.
19. R.P. Stanley, "Enumerative Combinatorics," Vol. 2, Chapter 5, manuscript.
20. R.P. Stanley, "Flag f-vectors and the cd-index," Math Z. 216 (1994), 483-499. SUNDARAM
21. J.R. Stembridge, "Some Permutation Representations of Weyl Groups Associated with the Cohomology of Toric Varieties," Adv. in Math., to appear (preprint, 1992).
22. S. Sundaram, "The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice," Adv. in Math. 104 (1994), 225-296.
23. G. Viennot, Interpretations combinatoires des nombres d'Euler et de Genocchi, in Seminaire de Theorie des Nombres, 1980-1981, expose no. 11 (1981).
24. ML. Wachs, "Bases for poset homology," in preparation (personal communication (1992)).
25. M.L. Wachs, Personal communication 1993.
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