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Enumeration of Chains in Semi-Primary Lattices
F. Regonati1, S.D. Sarti2
1University of Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy
2ITCG Paolini, via Gucciardini 2, 40026 Imola, Italy
Annals of Combinatorics 4 (1) p.109-124 March, 2000
AMS Subject Classification:05A15, 05A30, 06A08, 06C05
We show that some combinatorial properties of vector spaces (see [13]) and finite abelian p-groups (see [5, 6]) may be carried over to the purely synthetic setting of q-primary lattices (see [1,12]).
Let L be a q-primary lattice and c: 0 = c0 < c1 < #< cn = 1 a maximal chain such that the join of all the upper covers of any term of c is still a term of c, and for each x L, consider the subset of [n] (see [16]) S(x) = {i∈[n]; ci-1 x < cix}.
Our main results are:
  • the sets S(x); x L, form, with respect to set union and intersection, a 1-primary lattice L', i.e., a direct product of chains, and the mapping S is order- and type-preserving;
  • if X is any chain in L' and w = w1w2wn is the labeling of [n] induced by X, then the number of chains in L which are sent by S to X is qinv(w); where the exponent counts the ※minimal inversions§of w.
Our proof is based on the properties of radicals and socles in q-primary lattices [7, 21].
Keywords: semi-primary lattice, q每primary lattice, flag每victors, minimal inversions, Hall polynomials


1. R. Baer, A unified theory of projective spaces and finite abelian groups , Trans. Amer. Math. Soc. 52 (1942) 283每243.

2. G. Birkhoff, Subgroups of abelian groups , Proc. London Math. Soc. II Ser. 38 (1934) 385每401.

3. T.S. Blyth and M.F. Janowitz, Residuation Theory , International Series of Monographs in Pure and Applied Mathematics, Vol.102, Pergamon Press, Oxford, 1972.

4. L.M. Butler and A.W. Hales, Nonnegative Hall polynomials , J. Algebr. Combin. 2 (1993) 125每135.

5. L.M. Butler, Subgroup lattices and symmetric functions , Mem. Amer. Math. Soc. 539 (1994).

6. L.M. Butler, Order analogues and Betti polynomials , Adv. Math. 121 (1996) 62每79.

7. L. Giudici, Dintorni del teorema di coordinatizzazione di von Neumann ; Appendice D: Reticoli fortemente semimodulari semiprimari, Tesi di Dottorato, Universita' degli Studi di Milano, 1995.

8. G. Gratzer, General Lattice Theory , 2nd Edition, Birkh¨auser-Verlag, Basel, 1998.

9. C. Herrmann, S-Verklebte Summen von Verbaenden, Math. Z. 130 (1973) 255每274 .

10. E. Inaba, On primary lattices , J. Fac. Sci. Univ. Hokkaido 11 (1948) 39每107.

11. E. Inaba, Some remarks on primary lattices , Nat. Sci. Rep. Ochanomizu Univ. 2 (1951) 1每5.

12. B. Jonsson and G.S. Monk, Representation of primary arguesian lattices , Pacific J. Math. 30 (1969) 95每139.

13. D.E. Knuth, Subspaces , subsets , and partitions , J. Combin. Theory Ser. A 10 (1971) 178每180.

14. I.G. MacDonald, Symmetric Functions and Hall Polynomials , Clarendon Press, Oxford, 1995.

15. F.M. Maley, The Hall polynomial revisited , J. Algebra 184 (1996) 363每371.

16. R.P. Stanley, Supersolvable lattices , Algebra Universalis 2 (1972) 197每217

17. R.P. Stanley, Finite lattices and Jordan求Hoelder sets , Algebra Universalis 4 (1974) 361每371.

18. R.P. Stanley, Enumerative Combinatorics , Vols. 1&2, Cambridge Studies in Advanced Mathematics, Vol. 49, Cambridge University Press, Cambridge, 1997.

19. R.P. Stanley, Locally rank-symmetric and flag-symmetric posets , Elec. J. Combin. 3 (1996) 22.

20. M. Stern, Semimodular Lattices, Teubner-Texte zur Mathematik , Vol. 125, B.G. Teubner Verlagsgesellschaft, Stuttgart, 1991.

21. G.P. Tesler, Semi-primary lattices and tableau algorithms , Ph.D. Thesis, MIT, 1995.